Handbook of Exact Solutions for Ordinary Differential Equations [2 ed.] 9781420035339, 9781584882978, 1584882972

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HANDBOOK OF

EXACT SOLUTIONS for ORDINARY DIFFERENTIAL EQUATIONS SECOND EDITION

HANDBOOK OF

EXACT SOLUTIONS for ORDINARY DIFFERENTIAL EQUATIONS SECOND EDITION

Andrei D. Polyanin Valentin F. Zaitsev

CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

C2972_frame_DISC Page 1 Thursday, August 29, 2002 1:45 PM

Library of Congress Cataloging-in-Publication Data Polyanin, A. D. (Andrei Dmitrievich) Handbook of exact solutions for ordinary differential equations / Andrei D. Polyanin, Valentin F. Zaitsev.--2nd ed. p. cm. Includes bibliographical references and index. ISBN 1-58488-297-2 (alk. paper) 1. Differential equations--Numerical solutions. I. Zaitsev, V. F. (Valentin F.) II. Title. QA372 .P725 2002 515′.352--dc21

2002073735

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2003 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-297-2 Library of Congress Card Number 2002073735 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

CONTENTS Authors Foreword Notation and Some Remarks Introduction Some Definitions, Formulas, Methods, and Transformations 0.1. First-Order Differential Equations 0.1.1. General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems 0.1.1-1. Equations solved for the derivative. General solution 0.1.1-2. The Cauchy problem. The uniqueness and existence theorems 0.1.1-3. Equations not solved for the derivative. The existence theorem 0.1.1-4. Singular solutions 0.1.1-5. Point transformations 0.1.2. Equations Solved for the Derivative. Simplest Techniques of Integration 0.1.2-1. Equations with separated or separable variables 0.1.2-2. Equation of the form 
 =  (  +  ) 0.1.2-3. Homogeneous equations and equations reducible to them 0.1.2-4. Generalized homogeneous equations and equations reducible to them 0.1.2-5. Linear equation 0.1.2-6. Bernoulli equation 0.1.2-7. Equation of the form  
 = +  ( ) (  ) 0.1.2-8. Darboux equation 0.1.3. Exact Differential Equations. Integrating Factor 0.1.3-1. Exact differential equations 0.1.3-2. Integrating factor 0.1.4. Riccati Equation 0.1.4-1. General Riccati equation. Simplest integrable cases 0.1.4-2. Polynomial solutions of the Riccati equation 0.1.4-3. Use of particular solutions to construct the general solution 0.1.4-4. Some transformations 0.1.4-5. Reduction of the Riccati equation to a second-order linear equation 0.1.4-6. Reduction of the Riccati equation to the canonical form 0.1.5. Abel Equations of the First Kind 0.1.5-1. General form of Abel equations of the first kind. Simplest integrable cases 0.1.5-2. Reduction to the canonical form. Reduction to an Abel equation of the second kind 0.1.6. Abel Equations of the Second Kind 0.1.6-1. General form of Abel equations of the second kind. Simplest integrable cases 0.1.6-2. Reduction to the canonical form. Reduction to an Abel equation of the first kind 0.1.6-3. Use of particular solutions to construct self-transformations 0.1.6-4. Use of particular solutions to construct the general solution 0.1.7. Equations Not Solved for the Derivative 0.1.7-1. The method of “integration by differentiation.” 0.1.7-2. Equations of the form =  (
 )

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0.1.7-3. Equations of the form  =  (
 ) 0.1.7-4. Clairaut’s equation =  
 +  (
 ) 0.1.7-5. Lagrange’s equation =   (
 ) + (
 ) 0.1.8. Contact Transformations 0.1.8-1. General form of contact transformations 0.1.8-2. A method for the construction of contact transformations 0.1.8-3. Examples of contact transformations linear in the derivative 0.1.8-4. Examples of contact transformations nonlinear in the derivative 0.1.9. Approximate Analytic Methods for Solution of Equations 0.1.9-1. The method of successive approximations (Picard method) 0.1.9-2. The method of Taylor series expansion in the independent variable 0.1.9-3. The method of regular expansion in the small parameter 0.1.10. Numerical Integration of Differential Equations 0.1.10-1. The method of Euler polygonal lines 0.1.10-2. Single-step methods of the second-order approximation 0.1.10-3. Runge–Kutta method of the fourth-order approximation 0.2. Second-Order Linear Differential Equations 0.2.1. Formulas for the General Solution. Some Transformations 0.2.1-1. Homogeneous linear equations. Various representations of the general solution 0.2.1-2. Wronskian determinant and Liouville’s formula 0.2.1-3. Reduction to the canonical form 0.2.1-4. Reduction to the Riccati equation 0.2.1-5. Nonhomogeneous linear equations. The existence theorem 0.2.1-6. Nonhomogeneous linear equations. Various representations of the general solution 0.2.1-7. Reduction to a constant coefficient equation (a special case) 0.2.1-8. Kummer–Liouville transformation 0.2.2. Representation of Solutions as a Series in the Independent Variable 0.2.2-1. Equation coefficients are representable in the ordinary power series form 0.2.2-2. Equation coefficients have poles at some point 0.2.3. Asymptotic Solutions 0.2.3-1. Equations not containing 
 . Leading asymptotic terms 0.2.3-2. Equations not containing 
 . Two-term asymptotic expansions 0.2.3-3. Equations of special form not containing 
0.2.3-4. Equations not containing 
 . Equation coefficients are dependent on 0.2.3-5. Equations containing 
 0.2.3-6. Equations of the general form 0.2.4. Boundary Value Problems 0.2.4-1. The first, second, third, and mixed boundary value problems 0.2.4-2. Simplification of boundary conditions. Reduction of equation to the self-adjoint form 0.2.4-3. The Green’s function. Boundary value problems for nonhomogeneous equations 0.2.4-4. Representation of the Green’s function in terms of particular solutions 0.2.5. Eigenvalue Problems 0.2.5-1. The Sturm–Liouville problem 0.2.5-2. General properties of the Sturm–Liouville problem (1), (2) 0.2.5-3. Problems with boundary conditions of the first kind 0.2.5-4. Problems with boundary conditions of the second kind

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0.2.5-5. Problems with boundary conditions of the third kind 0.2.5-6. Problems with mixed boundary conditions 0.3. Second-Order Nonlinear Differential Equations 0.3.1. Form of the General Solution. Cauchy Problem 0.3.1-1. Equations solved for the derivative. General solution 0.3.1-2. Cauchy problem. The existence and uniqueness theorem 0.3.2. Equations Admitting Reduction of Order 0.3.2-1. Equations not containing explicitly 0.3.2-2. Equations not containing  explicitly (autonomous equations) 0.3.2-3. Equations of the form  (  +  , 
 , 

 ) = 0 0.3.2-4. Equations of the form  ( ,  
− , 

 ) = 0 0.3.2-5. Homogeneous equations 0.3.2-6. Generalized homogeneous equations 0.3.2-7. Equations invariant under scaling–translation transformations 0.3.2-8. Exact second-order equations 0.3.2-9. Reduction of quasilinear equations to the normal form 0.3.3. Methods of Regular Series Expansions with Respect to the Independent Variable or Small Parameter 0.3.3-1. Method of expansion in powers of the independent variable 0.3.3-2. Method of regular (direct) expansion in powers of the small parameter 0.3.3-3. Pad´e approximants 0.3.4. Perturbation Methods of Mechanics and Physics 0.3.4-1. Preliminary remarks. A summary table of basic methods 0.3.4-2. The method of scaled parameters (Lindstedt–Poincar e´ method) 0.3.4-3. Averaging method (Van der Pol–Krylov–Bogolyubov scheme) 0.3.4-4. Method of two-scale expansions (Cole–Kevorkian scheme) 0.3.4-5. Method of matched asymptotic expansions 0.3.5. Galerkin Method and Its Modifications (Projection Methods) 0.3.5-1. General form of an approximate solution 0.3.5-2. Galerkin method 0.3.5-3. The Bubnov–Galerkin method, the moment method, and the least squares method 0.3.5-4. Collocation method 0.3.5-5. The method of partitioning the domain 0.3.5-6. The least squared error method 0.3.6. Iteration and Numerical Methods 0.3.6-1. The method of successive approximations (Cauchy problem) 0.3.6-2. The Runge–Kutta method (Cauchy problem) 0.3.6-3. Shooting method (boundary value problems) 0.3.6-4. Method of accelerated convergence in eigenvalue problems 0.4. Linear Equations of Arbitrary Order 0.4.1. Linear Equations with Constant Coefficients 0.4.1-1. Homogeneous linear equations 0.4.1-2. Nonhomogeneous linear equations 0.4.2. Linear Equations with Variable Coefficients 0.4.2-1. Homogeneous linear equations. Structure of the general solution 0.4.2-2. Utilization of particular solutions for reducing the order of the original equation 0.4.2-3. Wronskian determinant and Liouville formula

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0.4.2-4. Nonhomogeneous linear equations. Construction of the general solution 0.4.3. Asymptotic Solutions of Linear Equations 0.4.3-1. Fourth-order linear equations 0.4.3-2. Higher-order linear equations 0.5. Nonlinear Equations of Arbitrary Order 0.5.1. Structure of the General Solution. Cauchy Problem 0.5.1-1. Equations solved for the highest derivative. General solution 0.5.1-2. The Cauchy problem. The existence and uniqueness theorem 0.5.2. Equations Admitting Reduction of Order 0.5.2-1. Equations not containing , 
 ,  , (  ) explicitly 0.5.2-2. Equations not containing  explicitly (autonomous equations) ( )  =0 0.5.2-3. Equations of the form  +  , 
 ,  ,  


 )  = 0 and its 0.5.2-4. Equations of the form  ,   − ,  ,  , ( generalizations 0.5.2-5. Homogeneous equations 0.5.2-6. Generalized homogeneous equations  0.5.2-7. Equations of the form    , 
  , 

  ,  , (  )   = 0  ) = 0 0.5.2-8. Equations of the form      ,  
 ,  2 

 ,  ,   ( 0.5.2-9. Other equations 0.5.3. A Method for Construction of Solvable Equations of General Form 0.5.3-1. Description of the method 0.5.3-2. Examples 0.6. Lie Group and Discrete-Group Methods 0.6.1. Lie Group Method. Point Transformations 0.6.1-1. Local one-parameter Lie group of transformations. Invariance condition 0.6.1-2. Group analysis of second-order equations. Structure of an admissible operator 0.6.1-3. Utilization of local groups for reducing the order of equations and their integration 0.6.2. Contact Transformations. B¨acklund Transformations. Formal Operators. Factorization Principle 0.6.2-1. Contact transformations 0.6.2-2. B¨acklund transformations. Formal operators and nonlocal variables 0.6.2-3. Factorization principle 0.6.3. First Integrals (Conservation Laws) 0.6.4. Discrete-Group Method. Point Transformations 0.6.5. Discrete-Group Method. The Method of RF-Pairs 1. First-Order Differential Equations 1.1. Simplest Equations with Arbitrary Functions Integrable in Closed Form 1.1.1. Equations of the Form 
 =  ( ) 1.1.2. Equations of the Form 
 =  ( ) 1.1.3. Separable Equations 
 =  ( ) ( ) 1.1.4. Linear Equation ( )
 =  1 ( ) +  0 ( ) 1.1.5. Bernoulli Equation ( )
 =  1 ( ) +   ( )  1.1.6. Homogeneous Equation 
 =  (  )

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1.2. Riccati Equation ( ) 
=  2 ( ) 2 +  1 ( ) +  0 ( ) 1.2.1. Preliminary Remarks 1.2.2. Equations Containing Power Functions 1.2.2-1. Equations of the form ( )
 =  2 ( ) 2 +  0 ( ) 1.2.2-2. Other equations 1.2.3. Equations Containing Exponential Functions 1.2.3-1. Equations with exponential functions 1.2.3-2. Equations with power and exponential functions 1.2.4. Equations Containing Hyperbolic Functions 1.2.4-1. Equations with hyperbolic sine and cosine 1.2.4-2. Equations with hyperbolic tangent and cotangent 1.2.5. Equations Containing Logarithmic Functions 1.2.5-1. Equations of the form ( )
 =  2 ( ) 2 +  0 ( ) 1.2.5-2. Equations of the form ( )
 =  2 ( ) 2 +  1 ( )
 +  0 ( ) 1.2.6. Equations Containing Trigonometric Functions 1.2.6-1. Equations with sine 1.2.6-2. Equations with cosine 1.2.6-3. Equations with tangent 1.2.6-4. Equations with cotangent 1.2.6-5. Equations containing combinations of trigonometric functions 1.2.7. Equations Containing Inverse Trigonometric Functions 1.2.7-1. Equations containing arcsine 1.2.7-2. Equations containing arccosine 1.2.7-3. Equations containing arctangent 1.2.7-4. Equations containing arccotangent 1.2.8. Equations with Arbitrary Functions 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives) 1.2.8-2. Equations containing arbitrary functions and their derivatives 1.2.9. Some Transformations 1.3. Abel Equations of the Second Kind 1.3.1. Equations of the Form !
 − =  ( ) 1.3.1-1. Preliminary remarks. Classification tables 1.3.1-2. Solvable equations and their solutions 1.3.2. Equations of the Form !
 =  ( ) + 1 1.3.3. Equations of the Form !
 =  1 ( ) +  0 ( ) 1.3.3-1. Preliminary remarks 1.3.3-2. Solvable equations and their solutions 1.3.4. Equations of the Form [ 1 ( ) + 0 ( )]
 =  2 ( ) 2 +  1 ( ) +  0 ( ) 1.3.4-1. Preliminary remarks 1.3.4-2. Solvable equations and their solutions 1.3.5. Some Types of First- and Second-Order Equations Reducible to Abel Equations of the Second Kind 1.3.5-1. Quasi-homogeneous equations 1.3.5-2. Equations of the theory of chemical reactors and the combustion theory 1.3.5-3. Equations of the theory of nonlinear oscillations 1.3.5-4. Second-order homogeneous equations of various types 1.3.5-5. Second-order equations invariant under some transformations

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1.4. Equations Containing Polynomial Functions of 1.4.1. Abel Equations of the First Kind "
 =  3 ( ) 3 +  2 ( ) 2 +  1 ( ) +  0 ( ) 1.4.1-1. Preliminary remarks 1.4.1-2. Solvable equations and their solutions 1.4.2. Equations of the Form ( # 22 2 + # 12  + # 11  2 + # 0 )
 = $ 22 2 + $ 12  + $ 11  2 + $ 1.4.2-1. Preliminary remarks. Some transformations 1.4.2-2. Solvable equations and their solutions 1.4.3. Equations of the Form ( # 22 2 + # 12  + # 11  2 + # 2 + # 1  )
 = $ 22 2 + $ 12  + $ 11  2 + $ 2 + $ 1  1.4.3-1. Preliminary remarks 1.4.3-2. Solvable equations and their solutions 1.4.4. Equations of the Form ( # 22 2 + # 12  + # 11  2 + # 2 + # 1  + # 0 )
 = $ 22 2 + $ 12  + $ 11  2 + $ 2 + $ 1  + $ 0 1.4.4-1. Preliminary remarks. Some transformations 1.4.4-2. Solvable equations and their solutions 1.4.5. Equations of the Form ( # 3 3 + # 2  2 + # 1  2 + # 0  3 +  1 +  0  )
 = $ 3 3 + $ 2 2 + $ 1 2 + $ 0 3 +  1 +  0

0

142

1.5. Equations of the Form  ( , )
 = ( , ) Containing Arbitrary Parameters 1.5.1. Equations Containing Power Functions 1.5.1-1. Equations of the form 
 =  ( , ) 1.5.1-2. Other equations 1.5.2. Equations Containing Exponential Functions 1.5.2-1. Equations with exponential functions 1.5.2-2. Equations with power and exponential functions 1.5.3. Equations Containing Hyperbolic Functions 1.5.4. Equations Containing Logarithmic Functions 1.5.5. Equations Containing Trigonometric Functions 1.5.6. Equations Containing Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions

1.6. Equations of the Form  ( , , 
 ) = 0 Containing Arbitrary Parameters 1.6.1. Equations of the Second Degree in 
1.6.1-1. Equations of the form  ( , )(
 )2 = ( , ) 1.6.1-2. Equations of the form  ( , )(
 )2 = ( , )
 + % ( , ) 1.6.2. Equations of the Third Degree in 
 1.6.2-1. Equations of the form  ( , )(
 )3 = ( , )
 + % ( , ) 1.6.2-2. Equations of the form  ( , )(
 )3 = ( , )(
 )2 + % ( , )
 + & ( , ) 180 1.6.3. Equations of the Form (
 )  =  ( ) + ( ) 1.6.3-1. Some transformations 1.6.3-2. Classification tables and exact solutions 1.6.4. Other Equations 1.6.4-1. Equations containing algebraic and power functions with respect to
 1.6.4-2. Equations containing exponential, logarithmic, and other functions with respect to 
1.7. Equations of the Form  ( , )
 = ( , ) Containing Arbitrary Functions 1.7.1. Equations Containing Power Functions 1.7.2. Equations Containing Exponential and Hyperbolic Functions 1.7.3. Equations Containing Logarithmic Functions 1.7.4. Equations Containing Trigonometric Functions 1.7.5. Equations Containing Combinations of Exponential, Logarithmic, and Trigonometric Functions

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1.8. Equations of the Form  ( , , 
 ) = 0 Containing Arbitrary Functions 1.8.1. Some Equations 1.8.1-1. Arguments of arbitrary functions depend on  and 1.8.1-2. Argument of arbitrary functions is "
 1.8.1-3. Arguments of arbitrary functions are linear with respect to
 1.8.1-4. Arguments of arbitrary functions are nonlinear with respect to
 1.8.2. Some Transformations 2. Second-Order Differential Equations 2.1. Linear Equations 2.1.1. Representation of the General Solution Through a Particular Solution 2.1.2. Equations Containing Power Functions 2.1.2-1. Equations of the form 

 +  ( ) = 0 2.1.2-2. Equations of the form 

 +  ( )
 + ( ) = 0 2.1.2-3. Equations of the form (  +  )

 +  ( )
 + ( ) = 0 2.1.2-4. Equations of the form  2 

 +  ( )
 + ( ) = 0 2.1.2-5. Equations of the form (  2 + ' + ( )

 +  ( )
 + ( ) = 0 2.1.2-6. Equations of the form (  3  3 +  2  2 +  1  +  0 )

 +  ( )
 + ( ) = 0 2.1.2-7. Equations of the form (  4  4 + ))) +  1  +  0 )

 +  ( )
 + ( ) = 0 2.1.2-8. Other equations 2.1.3. Equations Containing Exponential Functions 2.1.3-1. Equations with exponential functions 2.1.3-2. Equations with power and exponential functions 2.1.4. Equations Containing Hyperbolic Functions 2.1.4-1. Equations with hyperbolic sine 2.1.4-2. Equations with hyperbolic cosine 2.1.4-3. Equations with hyperbolic tangent 2.1.4-4. Equations with hyperbolic cotangent 2.1.4-5. Equations containing combinations of hyperbolic functions 2.1.5. Equations Containing Logarithmic Functions 2.1.5-1. Equations of the form  ( )

 + ( ) = 0 2.1.5-2. Equations of the form  ( )

 + ( )
 + % ( ) = 0 2.1.6. Equations Containing Trigonometric Functions 2.1.6-1. Equations with sine 2.1.6-2. Equations with cosine 2.1.6-3. Equations with tangent 2.1.6-4. Equations with cotangent 2.1.6-5. Equations containing combinations of trigonometric functions 2.1.7. Equations Containing Inverse Trigonometric Functions 2.1.7-1. Equations with arcsine 2.1.7-2. Equations with arccosine 2.1.7-3. Equations with arctangent 2.1.7-4. Equations with arccotangent 2.1.8. Equations Containing Combinations of Exponential, Logarithmic, Trigonometric, and Other Functions 2.1.9. Equations with Arbitrary Functions 2.1.9-1. Equations containing arbitrary functions (but not containing their derivatives) 2.1.9-2. Equations containing arbitrary functions and their derivatives 2.1.10. Some Transformations

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2.2. Autonomous Equations 

 =  ( , 
 ) 2.2.1. Equations of the Form 

 − 
 =  ( ) 2.2.2. Equations of the Form 

 +  ( ) 
+ = 0 2.2.2-1. Preliminary remarks 2.2.2-2. Solvable equations and their solutions 2.2.3. Lienard Equations 

 +  ( )
 + ( ) = 0 2.2.3-1. Preliminary remarks 2.2.3-2. Solvable equations and their solutions 2.2.4. Rayleigh Equations 

 +  (
 ) + ( ) = 0 2.2.4-1. Preliminary remarks. Some transformations 2.2.4-2. Solvable equations and their solutions

2.3. Emden–Fowler Equation 

 = #*  + 2.3.1. Exact Solutions 2.3.1-1. Preliminary remarks. Classification table 2.3.1-2. Solvable equations and their solutions 2.3.2. First Integrals (Conservation Laws) 2.3.2-1. First integrals with , = 2 2.3.2-2. First integrals with , = 3 2.3.2-3. First integrals with , = 4 2.3.2-4. First integrals with , = 5 2.3.3. Some Formulas and Transformations 2.4. Equations of the Form 

 = # 2.4.1. Classification Table 2.4.2. Exact Solutions

1

  1 +

1

+#

2

  2 +

2

2.5. Generalized Emden–Fowler Equation 

 = #*  + ( 
)2.5.1. Classification Table 2.5.2. Exact Solutions 2.5.3. Some Formulas and Transformations 2.5.3-1. A particular solution 2.5.3-2. Discrete transformations of the generalized Emden–Fowler equation 2.5.3-3. Reduction of the generalized Emden–Fowler equation to an Abel equation 2.6. Equations of the Form 

 = # 1   1 + 1 (
 )- 1 + # 2   2 + 2 (
 )- 2 2.6.1. Modified Emden–Fowler Equation 

 = # 1  −1 
 + # 2   + 2.6.1-1. Preliminary remarks. Classification table 2.6.1-2. Solvable equations and their solutions 2.6.2. Equations of the Form 

 = ( # 1   1 + 1 + # 2   2 + 2 )(
 )2.6.2-1. Classification table 2.6.2-2. Solvable equations and their solutions 2.6.3. Equations of the Form 

 = ./#*  + (
 )- + #*  −1 + +1 (
 )- −1 2.6.3-1. Classification table 2.6.3-2. Solvable equations and their solutions 2.6.4. Other Equations ( 0 1 ≠ 0 2 ) 2.6.4-1. Classification table 2.6.4-2. Solvable equations and their solutions 2.7. Equations of the Form 

 =  ( ) ( ) % ( 
) 2.7.1. Equations of the Form 

 =  ( ) ( ) 2.7.2. Equations Containing Power Functions ( %21 const) 2.7.3. Equations Containing Exponential Functions ( %31 const)

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2.7.3-1. Preliminary remarks 2.7.3-2. Solvable equations and their solutions 2.7.4. Equations Containing Hyperbolic Functions ( %31 const) 2.7.5. Equations Containing Trigonometric Functions ( %31 const) 2.7.6. Some Transformations 2.8. Some Nonlinear Equations with Arbitrary Parameters 2.8.1. Equations Containing Power Functions 2.8.1-1. Equations of the form  ( , )

 + ( , ) = 0 2.8.1-2. Equations of the form  ( , )

 + ( , )
 + % ( , ) = 0 2.8.1-3. Equations of the form  ( , )

 + ( , )(
 )2 + % ( , )
 + & ( , ) = 0 2.8.1-4. Other equations 2.8.2. Painlev´e Transcendents 2.8.2-1. Preliminary remarks. Singular points of solutions 2.8.2-2. First Painlev´e transcendent 2.8.2-3. Second Painlev´e transcendent 2.8.2-4. Third Painlev´e transcendent 2.8.2-5. Fourth Painlev´e transcendent 2.8.2-6. Fifth Painlev´e transcendent 2.8.2-7. Sixth Painlev´e transcendent 2.8.3. Equations Containing Exponential Functions 2.8.3-1. Equations of the form  ( , )

 + ( , ) = 0 2.8.3-2. Equations of the form  ( , )

 + ( , )
 + % ( , ) = 0 2.8.3-3. Equations of the form  ( , )

 + ( , )(
 )2 + % ( , )
 + & ( , ) = 0 2.8.3-4. Other equations 2.8.4. Equations Containing Hyperbolic Functions 2.8.4-1. Equations with hyperbolic sine 2.8.4-2. Equations with hyperbolic cosine 2.8.4-3. Equations with hyperbolic tangent 2.8.4-4. Equations with hyperbolic cotangent 2.8.4-5. Equations containing combinations of hyperbolic functions 2.8.5. Equations Containing Logarithmic Functions 2.8.5-1. Equations of the form  ( , )

 + ( , )
 + % ( , ) = 0 2.8.5-2. Other equations 2.8.6. Equations Containing Trigonometric Functions 2.8.6-1. Equations with sine 2.8.6-2. Equations with cosine 2.8.6-3. Equations with tangent 2.8.6-4. Equations with cotangent 2.8.6-5. Equations containing combinations of trigonometric functions 2.8.7. Equations Containing the Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions 2.9. Equations Containing Arbitrary Functions 2.9.1. Equations of the Form  ( , )

 + 4 ( , ) = 0 2.9.1-1. Arguments of arbitrary functions are algebraic and power functions of  and 2.9.1-2. Arguments of the arbitrary functions are other functions 2.9.2. Equations of the Form  ( , )

 + 4 ( , )
 + 5 ( , ) = 0 2.9.2-1. Argument of the arbitrary functions is  2.9.2-2. Argument of the arbitrary functions is 2.9.2-3. Other arguments of the arbitrary functions

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7

2.9.3. Equations of the Form  ( , )

 + 6

+

2.9.4.

2.9.5. 2.9.6.

2.9.7.

=0

4 + ( , )(
 ) + = 0 ( 8

= 2, 3, 4)

2.9.3-1. Argument of the arbitrary functions is  . 2.9.3-2. Argument of the arbitrary functions is 2.9.3-3. Other arguments of arbitrary functions Equations of the Form  ( , , 
 )

 + 4 ( , , 
 ) = 0 2.9.4-1. Arguments of the arbitrary functions depend on  or 2.9.4-2. Arguments of the arbitrary functions depend on  and 2.9.4-3. Arguments of the arbitrary functions depend on  , , and
 Equations Not Solved for Second Derivative Equations of General Form 2.9.6-1. Equations containing arbitrary functions of two variables 2.9.6-2. Equations containing arbitrary functions of three variables Some Transformations

3. Third-Order Differential Equations 3.1. Linear Equations 3.1.1. Preliminary Remarks 3.1.2. Equations Containing Power Functions 3.1.2-1. Equations of the form  3 ( )


 +  0 ( ) = ( ) 3.1.2-2. Equations of the form  3 ( )


 +  1 ( )
 +  0 ( ) = ( ) 3.1.2-3. Equations of the form  3 ( )


 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( ) 3.1.3. Equations Containing Exponential Functions 3.1.3-1. Equations with exponential functions 3.1.3-2. Equations with power and exponential functions 3.1.4. Equations Containing Hyperbolic Functions 3.1.4-1. Equations with hyperbolic sine 3.1.4-2. Equations with hyperbolic cosine 3.1.4-3. Equations with hyperbolic sine and cosine 3.1.4-4. Equations with hyperbolic tangent 3.1.4-5. Equations with hyperbolic cotangent 3.1.5. Equations Containing Logarithmic Functions 3.1.5-1. Equations with logarithmic functions 3.1.5-2. Equations with power and logarithmic functions 3.1.6. Equations Containing Trigonometric Functions 3.1.6-1. Equations with sine 3.1.6-2. Equations with cosine 3.1.6-3. Equations with sine and cosine 3.1.6-4. Equations with tangent 3.1.6-5. Equations with cotangent 3.1.7. Equations Containing Inverse Trigonometric Functions 3.1.8. Equations Containing Combinations of Exponential, Logarithmic, Trigonometric, and Other Functions 3.1.9. Equations Containing Arbitrary Functions 3.1.9-1. Equations of the form  3 ( )


 +  1 ( )
 +  0 ( ) = ( ) 3.1.9-2. Equations of the form  3 ( )


 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( )

3.2. Equations of the Form 


 = #* 9 : (
 ); (

 ) < 3.2.1. Classification Table 3.2.2. Equations of the Form 


 = # :

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3.2.3. Equations of the Form 


 = #* 9 3.2.4. Equations with |= | + | > | ≠ 0 3.2.5. Some Transformations

:

3.3. Equations of the Form 


 =  ( ) (
 ) % (

 ) 3.3.1. Equations Containing Power Functions 3.3.2. Equations Containing Exponential Functions 3.3.3. Other Equations 3.4. Nonlinear Equations with Arbitrary Parameters 3.4.1. Equations Containing Power Functions 3.4.1-1. Equations of the form  ( , ) 


 = ( , ) 3.4.1-2. Equations of the form 


 =  ( , , 
) 3.4.1-3. Equations of the form  ( , , 
 )


 + ( , , 
 )

 + % ( , , 
 ) = 0 3.4.1-4. Other equations 3.4.2. Equations Containing Exponential Functions 3.4.2-1. Equations of the form 


 =  ( , , 
 ) 3.4.2-2. Other equations 3.4.3. Equations Containing Hyperbolic Functions 3.4.3-1. Equations with hyperbolic sine 3.4.3-2. Equations with hyperbolic cosine 3.4.3-3. Equations with hyperbolic tangent 3.4.3-4. Equations with hyperbolic cotangent 3.4.4. Equations Containing Logarithmic Functions 3.4.4-1. Equations of the form 


 =  ( , , 
 ) 3.4.4-2. Other equations 3.4.5. Equations Containing Trigonometric Functions 3.4.5-1. Equations with sine 3.4.5-2. Equations with cosine 3.4.5-3. Equations with tangent 3.4.5-4. Equations with cotangent 3.5. Nonlinear Equations Containing Arbitrary Functions 3.5.1. Equations of the Form  ( , )


 + 4 ( , ) = 0 3.5.1-1. Arguments of the arbitrary functions are  or 3.5.1-2. Arguments of the arbitrary functions depend on  and 3.5.2. Equations of the Form  ( , , 
 )


 + 4 ( , , 
 ) = 0 3.5.2-1. Arguments of the arbitrary functions depend on  and 3.5.2-2. Arguments of the arbitrary functions depend on  , , and
 3.5.3. Equations of the Form  ( , , 
) 


 + 4 ( , , 
) 

 + 5 ( , , 
) = 0 3.5.3-1. The arbitrary functions depend on  or 3.5.3-2. Arguments of arbitrary functions depend on  and 3.5.3-3. Arguments of arbitrary functions depend on  , , and
 7 3.5.4. Equations of the Form  ( , , 
 )


 + 4 ( , , 
 )(

 ) 9 = 0

9

9

3.5.4-1. Arbitrary functions depend on  or 3.5.4-2. Arguments of arbitrary functions depend on  , , and
 3.5.5. Other Equations 3.5.5-1. Equations of the form  ( , , 
 , 

 )


 + 4 ( , , 
 , 

 ) = 0 3.5.5-2. Equations of the form  ( , , 
 , 

 , 


 ) = 0

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xvi

CONTENTS

4. Fourth-Order Differential Equations 4.1. Linear Equations 4.1.1. Preliminary Remarks 4.1.2. Equations Containing Power Functions 4.1.2-1. Equations of the form  4 ( ) 



 +  0 ( ) = ( ) 4.1.2-2. Equations of the form  4 ( ) 



 +  1 ( ) 
+  0 ( ) = ( ) 4.1.2-3. Equations of the form  4 ( )



 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( ) 4.1.2-4. Other equations 4.1.3. Equations Containing Exponential and Hyperbolic Functions 4.1.3-1. Equations with exponential functions 4.1.3-2. Equations with hyperbolic functions 4.1.4. Equations Containing Logarithmic Functions 4.1.5. Equations Containing Trigonometric Functions 4.1.5-1. Equations with sine and cosine 4.1.5-2. Equations with tangent and cotangent 4.1.6. Equations Containing Arbitrary Functions 4.1.6-1. Equations of the form  4 ( )



 +  1 ( )
 +  0 ( ) = ( ) 4.1.6-2. Equations of the form  4 ( )



 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( ) 4.1.6-3. Other equations 4.2. Nonlinear Equations 4.2.1. Equations Containing Power Functions 4.2.1-1. Equations of the form 



 =  ( , ) 4.2.1-2. Equations of the form 



 =  ( , , 
 4.2.1-3. Equations of the form 



 =  ( , , 
 4.2.1-4. Equations of the form 



 =  ( , , 
 4.2.2. Equations Containing Exponential Functions 4.2.2-1. Equations of the form 



 =  ( , ) 4.2.2-2. Other equations 4.2.3. Equations Containing Hyperbolic Functions 4.2.3-1. Equations with hyperbolic sine 4.2.3-2. Equations with hyperbolic cosine 4.2.3-3. Equations with hyperbolic tangent 4.2.3-4. Equations with hyperbolic cotangent 4.2.4. Equations Containing Logarithmic Functions 4.2.4-1. Equations of the form 



 =  ( , ) 4.2.4-2. Other equations 4.2.5. Equations Containing Trigonometric Functions 4.2.5-1. Equations with sine 4.2.5-2. Equations with cosine 4.2.5-3. Equations with tangent 4.2.5-4. Equations with cotangent 4.2.6. Equations Containing Arbitrary Functions 4.2.6-1. Equations of the form 



 =  ( , ) 4.2.6-2. Equations of the form 



 =  ( , , 
 4.2.6-3. Equations of the form 



 =  ( , , 
 4.2.6-4. Equations of the form 



 =  ( , , 
 4.2.6-5. Other equations

) , 

 ) , 

 , 


 )

) , 

 ) , 

 , 


 )

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5. Higher-Order Differential Equations 5.1. Linear Equations 5.1.1. Preliminary Remarks 5.1.2. Equations Containing Power Functions 5.1.2-1. Equations of the form   ( ) (  ) +  0 ( ) = ( ) 5.1.2-2. Equations of the form   ( ) (  ) +  1 ( ) 
+  0 ( ) = ( ) 5.1.2-3. Other equations 5.1.3. Equations Containing Exponential and Hyperbolic Functions 5.1.3-1. Equations with exponential functions 5.1.3-2. Equations with hyperbolic functions 5.1.4. Equations Containing Logarithmic Functions 5.1.5. Equations Containing Trigonometric Functions 5.1.5-1. Equations with sine and cosine 5.1.5-2. Equations with tangent and cotangent 5.1.6. Equations Containing Arbitrary Functions 5.1.6-1. Equations of the form   ( ) (  ) +  1 ( )
 +  0 ( ) = ( ) 5.1.6-2. Other equations 5.2. Nonlinear Equations 5.2.1. Equations Containing Power Functions 5.2.1-1. Fifth- and sixth-order equations 5.2.1-2. Equations of the form (  ) =  ( , ) 5.2.1-3. Equations of the form (  ) =  ( , , 
 , 

 ) 5.2.1-4. Other equations 5.2.2. Equations Containing Exponential Functions 5.2.2-1. Fifth- and sixth-order equations 5.2.2-2. Equations of the form (  ) =  ( , ) 5.2.2-3. Other equations 5.2.3. Equations Containing Hyperbolic Functions 5.2.3-1. Equations with hyperbolic sine 5.2.3-2. Equations with hyperbolic cosine 5.2.3-3. Equations with hyperbolic tangent 5.2.3-4. Equations with hyperbolic cotangent 5.2.4. Equations Containing Logarithmic Functions 5.2.4-1. Equations of the form (  ) =  ( , ) 5.2.4-2. Other equations 5.2.5. Equations Containing Trigonometric Functions 5.2.5-1. Equations with sine 5.2.5-2. Equations with cosine 5.2.5-3. Equations with tangent 5.2.5-4. Equations with cotangent 5.2.6. Equations Containing Arbitrary Functions 5.2.6-1. Fifth- and sixth-order equations 5.2.6-2. Equations of the form (  ) =  ( , ) 5.2.6-3. Equations of the form (  ) =  ( , , 
 ) 5.2.6-4. Equations of the form (  ) =  ( , , 
 , 

 ) 5.2.6-5. Equations of the form  ( , ) (  ) + ( , , 
 ) (  −1) =  −2)  %   , , 
 ,  , (?  −1)  5.2.6-6. Equations of the form (  ) =    , , 
 ,  , (? 
( )  =0 5.2.6-7. Equations of the general form    , ,  ,  , 

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Supplements S.1. Elementary Functions and Their Properties S.1.1. Trigonometric Functions S.1.1-1. Simplest relations S.1.1-2. Relations between trigonometric functions of single argument S.1.1-3. Reduction formulas S.1.1-4. Addition and subtraction of trigonometric functions S.1.1-5. Products of trigonometric functions S.1.1-6. Powers of trigonometric functions S.1.1-7. Addition formulas S.1.1-8. Trigonometric functions of multiple arguments S.1.1-9. Trigonometric functions of half argument S.1.1-10. Euler and de Moivre formulas. Relationship with hyperbolic functions S.1.1-11. Differentiation formulas S.1.1-12. Expansion into power series S.1.2. Hyperbolic Functions S.1.2-1. Definitions S.1.2-2. Simplest relations S.1.2-3. Relations between hyperbolic functions of single argument ( ≥ 0) S.1.2-4. Addition formulas S.1.2-5. Addition and subtraction of hyperbolic functions S.1.2-6. Products of hyperbolic functions S.1.2-7. Powers of hyperbolic functions S.1.2-8. Hyperbolic functions of multiple arguments S.1.2-9. Relationship with trigonometric functions S.1.2-10. Differentiation formulas S.1.2-11. Expansion into power series S.1.3. Inverse Trigonometric Functions S.1.3-1. Definitions and some properties S.1.3-2. Simplest formulas S.1.3-3. Relations between inverse trigonometric functions S.1.3-4. Addition and subtraction of inverse trigonometric functions S.1.3-5. Differentiation formulas S.1.3-6. Expansion into power series S.1.4. Inverse Hyperbolic Functions S.1.4-1. Relationships with logarithmic functions S.1.4-2. Relations between inverse hyperbolic functions S.1.4-3. Addition and subtraction of inverse hyperbolic functions S.1.4-4. Differentiation formulas S.1.4-5. Expansion into power series S.2. Special Functions and Their Properties S.2.1. Some Symbols and Coefficients S.2.1-1. Factorials S.2.1-2. Binomial coefficients S.2.1-3. Pochhammer symbol S.2.2. Error Functions and Exponential Integral S.2.2-1. Error function and complementary error function S.2.2-2. Exponential integral S.2.2-3. Logarithmic integral

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S.2.3. Gamma and Beta Functions S.2.3-1. Gamma function S.2.3-2. Logarithmic derivative of the gamma function S.2.3-3. Beta function S.2.4. Incomplete Gamma and Beta Functions S.2.4-1. Incomplete gamma function S.2.4-2. Incomplete beta function S.2.5. Bessel Functions S.2.5-1. Definitions and basic formulas S.2.5-2. Bessel functions for @ = A"BCA 12 , where B = 0, 1, 2,  S.2.5-3. Bessel functions for @ = A"B , where B = 0, 1, 2,  S.2.5-4. Wronskians and similar formulas S.2.5-5. Integral representations S.2.5-6. Asymptotic expansions S.2.5-7. Zeros and orthogonality properties of the Bessel functions S.2.5-8. Hankel functions (Bessel functions of the third kind) S.2.6. Modified Bessel Functions S.2.6-1. Definitions. Basic formulas S.2.6-2. Modified Bessel functions for @ = A"BCA 12 , where B = 0, 1, 2,  S.2.6-3. Modified Bessel functions for @ = B , where B = 0, 1, 2,  S.2.6-4. Wronskians and similar formulas S.2.6-5. Integral representations S.2.6-6. Asymptotic expansions as EDGF S.2.7. Degenerate Hypergeometric Functions S.2.7-1. Definitions. The Kummer’s series S.2.7-2. Some transformations and linear relations S.2.7-3. Differentiation formulas and Wronskian S.2.7-4. Degenerate hypergeometric functions for B = 0, 1, 2,  S.2.7-5. Integral representations S.2.7-6. Asymptotic expansion as | | DGF S.2.7-7. Whittaker functions S.2.8. Hypergeometric Functions S.2.8-1. Definition. The hypergeometric series S.2.8-2. Basic properties S.2.8-3. Integral representations S.2.9. Legendre Functions and Legendre Polynomials S.2.9-1. Definitions. Basic formulas S.2.9-2. Trigonometric expansions S.2.9-3. Some relations S.2.9-4. Integral representations S.2.9-5. Legendre polynomials S.2.9-6. Zeros of the Legendre polynomials and the generating function S.2.9-7. Associated Legendre functions S.2.10. Parabolic Cylinder Functions S.2.10-1. Definitions. Basic formulas S.2.10-2. Integral representations S.2.10-3. Asymptotic expansion as | H | DGF S.2.11. Orthogonal Polynomials S.2.11-1. Laguerre polynomials and generalized Laguerre polynomials S.2.11-2. Chebyshev polynomials S.2.11-3. Hermite polynomials

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S.2.11-4. Gegenbauer polynomials S.2.11-5. Jacobi polynomials S.2.12. The Weierstrass Function S.2.12-1. Definitions S.2.12-2. Some properties S.3. Tables of Indefinite Integrals S.3.1. Integrals Containing Rational Functions S.3.1-1. Integrals containing  + ' S.3.1-2. Integrals containing  +  and  +  S.3.1-3. Integrals containing  2 +  2 S.3.1-4. Integrals containing  2 −  2 S.3.1-5. Integrals containing  3 +  3 S.3.1-6. Integrals containing  3 −  3 S.3.1-7. Integrals containing  4 AI 4 S.3.2. Integrals Containing Irrational Functions S.3.2-1. Integrals containing  1 J 2 S.3.2-2. Integrals containing (  + ' )K J 2 S.3.2-3. Integrals containing ( 2 +  2 )1 J 2 S.3.2-4. Integrals containing ( 2 −  2 )1 J 2 S.3.2-5. Integrals containing (  2 −  2 )1 J 2 S.3.2-6. Reduction formulas S.3.3. Integrals Containing Exponential Functions S.3.4. Integrals Containing Hyperbolic Functions S.3.4-1. Integrals containing cosh  S.3.4-2. Integrals containing sinh  S.3.4-3. Integrals containing tanh  or coth  S.3.5. Integrals Containing Logarithmic Functions S.3.6. Integrals Containing Trigonometric Functions S.3.6-1. Integrals containing cos  S.3.6-2. Integrals containing sin  S.3.6-3. Integrals containing sin  and cos  S.3.6-4. Reduction formulas S.3.6-5. Integrals containing tan  and cot  S.3.7. Integrals Containing Inverse Trigonometric Functions References

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AUTHORS Andrei D. Polyanin, Ph.D., D.Sc., is a noted scientist of broad interests who works in various areas of mathematics, mechanics, and chemical engineering sciences. A. D. Polyanin graduated from the Department of Mechanics and Mathematics of the Moscow State University in 1974. He earned his Ph.D. degree in 1981 and D.Sc. degree in 1986 at the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1975, A. D. Polyanin has been a member of the staff of the Institute for Problems in Mechanics of the Russian Academy of Sciences. He is a member of the Russian National Committee on Theoretical and Applied Mechanics. Professor Polyanin has made important contributions to developing new exact and approximate analytical methods of the theory of differential equations, mathematical physics, integral equations, engineering mathematics, nonlinear mechanics, theory of heat and mass transfer, and chemical hydrodynamics. He obtained exact solutions for several thousand ordinary differential, partial differential, mathematical physics, and integral equations. Professor Polyanin is an author of 30 books in English, Russian, German, and Bulgarian, as well as over 120 research papers and three patents. He has written a number of fundamental handbooks, including A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (first edition), CRC Press, 1995; A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, 1998; A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002; A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, 2002; and A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Mathematical Physics Equations, Fizmatlit, 2002. Professor Polyanin is Editor of the book series Differential and Integral Equations and Their Applications, Taylor & Francis Inc., London. In 1991, A. D. Polyanin was awarded a Chaplygin Prize of the Russian Academy of Sciences for his research in mechanics. Address: Institute for Problems in Mechanics, RAS, 101 Vernadsky Avenue, Building 1, 119526 Moscow, Russia E-mail: [email protected]

Valentin F. Zaitsev, Ph.D., D.Sc., is a noted scientist in the fields of ordinary differential equations, mathematical physics, and nonlinear mechanics. V. F. Zaitsev graduated from the Radio Electronics Faculty of the Leningrad Polytechnical Institute (now St. Petersburg Technical University) in 1969 and received his Ph.D. degree in 1983 at the Leningrad State University. His Ph.D. thesis was devoted to the group approach to the study of some classes of ordinary differential equations. In 1992, Professor Zaitsev earned his Doctor of Sciences degree; his D.Sc. thesis was dedicated to the discretegroup analysis of ordinary differential equations. In 1971–1996, V. F. Zaitsev worked in the Research Institute for Computational Mathematics and Control Processes of the St. Petersburg State University. Since 1996, Professor Zaitsev has been a member of the staff of the Russian State Pedagogical University (St. Petersburg). Professor Zaitsev has made important contributions to new methods in the theory of ordinary and partial differential equations. He is an author of more than 130 scientific publications, including 18 books and one patent. Address: Russian State Pedagogical University, 48 Naberezhnaya reki Moiki, 191186 St. Petersburg, Russia E-mail: [email protected]

© 2003 by Chapman & Hall/CRC

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FOREWORD Exact solutions of differential equations play an important role in the proper understanding of qualitative features of many phenomena and processes in various areas of natural science. These solutions can be used to verify the consistencies and estimate errors of various numerical, asymptotic, and approximate analytical methods. This book contains nearly 6200 ordinary differential equations and their solutions. A number of new solutions to nonlinear equations are described. In some sections of this book, asymptotic solutions to some classes of equations are also given. When selecting the material, the authors gave preference to the following two types of equations: • Equations that have traditionally attracted the attention of many researchers: those of the simplest appearance but involving the most difficulties for integration (Abel equations, Emden– Fowler equations, Painlev´e equations, etc.) • Equations that are encountered in various applications (in the theory of heat and mass transfer, nonlinear mechanics, elasticity, hydrodynamics, theory of nonlinear oscillations, combustion theory, chemical engineering science, etc.) Special attention is paid to equations that depend on arbitrary functions. All other equations contain one or more arbitrary parameters (in fact, this book deals with whole families of ordinary differential equations), which can be fixed by a reader at will. In total, the handbook contains many more equations than any other book currently available (for example, the number of nonlinear equations of the second and higher order is ten times more than in the well-known E. Kamke’s Handbook on Ordinary Differential Equations). For the reader’s convenience, the introductory chapter of the book outlines basic definitions, useful formulas, and some transformations. In a concise form, it also presents exact, asymptotic, and approximate analytical methods for solving linear and nonlinear differential equations. Specific examples of utilization of these methods are considered. Formulations of existence and uniqueness theorems are also given. Boundary-value problems and eigenvalue problems are described. The handbook consists of chapters, sections, subsections, and paragraphs. Equations and formulas are numbered separately in each subsection. The equations within subsections and paragraphs are arranged in increasing order of complexity. The extensive table of contents provides rapid access to the desired equations. The main material is followed by some supplements, where basic properties of elementary and special functions (Bessel, modified Bessel, hypergeometric, Legendre, etc.) are described. Here are three main distinguishing features of the second edition vs. the first edition: • 1200 nonlinear equations with solutions have been added. • An introductory chapter that outlines exact, asymptotic, and approximate analytical methods for solving ordinary differential equations has been included. • The overwhelming majority of subsections are organized into paragraphs. As a result, the table of contents has been increased threefold to help the readers get faster access to desired equations. We would like to express our deep gratitude to Alexei Zhurov for fruitful discussions and valuable remarks. We are very grateful to Alain Moussiaux who tested a number of solutions, which allowed us to remove some inaccuracies and misprints. The authors hope that this book will be helpful for a wide range of scientists, university teachers, engineers, and students engaged in the fields of mathematics, physics, mechanics, control, chemistry, and engineering sciences. Andrei D. Polyanin Valentin F. Zaitsev

© 2003 by Chapman & Hall/CRC

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NOTATIONS AND SOME REMARKS 1. Throughout this book, in the original equations the independent variable is denoted by  , and the dependent one is denoted by . In the given solutions, the symbols L , L 0 , L 1 , L 2 ,  stand for arbitrary integration constants. Solutions are often represented in parametric form (e.g., see Subsections 1.3.1 and 2.3.1). 2. Notation for derivatives:


= M ,  M

2

3



 = M 2 ,  M




 = M 3 ,  M

3. Brief notation for partial derivatives:

   = N ,  N



4





 = M 4 ;  M

 (  ) = M   M

with

B ≥ 5.

2     = N , where  =  ( , ).  N N N  , which is defined by the recurrence 4. In some cases, we use the operator notation O M 3P M relation   −1 ( ) =  ( ) M O ( ) M O ( ) M (  )S . 2P RQ 2P M M M = N . 5. Brief operator notation corresponding to partial derivatives:  = N , N N   N N

  = N

,

2    = N 2 ,  N

6. In some sections of the book (e.g., see 1.3, 2.3–2.6, 3.2–3.3), for the sake of brevity, solutions are represented as several formulas containing the terms with the signs “A ” and “T .” Two formulas are meant—one corresponds to the upper sign and the other to the lower sign. For example, the solution of equation 1.3.1.6 can be written in the parametric form:

 = 

−1

exp(T"U 2 ),

= 

−1

V exp(T"U 2 ) A 2UW ,

where

 = X exp(T U 2 ) U − L , M

# = T 2 2.

This is equivalent so that the solutions of equation 1.3.1.6 are given by the two formulas:

 = 

−1

exp(−U 2 ),

= 

−1

V exp(−U 2 ) + 2UW ,

where

 = X exp(−U 2 ) U − L , M

V exp(U 2 ) − 2UW ,

where

 = X exp(U 2 ) U − L , M

# = −2

2

(for the upper signs) and

 = 

−1

exp(U 2 ),

= 

−1

# = 2

2

(for the lower signs).

 ( ) , it is often not stated that the assumption  ≠ 2 7. If a relation contains an expression like  −2 is adopted. 1   +1 can usually be defined so as to cover the 8. In solutions, expressions like Y  ( ) = B +1 case B = −1 in accordance with the rule Y −1 ( ) = ln | |. This is accounted for by the fact that such expressions arise from the integration of the power-law function Y  ( ) = XZ   . M

9. The order symbol [ is used to compare two functions  =  ( ) and = ( ), where is a small parameter. So  = [ ( ) means that |  | is bounded as \D 0, or  and are of the same order of magnitude as ID 0.

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10. When referencing a particular equation, a notation like “4.1.2.5” stands for “equation 5 in Subsection 4.1.2.” 11. The handbooks by Kamke (1977), Murphy (1960), Zaitsev and Polyanin (1993, 2001), Polyanin and Zaitsev (1995) were extensively used in compiling this book; references to these sources are frequently omitted.

]

^_

12. To highlight portions of the text, the following symbols are used throughout the book: indicates important information pertaining to a group of equations (Chapters 1–5); indicates the literature used in the preparation of the text in subsections, paragraphs, and specific equations.

© 2003 by Chapman & Hall/CRC

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Introduction

Some Definitions, Formulas, Methods, and Transformations 0.1. First-Order Differential Equations 0.1.1. General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems 0.1.1-1. Equations solved for the derivative. General solution. A first-order ordinary differential equation* solved for the derivative has the form


=  ( , ).

(1)

=  ( , )  . Sometimes it is represented in terms of differentials as M M A solution of a differential equation is a function ( ) that, when substituted into the equation, turns it into an identity. The general solution of a differential equation is the set of all its solutions. In some cases, the general solution can be represented as a function = Y ( , L ) that depends on one arbitrary constant L ; specific values of L define specific solutions of the equation (particular solutions). In practice, the general solution more frequently appears in implicit form, ` ( , , L ) = 0, or parametric form,  =  (a , L ), = (a , L ). Geometrically, the general solution (also called the general integral) of an equation is a family of curves in the  -plane depending on a single parameter L ; these curves are called integral curves of the equation. To each particular solution (particular integral) there corresponds a single curve that passes through a given point in the plane. For each point ( , ), the equation 
 =  ( , ) defines a value of 
, i.e., the slope of the integral curve that passes through this point. In other words, the equation generates a field of directions in the  -plane. From the geometrical point of view, the problem of solving a first-order differential equation involves finding the curves, the slopes of which at each point coincide with the direction of the field at this point. 0.1.1-2. The Cauchy problem. The uniqueness and existence theorems. 1 b . The Cauchy problem: find a solution of equation (1) that satisfies the initial condition =

0

at

 =  0,

(2)

where 0 and  0 are some numbers. Geometrical meaning of the Cauchy problem: find an integral curve of equation (1) that passes through the point ( 0 , 0 ). Condition (2) is alternatively written ( 0 ) = 0 or |  =  0 = 0 . * In what follows, we often call an ordinary differential equation a “differential equation” or even shorter an “equation.”

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2 b . Existence theorem (Peano). Let the function  ( , ) be continuous in an open domain c of the  -plane. Then there is at least one integral curve of equation (1) that passes through each point ( 0 , 0 ) dec ; each of these curves can be extended at both ends up to the boundary of any closed domain c 0 f c such that ( 0 , 0 ) belongs to the interior of c 0 . 3 b . Uniqueness theorem. Let the function  ( , ) be continuous in an open domain c and have in c a bounded partial derivative with respect to (or the Lipschitz condition holds: |  ( , ) −  ( , H )| ≤ 8 | − H |, where 8 is some positive number). Then there is a unique solution of equation (1) satisfying condition (2). 0.1.1-3. Equations not solved for the derivative. The existence theorem. A first-order differential equation not solved for the derivative can generally be written as

 ( , , 
) = 0.

(3)

Existence and uniqueness theorem. There exists a unique solution = ( ) of equation (3) satisfying the conditions |  =  0 = 0 and 
 |  =  0 = a 0 , where a 0 is one of the real roots of the equation  ( 0 , 0 , a 0 ) = 0, if the following conditions hold in a neighborhood of the point ( 0 , 0 , a 0 ): 1. The function  ( , , a ) is continuous in each of the three arguments. 2. The partial derivative hg exists and is nonzero. 3. There is a bounded partial derivative with respect to , |  | ≤ 8 .



The solution exists for | −  0 | ≤  , where  is a (sufficiently small) positive number. 0.1.1-4. Singular solutions. 1 b . A point ( , ) at which the uniqueness of the solution to equation (3) is violated is called a singular point. If conditions 1 and 3 of the existence and uniqueness theorem hold, then

 ( , , a ) = 0,

hg ( , , a ) = 0

(4)

simultaneously at each singular point. Relations (4) define a a -discriminant curve in parametric form. In some cases, the parameter a can be eliminated from (4) to give an equation of this curve in implicit form, i ( , ) = 0. If a branch = j ( ) of the curve i ( , ) = 0 consists of singular points and, at the same time, is an integral curve, then this branch is called a singular integral curve and the function = j ( ) is a singular solution of equation (3). 2 b . The singular solutions can be found by identifying the envelope of the family of integral curves, ` ( , , L ) = 0, of equation (3). The envelope is part of the L -discriminant curve, which is defined by the equations ` ( , , L ) = 0, `lk ( , , L ) = 0. The branch of the L -discriminant curve at which (a) there exist bounded partial derivatives, | `  | < 8 (b) | `  | + | ` | ≠ 0 is the envelope.

1

and | `



 | 0, under the conditions that is a positive ¡ integer and  ( ) ≠ 0. In this case, the leading term of the asymptotic solution, as D 0, in the vicinity of the point  = 0 is expressed in terms of a simpler model equation, which results from substituting the function  ( ) in equation (3) by the constant  (0) (the solution of the model equation is expressed in terms of the Bessel functions of order 1 , see equation 2.1.2.7). ¡ We specify below formulas by which the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (3) with  <  < 0 and 0 <  <  are related (excluding a small vicinity of the point  = 0). Three different cases can be extracted.

© 2003 by Chapman & Hall/CRC

Page 25

1 b . Let

be an even integer, and  ( ) > 0. Then,

¡

=

1

=

2






[  ( )]−1 J 4 exp



 ( )  S ü  M 1 −1 −1 J 4 , [  ( )] exp  ( )  S X Q 0 ü M  1 [  ( )]−1 J 4 exp − X  ( )  S Q 0 ü M 1 −1 J 4 , [  ( )] exp − X  ( )  S Q 0 ü M


 

1





where  =  ( ), , = sin O

Q

X

¡

. where  =  ( ), , = tan O 2 P ¡ be an odd integer. Then, 3 b . Let

¡

1

2





=

=

|  ( )|−1 J 4 cos −

 


1

if  < 0, if  > 0,



( )|  +

M

4 S

|  ( )|  −

M

( )|  −

M

( )|  +

M

if  < 0,

4 S

4 S

4 S

if  > 0, if  < 0, if  > 0,

4 S

if  < 0,

 ( )  S X Q 0 ü M  1 |  ( )|−1 J 4 cos − X |  ( )|  −  S Q 0 ü M 4 1 −1 J 4 , [  ( )] exp − X  ( )  S Q 0 ü M

if  > 0,

Q

1 , 2




if  > 0,

.

P ¡ be an even integer, and  ( ) < 0. Then,
  |  ( )|−1 J 4 cos Q − 1 X 0 ü | 
  1 = 1 , −1 |  ( )|−1 J 4 cos X Q 0 ü
1 −1 J 4  |  ( )| cos Q − X 0 ü | 
 =  2 1 , |  ( )|−1 J 4 cos X | Q 0 ü

2 b . Let

if  < 0,

0

where  =  ( ), , = sin O 2

¡

P

−1

X

0

[  ( )]−1 J 4 exp

ü

1

|  ( )|  +



M

if  < 0, if  > 0,

.

0.2.3-4. Equations not containing 
 . Equation coefficients are dependent on . Consider an equation of the form



 −  ( , ) = 0 (4) on a closed interval  ≤  ≤  under the condition that  ≠ 0. Assume that the following asymptotic

relation holds:

2

½  (  , ) = ÷   (  )  ,  =0

ID

0.

Then the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (4) are given by the formulas: 1 J  ) exp − 1 X  ( )  + X Q ü 0 M 2

1

=

−1 4 ( 0

2

=

−1 4 ( 0

1 J  ) exp 1 X  0 ( )  + X Q ü M 2

 1 ( ) V £  0 (  ) M  S 1 + [ ( )W ,  1 ( ) V £  0 (  ) M  S 1 + [ ( )W .

© 2003 by Chapman & Hall/CRC

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0.2.3-5. Equations containing 
 . 1 b . Consider an equation of the form



 + ( ) 
+  ( ) = 0 on a closed interval 0 ≤  ≤ 1. With ( ) > 0, the asymptotic solution of this equation, satisfying the boundary conditions (0) = L 1 and (1) = L 2 , can be represented in the form: = (L where , = exp X

Q

1 0

1

−, L

2 ) exp

V −

−1

(0) W + L

2

exp X  Q

 ( )  S + [ ( ), ( ) M

1

 ( )  S . ( ) M

2 b . Now let us take a look at an equation of the form 2



 +  ( ) 
+  ( ) = 0

(5)

on a closed interval  ≤  ≤  . Assume

c ( ) ≡ [ ( )]2 − 4  ( ) ≠ 0. Then the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (5), as pD 0, are expressed by 1

2

1 1 X c ( )  − X £ Q 2 ü M 2 1 
1 X = | c ( )|−1 J 4 exp c ( )  − X £ Q 2 ü M 2 c = | c ( )|−1 J 4 exp −


( )  S V 1 + [ ( )W , c ( ) M ( )  S V 1 + [ ( )W . ( ) M

0.2.3-6. Equations of the general form. The more general equation

2



 +  ( , ) 
+  ( , ) = 0

is reducible, with the aid of the substitution

= ¨ exp O −

1 XÈ  , to an equation of the form (4), 2 M P

2 ¨ 

 + (  − 14 2 − 12  
)¨ = 0,

to which the asymptotic formulas given above in Paragraph 0.2.3-4 are applicable.

^_

References for Subsection 0.2.3: W. Wasov (1965), F. W. J. Olver (1974), A. H. Nayfeh (1973, 1981), M. V. Fedoryuk (1993).

0.2.4. Boundary Value Problems 0.2.4-1. The first, second, third, and mixed boundary value problems (

1

≤  ≤  2 ).

We consider the second-order nonhomogeneous linear differential equation



 +  ( ) 
+ ( ) = % ( ).

(1)

1 b . The first boundary value problem: Find a solution of equation (1) satisfying the boundary conditions =  1 at  =  1 , =  2 at  =  2 . (2) (The values of the unknown are prescribed at two distinct points 

1

and  2 ).

© 2003 by Chapman & Hall/CRC

Page 27

2 b . The second boundary value problem: Find a solution of equation (1) satisfying the boundary conditions 
=  1 at  =  1 , 
=  2 at  =  2 . (3) (The values of the derivative of the unknown are prescribed at two distinct points 

1

and  2 ).

3 b . The third boundary value problem: Find a solution of equation (1) satisfying the boundary conditions 
− , 1 =  1 at  =  1 , (4) 
+ , 2 =  2 at  =  2 . 4 b . The third boundary value problem: Find a solution of equation (1) satisfying the boundary conditions 
=  2 at  =  2 . =  1 at  =  1 , (5) (The unknown itself is prescribed at one point, and its derivative at another point.) Conditions (1), (2), (3), and (4) are called homogeneous if  1 =  2 = 0. 0.2.4-2. Simplification of boundary conditions. Reduction of equation to the self-adjoint form. 1 b . Nonhomogeneous boundary conditions can be reduced to homogeneous ones by the change of variable H = # 2  2 + # 1  + # 0 + (the constants # 2 , # 1 , and # 0 are selected using the method of undetermined coefficients). In particular, the nonhomogeneous boundary conditions of the first kind (1) can be reduced to homogeneous boundary conditions by the linear change of variable  − H = − 2 1 ( −  1 ) −  1 .  2− 1 2 b . On multiplying by  ( ) = exp X

Q

 ( )  S , one reduces equation (1) to the self-adjoint form: M  (6) [ ( )
]
 +  ( ) = & ( ).

Without loss of generality, we further consider equation (6) instead of (1). We assume that the functions  , 
 ,  , and & are continuous on the interval  1 ≤  ≤  2 , and  is positive. 0.2.4-3. The Green’s function. Boundary value problems for nonhomogeneous equations. The Green’s function of the first boundary value problem for equation (6) with homogeneous boundary conditions (2) is a function of two variables 4 ( , ° ) that satisfies the following conditions: 1 b . 4 ( , ° ) is continuous in  for fixed ° , with 

1

≤ ≤

2

and 

1

≤ ° ≤  2.

2 b . 4 ( , ° ) is a solution of the homogeneous equation (6), with & = 0, for all  of the point  = ° .

1

( ) and rewrite the equation as 1 ‘'

‘ = > 2   O>H , , ‘
.

> P

(38)

The function > = > ( ) is selected so that the right-hand side of equation (38) has a nonzero limit ‘ value as ID 0, provided that H , , and 
are of the order of 1. Example 4. For

ê ( Ë , Ï , Ï ÒÍ

) = −ô

Ë^” Ï ÒÍ

+ Ï , where 0 ≤

Ï =2Í Í =

=−

•

< 1, the substitution

– 1+ ” ô ; ” Ï =Í + – 2 Ï .  

In order that the right-hand side of this equation has a nonzero limit value as

!

;

=

Ë

%N– ( ) brings equation (37) to

0, one has to set

– 1+ ” % 

= 1 or

– ” %  = const, where const is any positive number. It follows that – =  ” . ‡ leading asymptotic term of the inner expansion in the boundary layer, Ï = Ï 0 ( ; ) + 1131 , is determined by the equation ‡ ‡Ï Í Í + The ô ; ” Ï 0Í = 0, where the prime denotes differentiation with respect to ; . 0 1 1+

1+

If the position of the boundary layer is selected incorrectly, the outer amd inner expansions cannot be matched. In this situation, one should consider the case where an arbitrary boundary layer is located on the right (this case is reduced to the previous one with the change of variable  = 1 − H ). In example 4 above, the boundary layer is on the left if , > 0 and on the right if , < 0. There is a procedure for matching subsequent asymptotic terms of the expansion (see the seventh row and last column in Table 3). In its general form, this procedure can be represented as: inner expansion of the outer expansion ( -expansion for þD 0) ¶ = outer expansion of the inner expansion ( -expansion for HCDGF ).

x3y'z|{}~­³´€

The method of matched asymptotic expansions can also be applied to construct periodic solutions of singularly perturbed equations (e.g., in the problem of relaxation oscillations of the Van der Pol oscillator).

x3y'z|{}~‚µ €

Two boundary layers can arise in some problems (e.g., in cases where the right-hand side of equation (37) does not explicitly depend on "
 ).

x3y'z|{}~E¹€

The method of matched asymptotic expansions is also used for solving equations (in semi-infinite domains) that do not degenerate at = 0. In such cases, there are no boundary layers; the original variable is used in the inner domain, and an extended coordinate is introduced in the outer domain.

x3y'z|{}~Eº€

The method of matched asymptotic expansions is successfully applied for the solution of various problems in mathematical physics that are described by partial differential equations; in particular, it plays an important role in the theory of heat and mass transfer and in hydrodynamics.

^_

References for Subsection 0.3.4: M. Van Dyke (1964), A. Blaquiere (1966), G. D. Cole (1968), G. E. O. Giacaglia (1972), A. H. Nayfeh (1973, 1981), N. N. Bogolyubov and Yu. A. Mitropolskii (1974) J. Kevorkian and J. D. Cole (1981, 1996), P. A. Lagerstrom (1988), V. Ph. Zhuravlev and D. M. Klimov (1988), V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt (1993).

0.3.5. Galerkin Method and Its Modifications (Projection Methods) 0.3.5-1. General form of an approximate solution. Consider a boundary value problem for the equation

—

[ ] −  ( ) = 0

(1)

© 2003 by Chapman & Hall/CRC

Page 46

with linear homogeneous boundary conditions at the points  =  1 and  =  2 ( 1 ≤  ≤  2 ). Here, — is a linear or nonlinear differential operator of the second order (or a higher order operator); — = ( ) is the unknown function and  =  ( ) is a given function. It is assumed that [0] = 0. Let us choose a sequence of linearly independent functions

Y = Y  ( )

(B = 1, 2,  , Ä )

(2)

satisfying the same boundary conditions as = ( ). According to all methods that will be considered below, an approximate solution of equation (1) is sought as a linear combination

/

=

/

½ 

=1

#  Y  ( ),

(3)

with the unknown coefficients #  to be found in the process of solving the problem. The finite sum (3) is called an approximation function. The remainder term ² / obtained after the finite sum has been substituted into the equation (1) has the form

²

/

—

= [

/

]− .

If the remainder ² / is identically equal to zero, then the function equation (1). In general, ² / 1 0.

/

(4) is the exact solution of

0.3.5-2. Galerkin method. In order to find the coefficients #  in (3), consider another sequence of linearly independent functions j = j  ( ) ( , = 1, 2,  , Ä ). (5)

Let us multiply both sides of (4) by j  and integrate the resulting relation over the region ˜ = { ≤  ≤  }, in which we seek the solution of equation (1). Next, we equate the cor1 2 responding integrals to zero (for the exact solutions, these integrals are equal to zero). Thus, we obtain the following system of linear algebraic equations for the unknown coefficients #  :

X 



2 1

j  ²

/ M

 =0

( , = 1, 2,  , Ä ).

(6)

Relations (6) mean that the approximation function (3) satisfies equation (1) “on the  2 average” (i.e., in the integral sense) with weights j  . Introducing the scalar product 〈 , % 〉 = X  %  of 1

M

arbitrary functions and % , we can consider equations (6) as the condition of orthogonality of the remainder ² / to all weight functions j  . The Galerkin method can be applied not only to boundary value problems but also to eigenand seeks eigenfunctions  , together with value problems (in the latter case, one takes  = Œ eigenvalues Œ  ). Mathematical justification of the Galerkin method for specific boundary value problems can be found in the literature listed at the end of Subsection 0.3.5. Below we describe some other methods, which are in fact special cases of the Galerkin method. x3y'z|{}~/€ As a rule, one takes suitable sequences of polynomials or trigonometric functions as Y  ( ) in the approximation function (3). 0.3.5-3. The Bubnov–Galerkin method, the moment method, and the least squares method.

1 b . The sequences of functions (2) and (5) in the Galerkin method can be chosen arbitrarily. In the case of equal functions, Y  ( ) = j  ( ) ( , = 1, 2,  , Ä ), (7) the method is often called the Bubnov–Galerkin method.

© 2003 by Chapman & Hall/CRC

Page 47

2 b . The moment method is the Galerkin method with the weight functions (5) being powers of  ,

j  =  .

(8)

3 b . Sometimes, the functions j  are expressed in terms of Y  by the relations

—

j  = [Y  ]

—

( , = 1, 2,  ),

is the differential operator of equation (1). This version of the Galerkin method is called where the least squares method. 0.3.5-4. Collocation method. In the collocation method, one chooses a sequence of points   , , = 1,  , Ä , and imposes the condition that the remainder (4) be zero at these points,

²

/

 =  ( , = 1,  , Ä ). (9) When solving a specific problem, the points   , at which the remainder ² / is set equal to zero, are regarded as most significant. The number of collocation points Ä is taken equal to the number = 0 at

of the terms of the series (3). This allows one to obtain a complete system of algebraic equations for the unknown coefficients #  (for linear boundary value problems, this algebraic system is linear). Note that the collocation method is a special case of the Galerkin method with the sequence (5) consisting of the Dirac delta functions:

j  = > ( −   ). In the collocation method, there is no need to calculate integrals, and this essentially simplifies the procedure of solving nonlinear problems (although usually this method yields less accurate results than other modifications of the Galerkin method). 0.3.5-5. The method of partitioning the domain.

˜

The domain = { 1 ≤  ≤  2 } is split into Ä subdomains: In this method, the weight functions are chosen as follows:

˜

 = {  1 ≤  ≤   2 }, , = 1,  , Ä .

˜

1 for ‚d  , ˜ 0 for  ∉  .

™

j  ( ) =

˜

The subdomains  are chosen according to the specific properties of the problem under consider˜ ation and can generally be arbitrary (the union of all subdomains  may differ from the domain ˜ , and some ˜ and ˜ + may overlap).  0.3.5-6. The least squared error method. Sometimes, in order to find the coefficients #  of the approximation function (3), one uses the least squared error method based on the minimization of the functional:

` =X 



2 1

²

/

2

M

þD

min .

(10)

For given functions Y  in (3), the integral ` is a quadratic polynomial with respect to the coefficients #  . In this case, the necessary conditions of minimum in (10) have the form:

N

N

`

# 

=0

(B = 1,  , Ä ).

is a system of linear algebraic equations for the coefficients #  . ^This _

References for Subsection 0.3.5: L. V. Kantorovich and V. I. Krylov (1962), M. A. Krasnoselskii, G. M. Vainikko, P. P. Zabreiko et al. (1969), S. G. Mikhlin (1970), B. A. Finlayson (1972), D. Zwillinger (1989).

© 2003 by Chapman & Hall/CRC

Page 48

0.3.6. Iteration and Numerical Methods 0.3.6-1. The method of successive approximations (Cauchy problem). The method of successive approximations is implemented in two steps. First, the Cauchy problem



 =  ( , , 
) 
( 0 ) = 0
( 0 ) = 0 ,

(equation), (boundary conditions)

(1) (2)

is reduced to an equivalent system of integral equations by the introduction of the new variable ‡ ( ) = 
 . These integral equations have the form

 ‡ ( ) = 0
+ X    a , (a ), ‡ (a )  a , M 0

( ) =

0

+X 



0

‡ (a ) a . M

(3)

Then the solution of system (3) is sought by means of successive approximations defined by the following recurrence formulas:


+1 (  ) = 0 + X 

‡ 



0

  a ,  (a ), ‡  (a )  a , M

As the initial approximation, one can take

^_

 +1 ( ) = 0(

 )=

0,

0+X 



0

‡ 0 ( ) = 
0 .

‡  (a ) a ; M

B = 0, 1, 2, 

References for Paragraph 0.3.6-1: G. A. Korn and T. M. Korn (1968), N. S. Bakhvalov (1973), E. Kamke (1977).

0.3.6-2. The Runge–Kutta method (Cauchy problem). For the numerical integration of the Cauchy problem (1)–(2), one often uses the Runge–Kutta method. Let ¢C be sufficiently small. We introduce the following notation:




 = (  ),  =  (  ),   =  (  ,  ,  ); The desired values  and 
 are successively found by the formulas:
2  +1 =  +  ¢C + 16 (  1 +  2 +  3 )( ¢C ) ,

1  +1 =  + 6 (  1 + 2  2 + 2  3 +  4 ) ¢C ,   =

0

+ , ¢C ,

, = 0, 1, 2, 

where

 1 =     ,  , 
 ,  2 =     + 12 ¢C ,  + 12 
¢C , 
+ 12  1 ¢C  ,  3 = l  + 12 ¢C ,  + 12 
¢C + 14  1 ( ¢C )2 , 
+ 12  2 ¢C  ,  4 =     + ¢C ,  + 
¢C + 12  2 ( ¢C )2 , 
+  3 ¢C  . In practice, the step ¢C is determined in the same way as for first-order equations (see Remark 2 in Paragraph 0.1.10-3).

^_

References for Paragraph 0.3.6-2: G. A. Korn and T. M. Korn (1968), N. S. Bakhvalov (1973), E. Kamke (1977), D. Zwillinger (1989).

0.3.6-3. Shooting method (boundary value problems). In order to solve the boundary value problem for equation (1) with the boundary conditions ( 1 ) =

1,

( 2 ) =

2,

(4)

one considers an auxiliary Cauchy problem for equation (1) with the initial conditions ( 1 ) =

1,


( 1 ) =  .

(5)

© 2003 by Chapman & Hall/CRC

Page 49

(The solution of this Cauchy problem can be obtained by the Runge–Kutta method or some other numerical method.) The parameter  is chosen so that the value of the solution = ( ,  ) at the point  =  2 coincides with the value required by the second boundary condition in (4): ( 2 ,  ) =

2.

In a similar way one constructs the solution of the boundary value problem with mixed boundary conditions 
( 2 ) + , ( 2 ) = 2 . ( 1 ) = 1 , (6) In this case, one also considers the auxiliary Cauchy problem (1), (5). The parameter  is chosen so that the solution = ( ,  ) satisfies the second boundary condition in (6) at the point  =  2 .

^_

References for Paragraph 0.3.6-3: S. K. Godunov and V. S. Ryaben’kii (1973), N. N. Kalitkin (1978).

0.3.6-4. Method of accelerated convergence in eigenvalue problems. Consider the Sturm–Liouville problem for the second-order nonhomogeneous linear equation [  ( ) 
]
 + [ Œ ( ) − % ( )] = 0

(7)

with linear homogeneous boundary conditions of the first kind (0) = (1) = 0.

It is assumed that the functions  ,  
, , % are continuous and  > 0, > 0.

(8)

First, using the Rayleigh–Ritz principle, one finds an upper estimate for the first eigenvalue Œ 01 [this value is determined by the right-hand side of relation (6) from Paragraph 0.2.5-3]. Then, one solves numerically the Cauchy problem for the auxiliary equation [  ( ) 
]
 + [ Œ 01 ( ) − % ( )] = 0

with the boundary conditions

(0) = 0,


(0) = 1.

(9) (10)

satisfies the condition ( 0 , Œ where  0 < 1. The criterion of closeness The function ( , Œ the form of the inequality |1 −  0 | ≤ > , of the exact and approximate solutions, Œ 1 and Œ where > is a sufficiently small given constant. If this inequality does not hold, one constructs a refinement for the approximate eigenvalue on the basis of the formula: 0 1)

0 1 ) = 0, 0 1 , has

1 1

Œ



=Œ

0 1

− 0  (1)

[
 (1)]2

2

,

0 = 1 −  0,

(11)

1

2 = X ( ) 2 ( )  . Then the value Œ 11 is substituted for Œ 01 in the Cauchy problem where M 0 (9)–(10). As a result, a new solution and a new point  1 are found; and one has to check whether the criterion |1 −  1 | ≤ > holds. If this inequality is violated, one refines the approximate eigenvalue by means of the formula:

Œ

2 1

=Œ

1 1

− 1  (1)

[
 (1)]2

2

,

1 = 1 −  1,

(12)

and repeat the above procedure. x3y'z|{}~­³´€ Formulas of the type (11) are obtained by a perturbation method based on a transformation of the independent variable  (see Paragraph 0.3.4-1). If   > 1, the functions  , , and % are smoothly extended to the interval (1, ˆ ], where ˆ ≥   .

x3y'z|{}~‚µ €

The algorithm described above has the property of accelerated convergence

2 −4 to 10−8 +1 š  , which ensures that the relative error of the approximate solution becomes 10 after two or three iterations for 0 š 0.1. This method is quite effective for high-precision calcula-



tions, is fail-safe, and guarantees against accumulation of roundoff errors.

© 2003 by Chapman & Hall/CRC

Page 50

x3y'z|{}~E¹€

In a similar way, one can compute subsequent eigenvalues Œ + , ¡ that end, a suitable initial approximation Œ 0+ should be chosen).

= 2, 3,  (to

x3y'z|{}~Eº€

A similar computation scheme can also be used in the case of boundary conditions of the second and the third kinds, periodic boundary conditions, etc. (see the references below). Example 1. The eigenvalue problem for the equation

ÏÒ˜Í Í Ò

+ (1 + Ë

•

2 −2

)

Ï

=0

•

•

•

5

= 16 2 − 1. with the boundary conditions (8) admits an exact analytic solution and has eigenvalues 1 = 15, 2 = 63, ÿªÿ›ÿ , = sin( Ë ) yields the approximate According the Rayleigh–Ritz principle, formula (6) of Paragraph 0.2.5-3 for value 01 = 15.33728. The solution of the Cauchy problem (9)–(10) with â (Ë ) = 1, ã (Ë ) = (1 + Ë 2 )−2 , ( Ë ) = 0 yields Ë 0 = 0.983848, 1 − Ë 0 = 0.016152, ÖÏ 2 = 0.024585, Ï ÒÍ (Ë 0 ) = −0.70622822. The first iteration for the first eigenvalue is determined by (11) and results in the value 11 = 14.99245 with the relative 1 = 5 × 10−4 . error 1 2 < 10−6 . The second iteration results in 21 = 14.999986 with the relative error 1

•

;

œ ^œ

Ž• % •

•

6

›

E

•

•

Ž• % •

Example 2. Consider the eigenvalue problem for the equation ( 1+Ë

ž

Ï ÒÍ ) ÍÒ

•

+ Ï = 0

with the boundary conditions (8). The Rayleigh–Ritz principle yields 01 = 11.995576. The next two iterations result in the values 2 = 11.898458. For the relative error we have 2 < 10−5 . 1 1

^• _

•

Ž• % •

•

1 1

= 11.898578 and

References for Paragraph 0.3.6-4: L. D. Akulenko and S. V. Nesterov (1996, 1997).

]

For more details about finite-difference methods and other numerical methods, see, for instance, the books by Lambert (1973), Keller (1976), and Zwillinger (1998).

0.4. Linear Equations of Arbitrary Order 0.4.1. Linear Equations with Constant Coefficients 0.4.1-1. Homogeneous linear equations. An B th-order homogeneous linear equation with constant coefficients has the general form

 +  ( )

−1



( −1)

+ ))) + 

1


+ 

0

= 0.

(1)

The general solution of this equation is determined by the roots of the characteristic equation:

.

( Π) = 0,

where

.

(Π) = Π +  

−1

Π

−1

+ ))) +  1 Π+  0 .

(2)

The following cases are possible: 1 b . All roots Π1 , Π2 ,  , Π of the characteristic equation (2) are real and distinct. Then the general solution of the homogeneous linear differential equation (1) has the form: =L

1

exp( Π1  ) + L

2

exp( Π2  ) + ))) + L  exp( Π  ).

equal real roots Œ 1 = Œ 2 = ))) = Œ + 2 b . There are ¡ distinct. In this case, the general solution is given by:

(

¡

≤ B ), and the other roots are real and

= exp( Œ 1  )( L 1 + L 2  + ))) + L +  + −1 ) + L + +1 exp( Œ + +1  ) + L + +2 exp( Œ + +2  ) + ))) + L  exp( Œ   ). equal complex conjugate roots Œ = ƒ\AC™Ýý (2 ≤ B ), and the other roots are real 3 b . There are ¡ ¡ and distinct. In this case, the general solution is:

 + ))) + # +  + −1 ) + exp( ƒ… ) sin(ý± )( $ 1 + $ 2  + ))) + $ +  + −1 ) + L 2 + +1 exp( Œ 2 + +1  ) + L 2 + +2 exp( Œ 2 + +2  ) + ))) + L  exp( Œ   ), where # 1 ,  , # + , $ 1 ,   , $ + , L 2 + +1 ,  , L  are arbitrary constants. = exp( ƒ… ) cos(ý± )( #

1

+#

2

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4 b . In the general case, where there are & different roots Œ 1 , Œ 2 ,  , ŒŸ of multiplicities , ,  , Ÿ , respectively, the right-hand side of the characteristic equation (2) can be rep¡ 1 ¡ 2 ¡ resented as the product

.

where

¡

1+

¡

2+

))) +

Ÿ ¡

=

(Œ ) = (Œ − Œ 1)

+ 1 ( Œ − Œ )+ 2

2

+  ,  ( Œ − ŒŸ ) 

= B . The general solution of the original equation is given by the formula:

Ÿ

½ 

=1

exp( Π  )( L 

,0

+ L  ,1  + ))) + L  , +‹¼

−1

 +‹¼

−1

),

where L  , are arbitrary constants. If the characteristic equation (2) has complex conjugate roots, then in the above solution, one should extract the real part on the basis of the relation exp( ƒeAI™Ýý ) =  9 (cos ýeAI™ sin ý ). 0.4.1-2. Nonhomogeneous linear equations. An B th-order nonhomogeneous linear equation with constant coefficients has the general form

(  ) +  

( 

−1

−1)

+ ))) + 

1


+ 

0

=  ( ).

(3)

The general solution of this equation is the sum of the general solution of the corresponding homogeneous equation (see Paragraph 0.4.1-1) and any particular solution of the nonhomogeneous equation. If all the roots Π1 , Π2 ,  , Π of the characteristic equation (2) are different, equation (3) has the general solution: =

½  Á

=1

  ½ LlÁ   ¡ + . 
 (¡ Œ"Á ) ù Á =1 

 ( )  −  ¡

 M



(for complex roots, the real part should be taken). Table 4 lists the forms of particular solutions corresponding to some special forms of functions on the right-hand side of the linear nonhomogeneous equation. In the general case, a particular solution can be constructed on the basis of the formulas from Paragraph 0.4.2-3.

^_

References for Subsection 0.4.1: G. A. Korn and T. M. Korn (1968), E. Kamke (1977), A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov (1980), D. Zwillinger (1989).

0.4.2. Linear Equations with Variable Coefficients 0.4.2-1. Homogeneous linear equations. Structure of the general solution. The general solution of the B th-order homogeneous linear differential equation

  ( ) (  ) +  

−1 (

 ) ( 

−1)

+ ))) +  1 ( ) 
+  0 ( ) = 0

(1)

has the form: =L

1 1(

 )+L

2 2(

 ) + ))) + L   ( ).

(2)

Here, 1 ( ), 2 ( ),  ,  ( ) is a fundamental system of solutions (the  are linearly independent particular solutions,  1 0); L 1 , L 2 ,  , L  are arbitrary constants.

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TABLE 4 Forms of particular solutions of the constant coefficient nonhomogeneous linear equation (  ) +   −1 (  −1) + ))) +  1 
 +  0 =  ( ) that correspond to some special forms of the function  ( ) Form of the function  ( )

. .

.¶

Zero is not a root of the characteristic equation (i.e.,  0 ≠ 0)

+ ( )

+ ( )  9

Form of a particular ¶ solution = ( )

Roots of the characteristic equation Œ  +   −1 Œ  −1 + ))) +  1 Œ +  0 = 0

Zero is a root of the characteristic equation (multiplicity & )

.¶

characteristic equation

( ƒ is a real constant)

ƒ is a root of the



characteristic equation (multiplicity & )

.¶

™Ýý is not a root of the

.

+ ( )

+ ( )  9

Ÿ .¶

+ ( )  9

 

Á¶ ( ) cos ý± + ÆIÁ ( ) sin ý±

characteristic equation

+ ( ) cos ý± + Æ  ( ) sin ý±

Ÿ .¶ 

ƒ is not a root of the



+ ( )

.¶  Ÿ [¶ Á ( ) cos ý± characteristic equation (multiplicity & ) + ÆIÁ ( ) sin ý± ] .¶ ƒ + ™Ýý is not a root of the ¶ [ Á ( ) cos ý±  . characteristic equation + ÆIÁ ( ) sin ý± ] 9 [ + ( ) cos ý±  .¶ 9 + Æ  ( ) sin ý± ] ƒ + ™Ýý is a root of the  ¶ Ÿ [ Á ( ) cos ý±  characteristic equation (multiplicity & ) + ÆIÁ ( ) sin ý± ] 9 . .¶ .¶ Notation: + and Æ  are polynomials of degrees and B with given coefficients; + , Á , ¶ ¡ and ÆIÁ are polynomials of degrees and @ whose coefficients are determined by substituting ¡ the particular solution into the basic equation; @ = max( , B ); and ƒ and ý are real numbers, ¡ ™ 2 = −1. ™Ýý is a root of the

0.4.2-2. Utilization of particular solutions for reducing the order of the original equation. 1 b . Let

1

=

1(

 ) be a nontrivial particular solution of equation (1). The substitution

 ) XZH ( )  M results in a linear equation of order B − 1 for the function H ( ). 2 b . Let 1 = 1 ( ) and 2 = 2 ( ) be two nontrivial linearly independent solutions of equation (1). =

The substitution

=

1

X

1(

2

¨

 −

M

results in a linear equation of order B − 2 for ¨ ( ).

2

X

1

¨ M



linearly independent solutions 1 ( ), 2 ( ),  , + ( ) of equation (1) are 3 b . Suppose that ¡ known. Then one can reduce the order of the equation to B − by successive application of the ¡ following procedure. The substitution = + ( ) X H ( )  leads to an equation of order B − 1 for M the function H ( ) with known linearly independent solutions:

H 1= O

1




, + P 

H 2= O

2




, + P 

 ,

H +

−1

= O

+ +

−1




. P 

© 2003 by Chapman & Hall/CRC

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The substitution H = H + −1 ( ) X ¨ ( )  yields an equation of order B − 2. Repeating this procedure M times, we arrive at a homogeneous linear equation of order B − .

¡

¡

0.4.2-3. Wronskian determinant and Liouville formula. The Wronskian determinant (or simply, Wronskian) is the function defined as: 1 ( ) û 
( ) 1 ( ) = ûû ))) û (  −1) û û 1 ( ) û

ú

))) ))) ))) )))

 ( ) 
( ) û  û, ))) ûû (  −1) ( ) ûû  û

(3)

1 (  ),  ,  (  ) is a fundamental system of solutions of the homogeneous equation (1); +3 + ) ( ) = M  , = 1,  , B − 1; , = 1,  , B .  ¡  + M The following Liouville formula holds:

where (

ú

( ) =

ú

( 0 ) exp ¬ − X 

   −1 (a ) aÝ® .  0  (a ) M

0.4.2-4. Nonhomogeneous linear equations. Construction of the general solution. 1 b . The general nonhomogeneous B th-order linear differential equation has the form

  ( ) (  ) +  



( −1)

−1

+ ))) +  1 ( ) 
+  0 ( ) = ( ).

(4)

The general solution of the nonhomogeneous equation (4) can be represented as the sum of its particular solution and the general solution of the corresponding homogeneous equation (1). 2 b . Let 1 ( ),  ,  ( ) be a fundamental system of solutions of the homogeneous equation (1), ú and let ( ) be the Wronskian determinant (3). Then the general solution of the nonhomogeneous linear equation (4) can be represented as: =

ú

½ Á

LlÁ Á ( ) + =1

½  Á

Á ( ) X =1

ú

Á ( )  ú M ,   ( ) ( )

where Á ( ) is the determinant obtained by replacing the @ th column of the matrix (3) by the column vector with the elements 0, 0,  , 0, .

^_

References for Subsection 0.4.2: G. A. Korn and T. M. Korn (1968), E. Kamke (1977), A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov (1980), D. Zwillinger (1989).

0.4.3. Asymptotic Solutions of Linear Equations This subsection presents asymptotic solutions, as ÑD 0 ( > 0), of some higher-order linear ordinary differential equations containing arbitrary functions (sufficiently smooth), with the independent variable being real. 0.4.3-1. Fourth-order linear equations. 1 b . Consider the equation

4





 −  ( ) = 0

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on a closed interval  ≤  ≤  . With the condition  > 0, the leading terms of the asymptotic expansions of the fundamental system of solutions, as pD 0, are given by the formulas: 1 1 −3 J 8 exp ™ − ù [  ( )]1 J 4 ˆ¢ , 2 = [  ( )]−3 J 8 exp ™ ù [  ( )]1 J 4 ˆ¢ , 1 = [  (  )]

3

= [  ( )]−3 J 8 cos ™

M

1

[  ( )]1 J 4 ˆ¢ ,

ù M

4

= [  ( )]−3 J 8 sin ™

M

1

[  ( )]1 J 4 ˆ¢ .

ù M

2 b . Now consider the “biquadratic” equation:

4 



 − 2 2 ( ) 

 −  ( ) = 0. (1) Introduce the notation: c ( ) = [ ( )]2 +  ( ). In the range where the conditions  ( ) ≠ 0 and c ( ) ≠ 0 are satisfied, the leading terms of the asymptotic expansions of the fundamental system of solutions of equation (1) are described by the formulas: 1 1 [ Œ ( )]
 −1 J 2 −1 J 4 Œ  ( )  − ù £  ˆ¢ ; , = 1, 2, 3, 4, exp ™ ù  = [ Œ  ( )] [ c ( )]

M 2 c ( ) M where Œ 1 ( ) = ( ) + £ c ( ), Œ 2 ( ) = − ( ) + £ c ( ),

Œ 3 ( ) = ü

ü

ü Œ 4 ( ) = − ( ) − £ c ( ). ü

( ) − £ c ( ),

0.4.3-2. Higher-order linear equations. 1 b . Consider an equation of the form

 (  ) −  ( ) = 0 on a closed interval  ≤  ≤  . Assume that  ≠ 0. Then the leading terms of the asymptotic expansions of the fundamental system of solutions, as pD 0, are given by: 1   + −1 + 1 V  ( )W  ˆ¢ V 1 + [ ( )W , + = V  ( )W 2 2  exp ™ ù

M  where   1 ,   2 ,  ,    are roots of the equation   = 1: 2

2

  + = cos O /¡ + ™ sin O /¡ , B P B P

¡

= 1, 2,  , B .

2 b . Now consider an equation of the form

 (  ) +  −1   −1 ( ) (  −1) + ))) + ´ 1 ( ) 
+  0 ( ) = 0 (2) on a closed interval  ≤  ≤  . Let Œ + = Œ + ( ) ( = 1, 2,  , B ) be the roots of the characteristic ¡ equation: . ( , Œ ) ≡ Œ  +  ( ) Œ  −1 + ))) +  ( ) Œ +  ( ) = 0.  −1 1 0 Let all the roots of the characteristic equation be different on the interval  ≤  ≤  , i.e., the ≠ , , are satisfied, which is equivalent to the fulfillment of the conditions Œ + ( ) ≠ Œ  ( ), . ( , Œ + ) ≠ 0. Then ¡ the leading terms of the asymptotic expansions of the fundamental conditions  system of solutions of equation (2), as ID 0, are given by: .  , Œ ( )  1 1 +
 ˆ¢ , + = exp ™ ù Œ + ( )  − ù [ Œ + ( )] . 

M 2 , Π+ ( )  M     where .

.

 −1 + (B − 1)   −1 ( ) Œ  −2 + ))) + 2 Œ" 2 ( ) +  1 ( ),  ( , Œ ) ≡ N Œ = B…Œ . ( , Œ ) ≡ N N 2 . = B (B − 1) Œ  −2 + (B − 1)(B − 2)  ( ) Œ  −3 + ))) + 6 Œ" ( ) + 2  ( ). 3 2  −1  Œ 2 ^_ N References for Subsection 0.4.3: W. Wasov (1965), M. V. Fedoryuk (1993).

© 2003 by Chapman & Hall/CRC

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0.5. Nonlinear Equations of Arbitrary Order 0.5.1. Structure of the General Solution. Cauchy Problem 0.5.1-1. Equations solved for the highest derivative. General solution. An B th-order differential equation solved for the highest derivative has the form

  =  ( , , 
,  , (  ( )

−1)

).

(1)

The general solution of this equation depends on B arbitrary constants L 1 ,  , L  . In some cases, the general solution can be written in explicit form as = Y ( , L 1 ,  , L  ). 0.5.1-2. The Cauchy problem. The existence and uniqueness theorem. 1 b . The Cauchy problem: find a solution of equation (1) with the initial conditions ( 0 ) =

0,


( 0 ) =

(1) 0 ,

 ,

( 

−1)

( 0 ) =



( −1) . 0

(2)

(At a point  0 , the values of the unknown function ( ) and all its derivatives of orders ≤ B − 1 are prescribed.) 2 b . The existence and uniqueness theorem. Suppose the function  ( , , H 1,  , H  −1 ) is continuous in all its arguments in a neighborhood of the point ( 0 , 0 , 0(1) ,  , 0(  −1) ) and has bounded derivatives with respect to , H 1 ,  , H  −1 in this neighborhood. Then a solution of equation (1) satisfying the initial conditions (2) exists and is unique.

^_

References for Subsection 0.5.1: G. A. Korn and T. M. Korn (1968), I. G. Petrovskii (1970), E. Kamke (1977), A. N. Tikhonov, A. B. Vasil’eva, and A. G. Sveshnikov (1980).

0.5.2. Equations Admitting Reduction of Order 0.5.2-1. Equations not containing , 
 ,  , (  ) explicitly. An equation that does not explicitly contain the unknown function and its derivatives up to order , inclusive can generally be written as

   , ( 

+1)

 ) = 0 ,  , (

(1 ≤ , + 1 < B ).

(1)

+const (the Such equations are invariant under arbitrary translations of the unknown function, D form of such equations is also preserved under the transformation ‡ ( ) = +     + ))) +  1  +  0 , where the  + are arbitrary constants). The substitution H ( ) = (  +1) reduces (1) to an equation whose order is by , + 1 smaller than that of the original equation, Ê , H , H 
,  , H (  −  −1)  = 0. 0.5.2-2. Equations not containing  explicitly (autonomous equations). An equation that does not explicitly contain  has in the general form

 )  = 0.   , 
,  , (

(2)

Such equations are invariant under arbitrary translations of the independent variable, CD  + const . The substitution 
= ¨ ( ) (where plays the role of the independent variable) reduces by one the order of an autonomous equation. Higher derivatives can be expressed in terms of ¨ and its derivatives with respect to the new independent variable, "

 = ¨*¨2
, 


 = ¨ 2 ¨2

+ ¨ (¨2
)2 , 







© 2003 by Chapman & Hall/CRC

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( )  0.5.2-3. Equations of the form    +  , 
 ,  ,  = 0.

Such equations are invariant under simultaneous translations of the independent variable and the − !( , where ( is an arbitrary constant. unknown function, ED„ + ˜( and D For  = 0, see equation (1). For  ≠ 0, the substitution ¨ ( ) = +(  ) leads to an autonomous equation of the form (2).

 )  = 0 and its generalizations. 0.5.2-4. Equations of the form  ,  
 − , 

 ,  , (

The substitution ¨ ( ) =  
− reduces the order of this equation by one. This equation is a special case of the equation

   ,  
− , ( + ¡ The substitution ¨ ( ) =  
− ¡

+1)

,  , (  )  = 0,

where

¡

= 1, 2,  , B − 1.

(3)

reduces by one the order of equation (3).

0.5.2-5. Homogeneous equations. 1 b . Equations homogeneous in the independent variable are invariant under scaling of the independent variable, |D†ƒ… , where ƒ is an arbitrary constant (ƒ ≠ 0). In general, such equations can be written in the form  )  = 0.   ,  
,  2 

 ,  ,   ( The substitution H ( ) = 


reduces by one the order of this equation.

2 b . Equations homogeneous in the unknown function are invariant under scaling of the unknown function, D ƒ , where ƒ is an arbitrary constant ( ƒ ≠ 0). Such equations can be written in the general form   , 
 , 

  ,  , (  )   = 0.

The substitution H ( ) = 
 reduces by one the order of this equation.

3 b . Equations homogeneous in both variables are invariant under simultaneous scaling (dilatation) of the independent and dependent variables, CD ƒ… and D ƒ , where ƒ is an arbitrary constant ( ƒ ≠ 0). Such equations can be written in the general form

 )  = 0.     , 
,  

 ,  ,   −1 ( The transformation a = ln | |, ¨ =   leads to an autonomous equation considered in Paragraph 0.5.2-2.

0.5.2-6. Generalized homogeneous equations. 1 b . Generalized homogeneous equations (equations homogeneous in the generalized sense) are invariant under simultaneous scaling of the independent variable and the unknown function, uD ƒ… and D ƒ  , where ƒ ≠ 0 is an arbitrary constant and , is a given number. Such equations can be written in the general form ( )     −  ,  1−  
,  ,   −   = 0. ¨ =  − leads to an autonomous equation considered in ParaThe transformation a = ln  ,

graph 0.5.2-2.

2 b . The most general form of generalized homogeneous equations is

¸    + ,  
 ,  ,   (  )   = 0. The transformation H =   + , ‡ =  
  reduces the order of this equation by one. © 2003 by Chapman & Hall/CRC

Page 57

  
 



,  ,  ,  , (  )   = 0.

0.5.2-7. Equations of the form  

Such equations are invariant under simultaneous translation and scaling of variables, eD  + ƒ and  DGý , where ý = exp(−ƒsŒ B ), and ƒ is an arbitrary constant. The transformation H = !  ,
¨ =   leads to an equation of order B − 1. 0.5.2-8. Equations of the form       ,  
 ,  2 

 ,  ,  

   = 0.  ( )

Such equations are invariant under simultaneous scaling and translation of variables,  D ƒ… and + ý , where ƒ = exp(−ýhŒ B ), and ý is an arbitrary constant. The transformation H =    , D ¨ =  
 leads to an equation of order B − 1. 0.5.2-9. Other equations. Consider the nonlinear differential equation

   , L1 [ ],  , L  [ ]  = 0, À

(4)

forms, where the L [ ] are linear homogeneous differential À

À

L [ ]= Let

0

=

0(

½  £

+

=0

Y (+ ) ( ) ( + ) ,

° = 1,  , , .

 ) be a common particular À solution of the linear equations: ( ° = 1,  , , ).

L [ 0] = 0 Then the substitution

¨ = Y ( ) V ( ) 
−
( ) W 0 0

(5)

with an arbitrary function Y ( ) reduces by one the order of equation (4). Example 1. Consider the third-order equation

Ë ÏÒ˜Í Í Ò˜Í Ò

= â (Ë

ÏÒÍ

− 2Ï ).

It can be represented in the form (4) with

ô

= 2,

ê (Ë

¤ 

, , )=

The linear equations L [ Ï ] = 0 are

õ

Ë



¤

− â ( ),

ÏÒ˜Í Í Ò˜Í Ò

= 0,

Ë ÏÒÍ

ÏÒ˜Í Í Ò˜Í Ò

L2 [ Ï ] =

,

Ë ÏÒÍ

− 2Ï = 0.

− 2Ï .

2 . Therefore, the substitution = Ë Ï ÒÍ − 2 Ï These equations have a common particular solution Ë Ë ˜ Ò Ò with ( ) = 1 ) leads to an autonomous second-order equation of the form 2.9.1.1: Í Í = â ( ).

ä

%

5

Example 2. The th-order equation

Ï (Ò E

can be represented in the form (4) with

ô

= 2,

ê (Ë

The linear equations have a common particular solution, leads to an ( − 1)st-order equation.

^_

5

¤ 

, , )=

¤

L1 [ Ï ] ≡

Ï0=ÕÒ

Ï 0=Ë

L1 [ Ï ] =

)

= â (Ë , Ï

− â ( Ë , ),



Ï (Ò E

)

− Ï = 0,





Ò˜Í Í Ò



(see formula (5)

−Ï )+Ï

L1 [ Ï ] =

Ï (Ò E

L2 [ Ï ] ≡

Ï Ò˜Í Í Ò

. Therefore, the substitution

−Ï ,

)



L2 [ Ï ] =

−Ï =0 =

Ï ÒÍ

−

Ï

ÏÒ˜Í Í Ò

−Ï .

(see formula (5) with

ä

(Ë ) =

Õ −Ò

)

References for Subsection 0.5.2: G. M. Murphy (1960), G. A. Korn and T. M. Korn (1968), E. Kamke (1977), V. F. Zaitsev and A. D. Polyanin (1993, 2001), A. D. Polyanin and V. F. Zaitsev (1995), D. Zwillinger (1998).

© 2003 by Chapman & Hall/CRC

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0.5.3. A Method for Construction of Solvable Equations of General Form 0.5.3-1. Description of the method. Consider a function

=  ( , L

L 2 ,  , L 

1,

+1 )

(1)

depending on B + 1 free parameters L  . Differentiating relation (1) B times, we obtain the following sequence of equations:

  =   (  ) ( , L 1 , L 2 ,  , L  ( )

, = 1, 2,  , B .

+1 ),

(2)

Treating relations (1), (2) as an algebraic (transcendental) system of equations for the parameters L 1 , L 2 ,  , L  +1 and solving this system, we obtain

L  = Y    , , 
,  , (  )  ,

, = 1, 2,  , B + 1.

(3)

Consider a general B th-order equation of the form

 ( Y 1 , Y 2 ,  , Y 

+1 )

= 0,

(4)

 )  are the functions where  is an arbitrary function of (B +1) variables and Y  = Y    , , 
 ,  , ( from (3). Equation (4) is satisfied by the function (1), where the (B + 1) arbitrary parameters L 1 , L 2 ,  , L  +1 are related by a single constraint:  ( L 1 , L 2 ,  , L 

+1 )

= 0.

x3y'z|{}~­³´€

Equation (4) may also have singular solutions depending on a smaller number of arbitrary constants. In order to examine these solutions, one should differentiate equation (4); see Example 1.

x3y'z|{}~‚µ €

Instead of (4), one can consider a more general equation

 (j 1 , j 2 ,  , j 

+1 )

= 0,

j  = j  ( Y 1 , Y 2 ,  , Y 

where

+1 ).

x3y'z|{}~E¹€ The original expression (1) can be specified in an implicit form. x3y'z|{}~Eº€ The original expression (1) can be written as an th-order differential equation ¡ ( < B ) with B − + 1 free parameters L  . The solution of the B th-order differential equation ¡obtained in this way¡ can be expressed in terms of the solution of an th-order differential equation ¡ (see Example 4). 0.5.3-2. Examples. Example 1. Consider the function

Ï

By differentiation we obtain Let us solve equations (5)–(6) for the parameters



ÏÒÍ



1

1

=



=−

and

Ò Õ Ï ÒÍ



Õ

1

=

−



Ò

1



+

Õ

−

Ò

2.

(5)

.

(6)

2 . We have

,



2

=

ÏÒÍ

+Ï .

Using the above method, we construct an equation in accordance with (4):

Ò ê Õ Ï ÒÍ , Ï ÒÍ ë

This equation admits a solution of the form (5) with constants



+Ï 1

ì

= 0.

and



2

(7) related by the constraint

ê

(

 1, 

2)

= 0.

© 2003 by Chapman & Hall/CRC

Page 59

Ë

, we get

Singular solution. Differentiating equation (7) with respect to

¤

Ò

Í Í Ò + ÏÒÍ )( Õ ( ÏҘ

¨

ꦥ

¦§ ) = 0,

+ê

(8)

¤ ¨

where the subscripts and indicate the respective partial derivatives of the function ê = ê ( , ). Equating the first factor in (8) to zero, we obtain solution (5). Equating the second factor to zero, we obtain an expression which, combined with equation (7), yields a singular solution in parametric form:

One should eliminate

:

=

Ï ÒÍ

Ò Õ ¦ê ¥

¤ ¨

ê

( , ) = 0,

¦§

+ê

= 0,

¤

where

Ò

Õ

=

:, ¨ =:

+Ï .

from these expressions.

Example 2. Consider the function

Ï

Differentiating this function twice, we get

Solving (9)–(10) for the parameters



1

=

 1 2

õ

=



1

ÏÒÍ Ï Ò˜Í Í Ò

Ë

+





1

2

=2 =2



2

Ë

Ë

+



+



2,

3.

(9) (10)

1.

, we find that

ÏÒ˜Í Í Ò



,

2

=

ÏÒÍ

−Ë

ÏÒ˜Í Í Ò



,

3

−Ë

Ï

=

ÏÒÍ

+

1 2

Ë 2 Ï Ò˜Í Í Ò

.

These relations lead to a second-order equation of general form:

ê

1 2

ë

−Ë

ÏÒ˜Í Í Ò , Ï ÒÍ

−Ë

ÏÒ˜Í Í Ò , Ï

ÏÒÍ

+

1 2

Ë 2 Ï Ò˜Í Í Òì

= 0,

 1 ,  2 , and  3 related by the constraint ê (  1 ,  2 ,  Example 3. In Example 2, one can choose the functions  of the form (see Remark 2) õ  1 = 2 ä 1 ,  2 = − ä 2 ,  3 = 4 ä 1 ä 3 − ä 22 ,

which has a solution of the type (9) with the three constants

where

ä

1

=

1 2

Ï Ò˜Í Í Ò

,

ä

2

=

Ï ÒÍ

−Ë

Ï Ò˜Í Í Ò

,

ä

3

©

=

ë

Ï

−Ë

Ï ÒÍ

+

Ï Ò˜Í Í Ò , Ë Ï Ò˜Í Í Ò

Its solution is given by (9) with three constants

 1, 

1 2

Ë 2 Ï Ò˜Í Í Ò ÒÍ

−Ï

2 , and

Example 4. Consider the autonomous equation

Ï Ò˜Í Í Ò

=





1

= 0.

. As a result, we obtain the differential equation:

, 2ÏÏ 3

3)

Ò˜Í Í Ò

Ò Í )2 ì

− (Ï

= 0.

related by a single constraint

Ï −×

+



© (2 



 

1, − 2, 4 1

3−



2 3)

2.

= 0.

(11)

Its solution can be represented in implicit form (see 2.4.2.1 and 2.9.1.1). Differentiating (11), we obtain

D 1 Ï −× −1 ÏÒÍ . Let us solve equations (11)–(12) for the parameters  1 and  2 :  1 = −Ï × +1 Ï ÓÒ˜Í ÏÍ Ò˜Í ÒÍ Ò ,  2 = Ï Ò˜Í Í Ò + Ï Taking  1 = − Ó 1 and  2 = Ó 2 (see Remark 2), we obtain the equation: ä ä ÏÒ˜Í Í Ò˜Í Ò

êCñÏ ×

+1

= −Ó

Ï ˜ÒÍ Í Ò˜Í Ò , Ï Ï Ò˜Í Í Ò˜Í Ò Ï ÒÍ Ï ÒÍ

+ ӁÏ

Ò˜Í Í Ò ò

(12)

Ï ˜ÒÍ Í Ò˜Í Ò ÓÏ ÒÍ

.

= 0.

This equation is satisfied by the solutions of a second-order autonomous equation of the form (11), where the constants and 2 are related by the constraint ê (− Ó 1 , Ó 2 ) = 0.



D D



1

0.6. Lie Group and Discrete-Group Methods 0.6.1. Lie Group Method. Point Transformations 0.6.1-1. Local one-parameter Lie group of transformations. Invariance condition. Here, we examine transformations of the ordinary differential equation

  =    , ,  , (  ( )

−1)

.

(1)

© 2003 by Chapman & Hall/CRC

Page 60

Consider the set of transformations Tª =

 ¯ |ª

 ¯ = Y ( , , ), ¯ = j ( , , ),

™

=0

¯|ª

=0

= , = ,

(2)

where Y , j are smooth functions of their arguments and is a real parameter. The set T ª is called a continuous one-parameter Lie group of point transformations if, for any 1 and 2 , the following relation holds: T ª 1 « Tª 1 = Tª 1 +ª 2 , i.e., consecutive application of two transformations of the form (1) with parameters 1 and 2 is equivalent to a single transformation of the same form with parameter

1 + 2. In what follows, we consider local continuous one-parameter Lie groups of point transformations (briefly called point groups) corresponding to an infinitesimal transformation (2) for ·D 0. Taylor’s expansion of  ¯ and ¯ in (2) with respect to the parameter about = 0 yields:

­ ¯ ¬v + ˆ ( , ) ,

¯

¬

+ ‰ (  , ) ,

(3)

where

Y ( , , ) j ( , , ) , ‰ ( , ) = N û û ª =0 . ª

û =0 û N N û to the curve described û by the transformed points At each point ( , ), the vector (ˆ , ‰ ) is tangent ( ¯ , ¯ ). ˆ ( , ) = N

The first-order linear differential operator

X = ˆ ( , ) N

+ ‰ ( , ) N



N

(4)

N

corresponding to the infinitesimal transformation (3), is called the infinitesimal operator (or infinitesimal generator) of the group. By definition, the universal invariant (briefly, invariant) of the group (2) and the operator (4) is a function ® 0 ( , ), satisfying the condition ® 0 ( ¯ , ¯ ) = ® 0 ( , ). Taylor’s expansion with respect to the small parameter yields the following linear partial differential equation for ® 0 : X®

0

= ˆ ( , ) N

N

®0

+ ‰ ( , ) N



®0

N

= 0.

Equation (1) will be treated as a relation for B +2 variables  , , 
,  , (  constraints



( +1)

= M



( )

M



(5) )

with the differential

.

(6)

The space of these B + 2 variables is called the space of B th prolongation; and in order to work with differential equations, one has to define the action of operator (4) on the “new” variables 
 ,  , (  ) , taking into account the differential constraints (6). For example, let us calculate the infinitesimal transformation of the first derivative. We have

M

M

¯ c  ( + ‰ ) =  ¯ c  ( + ˆ )

¬


 + (‰  + ‰ 
 )  1 + ( ˆ  + ˆ 
) 

( c  =  + 
 + ))) is the operator of total derivative). Expanding the right-hand side into a power N N  series with respect to the parameter and preserving  the first-order terms, we obtain where




¯ ¯

 1

= ‰  + (‰

¬


+ 1 (  , , 
) ,

 
 
2 
  − ˆ ) − ˆ  ( ) = c (‰ ) − c (ˆ ).

The action of the group on higher-order  derivatives  is determined by the recurrence formula:



+1

= c  (  ) − ( 

+1)

c  (ˆ ).

© 2003 by Chapman & Hall/CRC

Page 61

To a prolonged group there corresponds a prolonged operator: X = ˆ ( , ) N



+ ‰ ( , ) N



N

+

N



½ 


( )     , , ,  , È

=1

N

N ( ) . 

(7)

The ordinary differential equation (1) admits the group (2) if X V (  ) −    , , 
,  , ( 



−1)

W

= 0. û  (œ n ) = Ž û

(8)

Relation (8) is called the invariance condition. x3y'z|{}~/€ The invariant ® 0 , which is a solution of equation (5), also satisfies the equation X ® 0 = 0.



0.6.1-2. Group analysis of second-order equations. Structure of an admissible operator. For second-order nonlinear equations



 =  ( , , 
),

(9)

the invariance condition (8) is written in the form

‰  + (2‰   − ˆ   ) 
+ (‰   − 2ˆ   )( 
)2 − ˆ  ( 
)3


= (2ˆ  − ‰ + 3ˆ  )  + ˆž  + ‰ + [‰  + (‰ − ˆ  )  − ˆ (  )2 ]  ,      œ
where  =  ( , ,  ). This condition is in fact a second-order partial differential equation for two unknown functions ˆ ( , ) and ‰ ( , ). Since the unknown functions do not depend on the derivative 
, this equation can be represented (after  has been expanded in a power series with respect to 
, unless it is already a polynomial) in the form ½ 

÷ =0

`  ( 
)  = 0,

(10)

with the `  independent of 
. In order to ensure that condition (10) holds identically, one should set `  = 0, , = 0, 1,  Thus, the invariance condition for a second-order equation can be “split” and represented as a system of equations (whose number can generally be infinite).

Example 1. If ê = ê ( Ë , Ï ), i.e., the right-hand side of equation (9) does not depend on equation can be “split” and represented as the system:

~2N = 0, N − 2~ Ò  = 0, 2  Ò  − ~ Ò˜Ò − 3 ê ( Ë , Ï ) ~2 = 0,  Ò˜Ò + ( − 2~ Ò )ê (Ë , Ï ) − ê Ò (Ë

~ ¦ (Ë

Ï ÒÍ

, then the determining



,Ï ) − ê

, Ï ) = 0.

From the first two equations we find that where Ó ( Ë ), Ô ( Ë ), equations, we get

¯ (Ë

~

), and 3 ÓÍ Í

Ó ÍÍÍÏ

Ï

æ 2

= Ó ( Ë ) Ï + Ô ( Ë ),



=

ÓÍ (Ë )Ï

+ ( Ë ) Ï + ( Ë ),

¯

2

æ

(Ë ) are arbitrary functions. Substituting these expressions into the third and the fourth + 2 ÖÍ − ÔÝÍ Í − 3 ê ( Ë , Ï ) Ó = 0,

¯

+

¯ ÍÍÏ

+

æ

ÍÍ

¯

+ ( − 2Ô Í )ê

− ( ÓÏ + Ô ) ê

Ò

− (Ó

ÍÏ

2

¯

¦

+ ÖÏ + ) ê

æ

= 0.

(11)

In what follows, it is assumed that the function ê ( Ë , Ï ) is nonlinear with respect to the second argument. Then from the is an arbitrary constant. The second equation in (11) first equation in (11), we find that Ó = 0 and = 12 Ô Í + , where becomes 1 ÔÝÍ Í Í Ï + Í Í + − 32 ÔÝÍ ì ê − ÔÖê Ò − 12 ÔÝÍ + ì Ï + î ê = 0. (12) 2

æ

ë

°

¯

°

Equation (12) allows us to solve two different problems.

°

íë

°

æ

¦

© 2003 by Chapman & Hall/CRC

Page 62

1 . If the function ê ( Ë , Ï ) is given, then, splitting equation (12) with respect to powers of Ï (the unknown functions Ô and are independent of Ï ), we obtain a new system, from which Ô , , and can be found; i.e., we ultimately obtain an æ æ admissible operator.

O

°

°

O

2 . Assuming that the functions Ô , and the constant are known but arbitrary, one can regard relation (12) as an equation æ this equation, we obtain a class of equations admitting a point operator. for the unknown function ê ( Ë , Ï ). Solving

1 2

 Ë E ÏD± , i.e., we are dealing with the Emden–Fowler equation. Then equation (12) becomes + Í Í + ° − 32 ÔÝÍ ì  Ë Ï ± − Ô²5³ Ë −1 Ï ± − 12 ÔÝÍ + ° ì Ï + îµ´  Ë Ï ± −1 = 0. E E E æ ë íë æ

ê (Ë , Ï

Example 2. Let

ÔÝÍ Í Í Ï

)=

This relation must be satisfied identically by any function Ï = Ï ( Ë ), and therefore, the coefficients of different powers of must be equal to zero. As a result, we obtain a new system whose structure essentially depends on the value of .

ê (Ë , Ï

O

1 . It was assumed above that Then the system has the form:

´

) is nonlinear in its second argument, and therefore,

≠ 0 and

´

´

≠ 1. Let

´

Ï

≠ 2.

Í Í = 0, Ô Í Í Í = 0, æ æ

It follows that

æ

= 0 and Ô ( Ë ) = (

Ô 2Ë

´

+

+Ô

2

5

1

+ 3) Ô

Ë

= 0,

° í

)−

1 (3 2

´

−

îË

) ÔÝÍ

5

− !Ô = 0.

+ Ô 0 , and the last equation of the system can be written in the form

Ë

2

´

(1 −

2

+

1 ( 2

í

´

5

+ 2 + 3) Ô

1

° ´

Ë

− 1)î

+ (

5

+ !Ô

0

= 0.

(13)

To ensure relation (13), we equate all coefficients of this quadratic trinomial to zero to obtain (

´

+

5

+ 3) Ô

2

1 ( 2

= 0,

´

€

5

+ 2 + 3) Ô

€

° ´

+ (

1

5!Ô 0 = 0.

− 1) = 0,

(14)

Analysis of system (14) yields solutions of the determining system corresponding to three different operators:

€ X1 = ( ´ X2 =

O

2 . Let

´

X3 =

Ë

€ Ò

2

− 1) Ë

Ò

Ò

+Ë

5



− ( + 2) Ï

€

if if



Ï

5 5

if

´

°

ìË

and = 0, +

´

are arbitrary,

5

+ 3 = 0.

= 2. Then equation (12) becomes

æ Equating the term

æ

æ ÍÍ

Ô

(Ë ) =

æ

ÍÍ

+

ë

1 2

Ô ÍÍÍ

−2

Ï

and the coefficient of 1

Ë

+

æ

 æ

Ë ìÏ E

−

5 2

íë

ÔÍ

+

in parentheses to zero, we get

0,

4 Ó 0 Ë +3 4 Ó 1 Ë +4 + +Ô (Ë ) = æ æ ( + 2)( + 3)( + 4) ( + 1)( + 2)( + 3)

5

5 î  Ë E −1 Ï 2 = 0.

+ !Ô

5 E 5

5 E 5

5

2

Ë

2

+Ô

1

Ë

+ Ô 0,

5

The expression in square brackets (the coefficient of Ï 2 ) can be split with respect to powers of system which, to within nonzero coefficients, has the form: (7 (7 ( (2

5

5!Ô

5

5

5

≠ −1, −2, −3, −4.

Ë

and we obtain an algebraic

+ 20) 1 = 0, æ + 15) 0 = 0, æ + 5) Ô 2 = 0, + 5) Ô 1 + 2 = 0, 0 = 0.

°

The last three equations coincide with the corresponding equations of system (14), whose solutions are already known. The first two equations yield two cases of prolongation of the admissible group:

€  X1 = 343  8 7 € Ò  X2 = 343  Ë 6 7 Ò Ë

€  Ë + 4 49  1 7 Ï − 3 ì €  ë  + 3 49  Ë −1 7 Ï + 4 ì  ë Ë

if if

5

5

= − 20 , 7 = − 15 . 7

Page 63

0.6.1-3. Utilization of local groups for reducing the order of equations and their integration. Suppose that an ordinary differential equation (1) admits an infinitesimal operator X of the form (4). Then the order of the equation can be reduced by one. Below we describe two methods for reducing the order of an equation. 1 b . The first method. The transformation

a =  ( , ),

‡ = ( , ),

(15)

with  and ( *1 0) being arbitrary particular solutions of the first-order linear partial differential equations

ˆ ( , ) N



N



+ ‰ ( , ) N





N

=, , (16)



ˆ ( , ) N + ‰ ( , ) N  N N

= 0,

reduces equation (1) to an autonomous equation (the constant , ≠ 0 can be chosen arbitrarily). The function = ( , ) is a universal invariant of the operator X. Suppose that the general solution of the characteristic equation



M

has the form

M ‰ ( , )

=

ˆ ( , )

’ ( , ) = L ,

where L is an arbitrary constant. Then the general solutions of equations (16) are given by (see A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, 2002):

 = ,‹X



+ i 1 ( ’ ), ˆ ( , ’ ) = i 2 ( ’ ), ’ = ’ ( , ),

M

where i 1 ( ’ ) and i 2 ( ’ ) are arbitrary functions, ˆ   , ’ ( , )  ≡ ˆ ( , ), and ’ regarded as a parameter.

€

Ï ˜ÒÍ Í Ò

€

Example 3. The Emden–Fowler equation Example 2): X = (Ë , Ï )

~

Equations (16) for

Solving (15) for

Ë

ô

= 49



€

+ (Ë , Ï )



Ë

€

,

Ï

Ï

~ (Ë

where

=



Ë

1 7

,

ã

Ï

−15 7 2

=

Ë



−3 7

Ï

+

 Ë 6  7 ,  (Ë

6 49



, we obtain the transformation

Ë

=

: 7, Ï

=

: 3¤

= 49

¤

2

Ë



−2 7

, Ï ) = 147



Ë



−1 7

Ï

+ 12.

.

 :,

6 49

−

which reduces the original equation to the autonomous equation

¤ Í89Í 8

O

admits the operator (cf. the operator X2 in Item 2 of

, Ï ) = 343

admit the particular solutions

â and



Ë



=

in the integral is

,

which can easily be integrated by quadrature.

2 b . The second method. Suppose that we know two invariants of the admissible operator X:

® 0 = ® 0 ( ® 1 = ® 1 (

´

, ) , , 
)

(universal invariant), (first differential invariant*).

(17) (18)

* By definition, an th-order differential invariant of the operator X is a function è = = 0 with the operator X defined by (7). satisfying the linear partial differential equation X è

± ±

±

±

è

± ë

Ë , Ï , Ï ÒÍ , ªÿ ÿ›ÿ , Ï (Ò ± ) ì

,

© 2003 by Chapman & Hall/CRC

Page 64

Then the second differential invariant can be found by differentiation,

® 1,

, , 
, 

 ) = M

® 2 (

(19)

®0

M

where ® + = ( c  ® + )  . Using (18)–(19), let us eliminate the derivatives 
and 

 from the M M original equation and take into account relation (17). Thus we obtain the first-order equation:

®1 M M

®0

Example 4. The Emden–Fowler equation whose first prolongation has the form: X= 1

This operator admits the invariants:

Ï Ò˜Í Í €Ò Ë

2

 Ë −6€ Ï 3 (see 2.3.1.3, € the special case ´ + Ë Ï  + ( Ï − Ë ÏÍ )  . 

Ò

=

è0=Ï % Ë

€

= 4 ( ® 0 , ® 1 ).

€

è 1 = Ë ÏÒÍ

,

= 3) admits an operator

−Ï ,

(20)

€

which form an integral basis of the first-order linear partial differential equation

Ë

€

2

Ë

è

+Ë

€

Ï è Ï

+ (Ï − Ë

€

Ï Í) è ÏÍ

= 0.

Using (19) and (20), we find the second invariant:

è2= æ è1 è æ 0

=

Ë 3 Ï ˜ÒÍ Í Ò Ë Ï ÒÍ − Ï

.

(21)

Let us express the unknown function and its derivatives from (20)–(21) to obtain

Ï

=

¤

Ë

,

ÏÒÍ

=

Ë

¤

Ë

+



,

ÏÒ˜Í Í Ò

=

„ ¥Í Ë 3

,

where

¤

Substituting these expressions into the original equation, we see that the variable form hÍ = 3 ,

è 0,

=

Ë



=

è 1.

is canceled and the equation takes the

„ ¥ ¤

i.e., it becomes a first-order separable equation.

^_

References for Subsection 0.6.1: G. W. Bluman and J. D. Cole (1974), L. V. Ovsiannikov (1982), J. M. Hill (1982), P. J. Olver (1986), G. W. Bluman and S. Kumei (1989), H. Stephani (1989), N. H. Ibragimov (1994).

¨ 0.6.2. Contact Transformations. Backlund Transformations. Formal Operators. Factorization Principle 0.6.2-1. Contact transformations. The set of transformations Tª =



 ¯ = Y ( , , 
 , ), ¯ = j ( , , 
 , ), 
¯  = ¶ ( , , 
 , ), ¯

 ¯ |ª

=0 =  ¯ |ª =0 = 
¯  |ª = ¯ =0

, ,




(1)

(here, Y , j , ¶ are smooth functions of their arguments and is a real parameter) is called a continuous one-parameter Lie group of tangential transformations (or simply, a tangential or contact group) if T ª 1 « Tª 2 = Tª 1 +ª 2 , i.e., if successive application of transformations (1) with parameters 1 and 2 is equivalent to the same transformation with parameter 1 + 2 . The transformed derivative ¯ 
¯ depends only on the first derivative 
and does not depend on the second derivative. Thus, the functions Y and j in (1) cannot be arbitrary but are related by (see Paragraph 0.1.8-1):

Y j j Y Y j O N + 
N N
O…N + 
N − N
… = 0,    P  P N N N N N N © 2003 by Chapman & Hall/CRC

Page 65

where the function

¶

is defined by

¶

j

= N
¸·

N

N

Y N
. 

Proceeding as in Paragraph 0.6.1-1, we consider the Taylor expansions of  ¯ , ¯ , and /
¯  ¯ in (1) with We have respect to the parameter about = 0, preserving only the first-order terms. 

¹ ¯ ¬¯ + ˆ ( , , 
) ,

where

Y ( , , 
, ˆ ( , , 
) = N

N

+ ‰ (  , , 
) ,

¬

¯

¯ 
¯



) = N j ( , ,  ,  , ‰ (  , , û ª =0

û N û

)

On the other hand,

¯ c  ( + ‰ ) =  ¯ c  ( + ˆ )

¯
¯ ≡ M

M


+ (  , , 
) ,

¬



)

ûª û û

=0

,


( , ,  ) = N


 + (‰  + ‰ 
 + ‰  œ 1+  (ˆ  + ˆ  
+ ˆ  œ

¬

¶

( , , 
, )

N



 ) . 

 )

û

û

ûª

=0

.

(2)

 Expanding (2) with respect to and requiring that be independent of

 , we find that three the ú functions ˆ , ‰ , and are expressed in terms of a single function ( , ,
 ) as follows: ˆ =−N N

ú




‰ =

,

ú

− 
N



N

ú




= N

,

N

ú



ú

+ 
N

.

(3)

N

To an infinitesimal tangential transformation (1) there corresponds the infinitesimal operator: X = ˆ ( , , 
) N + ‰ ( , , 
) N  N N



+ ( , , 
) N


(4)

N

whose coordinates satisfy relations (3). The action of the group on higher  derivatives  is determined by the recurrence formula:



 

+1

= c  (  ) − ( 

+1)

c  (ˆ ),

where 1 = . The invariance condition and the algorithm of finding tangential operators (4) admitted by ordinary differential equations are similar as those for point operators. The only difference is that the coordinates of the tangential operator depend on the first derivative; therefore, the determining equation can be split and reduced to a system only in the case of equations whose order is greater or equal to three. x3y'z|{}~/€ There are no tangential transformations of finite order , > 1 other than prolonged point transformations and contact transformations [these transformations are described by formulas similar to (1) and, in addition to 
 , 
¯  ¯ , contain higher derivatives of up to order , inclusive]. 0.6.2-2. B¨acklund transformations. Formal operators and nonlocal variables. 1 b . If the coordinates of the infinitesimal operator are allowed to depend on the derivatives of arbitrary (up to infinity) orders, we obtain Lie–B¨acklund groups (of tangential transformations of infinite order). However, on the manifold determined by an ordinary differential equation, all higher derivatives are expressed through finitely many lower derivatives, as dictated by the structure of the equation itself and the differential relations obtained from the equation. The substitution of the right-hand side of equation (1) into an infinite series with derivatives usually results in very cumbersome formulas hardly suitable for practical calculations. For this reason, the Lie–B a¨ cklund groups are widely used only for the investigation of partial differential equations, whereas in the case of ordinary differential equations, a more effective approach is that based on the canonical form of an operator and the notion of a formal operator.

© 2003 by Chapman & Hall/CRC

Page 66

¶

2 b . The canonical form X is defined by the relation

¶
X = X − ˆ (  , ) c  = [ ‰ (  , ) − ˆ ( , )  ] N , N

where X = ˆ ( , ) º  + ‰ ( , ) º is the infinitesimal operator of the group [see formula (4) in ¶ º º Paragraph 0.6.1-1], and c  is the operator of total derivative. The operators X and X are equivalent in the sense that if one of them is admissible for the equation, then the other is also admissible (the operator of total derivative is admissible for any ordinary differential equation). The function ® 0 ( , ) ≡  is an invariant of any operator in canonical form.  order derivatives  The action of the group on higher for an operator in canonical form is determined ¶ ¶ by the simple recurrence formula  +1 = c  (  ). The order of an equation that admits an operator in canonical form can be reduced on the basis of the algorithm described in Paragraph 0.6.1-3 (see Item 2 b , the second method). 3 b . By definition, a formal operator is an infinitesimal operator of the form

m

=`

, N 

(5)

where the function ` depends on  , , 
 ,  , (  ) (with , smaller than the order of the equation under investigation) and auxiliary variables whose definition involves the symbol of indefinite  integral, for instance, X ( , , 
) 

M

(the integration is with respect to the variable  which is involved both explicitly and implicitly, through the dependence of on  ). Such auxiliary variables are called nonlocal, in contrast to the coordinates of the prolonged space defined pointwise. The nonlocal variables depend on derivatives of arbitrarily high order, for instance,

X

½  + +1 ( + )  .  = ÷ (−1) + ( + 1)! M + =0 ¡

This formula is obtained by successive integration by parts of its left-hand side. Thus, a nonlocal variable can be represented as an infinite formal series; and this allows us to express the coordinates of the Lie–B¨acklund operator in concise form. A formal operator is a far-reaching generalization of an operator in canonical form. The function ® 0 ( , ) ≡  is an invariant of the formal operator (5) for any ` . When solving the direct problem, one usually prescribes the nonlocal operator in the general   form  + ‰ 2S or X = O ‰ 1 X  +‰ 2 , (6) X = ‰ 1 exp O X

Q

N 

M P

P/N 

M

and then, in order to find an admissible operator, one uses a search algorithm similar to that    described in Paragraph 0.6.1-2. The coordinates of the prolonged operator are found by the formulas   = c  (  −1 ), where 0 = ` . In contrast to the method of finding a point operator, in the present case, there are three unknown functions (‰ 1 , ‰ 2 , ); and the splitting procedure to obtain a system can be realized with respect to all “independent” variables, in particular, the nonlocal variables. Suppose that the differential equation

  =   , , 
,  , can be written in new variables  = ® 0 , H = ® 1 ( , , 
 ), H ( )

  ? 
, H 

 ,  , H (  ( −1)

invariants of an admissible operator of the form (5). Then the coordinate ` the equation

N N

® 1`

+ N

N

®1c 


(7) , where ® 0 and ® 1 are of this operator satisfies

−1)

 [ ` ] = 0,

© 2003 by Chapman & Hall/CRC

Page 67

which is an analogue of a linear ordinary differential equation for a function of several variables, since it involves the total derivative of the unknown function (exact differential equation). Its solution has the form:

` = exp O − X N

N

® 1

® 1

N

N
 ,  M P

(8)

where the integral is taken with respect to  involved explicitly and implicitly (through the dependence of , 
 ,  on  ), which means that this representation of an operator through a nonlocal variable is most universal. The function (8) generates a nonlocal exponential operator of the form (5) [the class of nonlocal exponential operators is specified by the first expression in (6) with ‰ 2 ≡ 0]. Example 1. The equation

ÏÒ˜Í Í Ò

admits two Lie–B¨acklund operators: X1 = ( Ë , Ï , ÏÒÍ )

~

~

» Ò

,

X2 =

›

±

¼’ =0

=0

» Ò± … Ïã (Ï

−Ë

ÏÒÍ , Ï ÒÍ

) + (Ï − Ë

›

ÏÒÍ , Ï ÒÍ

)

†

€

€ Ï (Ò ±

)

,

where , ã , are arbitrary functions of their variables. The first operator is trivial (the operator of total derivative is admissible for any differential equation), while the second operator determines the maximal group of contact transformations admitted by the equation under consideration.

A Lie–B¨acklund operator admitted by an ordinary differential equation can by found by three methods: (i) in the form of an infinite formal series; (ii) by passing to an equivalent system of ordinary first-order differential equations:


=

1

2,


=

2

3,

 ,


 =  ( , 1 , 2 ,  ,  ),

and finding an admissible point group; (iii) by its representation as a formal operator whose coordinates depend on nonlocal variables (the general form of the operator is chosen by the investigator). In all cases, the search algorithm amounts to solving the determining system which is constructed by a procedure similar to that of Subsection 0.6.1. From the standpoint of simplicity and the possibility of integrating equations, the third method seems to be the most effective if one takes into account that an equation admitting an operator can be written in terms of new variables—invariants of the admissible operator—as a new ordinary differential equation whose order is by one less than that of the original equation. 0.6.2-3. Factorization principle. The use of formal operators allows us to formulate universal principles for reducing the order of an equation, independently of the specific structure of the operator (it can be a point operator, a tangential or nonlocal operator, or a Lie–B¨acklund operator). Theorem 1. An arbitrary B th-order differential equation (7) can be factorized to a system of special form H (  −1) = 4  , H , H 
,  , H (  −2)  , (9) H = 5 ( , , 
), if and only if equation (7) admits the nonlocal exponential operator: X = exp O − X

5   N . 5   M P œ N

(10)

© 2003 by Chapman & Hall/CRC

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The function 5 ( , , 
) is the first differential invariant of the operator (10). Therefore, having found an admissible operator (10) of the form X=`

N

` ≡ exp X Æ ( , , 
)  S , Q M

N ,

(11)

we can calculate 5 by solving the first-order linear partial differential equation 5  + Æp5  œ = 0. The function Æ ( , , 
) is found as a solution of the determining system obtained by “splitting” the invariance condition for operator (11):

 −1)  W X V (  ) −    , , 
,  , (?

= 0, û  (œ n ) = Ž û



where X=



½ 

=0

` 

`  =c  ` 

N

N ( ) , 

−1 ,

`

c  = N + 
N  N N

=` ,

0



+   N

N
+ ))) . 

Theorem 1 generalizes the classical Lie algorithm, which is restricted to the case of unconditional solvability of the second equation of system (9). On the other hand, the introduction of the factor system (9) allows for two more cases, since the first equation is independent of . These cases are the following: 1. The first equation of system (9) allows for the reduction of the order or is solvable. 2. The first equation of system (9) has some special properties, for instance, admits a fundamental system of solutions. Example 2. The equation

â (Ë

= â ( Ë ) Ï + ãÒÍ ( Ë ) Ï

ÏÒ˜Í Í Ò

ã (Ë

−1

− [ ã ( Ë )]2 Ï

−3

(12)

) and ) is the only equation of the form (its uniqueness is to within a Kummer–Liouville for arbitrary functions equivalence transformation; see Paragraph 0.2.1-8)

Ï Ò˜Í Í Ò

€

=

ê (Ë , Ï

)

admitting the nonlocal exponential operator:

€

€

€

 Ò +  Ï ÒÍ Ë † ,  =  (Ë , Ï ), ½ = ½ (Ë , Ï ). … + X ½ Ø  Ù æ Ï Ï The second prolongation€ of the operator X has€ the form:   + ( Ò +  Ï ÒÍ +  ½ )  œ X = exp X ½ Ë € 2 æ Ù¿¾ Ø +  Ò˜Ò + 2 ½  Ò +  ½ Ò + ½ 2  + (2  Ò  + 2 ½  +  ½  ) Ï ÒÍ + N ( Ï ÒÍ )2 + Ï Ò˜ ÍÍÒ î   . í œœ¦À X = exp

Ë Ø X}½ æ Ù



Applying this operator to the equation invariance condition in the form:

 ҘÒ

+2

= exp

Ï Ò˜Í Í Ò

=

ê (Ë , Ï

) and replacing all instances of

½  Ò +  ½ Ò + ½ 2  +  ê −  ê¦

+ (2

Ò

+2

by

ê

=

ê (Ë , Ï

), we obtain the

½  +  ½  )Ï ÒÍ +  N (Ï ÒÍ )2 = 0.

Splitting this relation with respect to powers of the “independent” variable equations for the functions , , and ê :

 ½

Ï Ò˜Í Í Ò

N = 0,  Ò  + 2½  +  ½  = 0,  Ò˜Ò + 2½  Ò +  ½ Ò + ½ 2  + ê −  ê¦

Ï ÒÍ

, we obtain the following system of three

2

= 0.

From the first two equations it follows that = Ó ( Ë ) Ï + Ô ( Ë ),

 ½

=−

ÓÓ Í Ï

2

+ 2 Ó Í ÔÝÏ + ( Ë ) , ( ÓÏ + Ô )2

¯

© 2003 by Chapman & Hall/CRC

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where Ó = Ó ( Ë ), Ô = Ô ( Ë ), and = ( Ë ) are arbitrary functions. The third equation can be treated as a first-order linear differential equation for the unknown function ê = ê (Ë , Ï ):

¯ ¯



Substituting the above expressions of

ê (Ë , Ï

) = ( ÓÏ + Ô ) â ( Ë ) +

[ ÓÓ

ê æ Ï æ

−

½

and

− 2( Ó Í )2 ] Ô − ( ÓÔ

ÍÍ

Ó

3

 

ê

1 Ò˜Ò  (

=

+2

½  Ò +  ½ Ò +  ½ 2 ).

into this relation and integrating the result, we obtain − 2Ó

ÍÍ

Í Ô Í )Ó

−

[ ÓÓ

− 3( Ó Í )2 ] Ô

ÍÍ

i¯ Í − 2Ó Í ¯ )Ó

+ 2 ÓÓ Í ÔÖÔ Í − ( Ó 2 Ó 3 ( ÓÏ + Ô )

2

where â ( Ë ) is an arbitrary function. The differential invariant of the operator X satisfies the linear partial differential equation

€ €;  Ï

;

(obtained after the division by exp(  equation

where



=

Ï ÒÍ

½

Ë



æ

æ Ï æ

€ € ; + (  Ò + Ï ÒÍ +  ½ ) Ï ÒÍ

)). Substituting the above =

ӁÏ

Ó

+Ô

+

2 ÓÓ

ÍÏ

2

+ (3 Ó



−

i¯

( Ó Í Ô 2 − Ó )2 , 4 Ó 3 ( ÓÏ + Ô )3

=0

½

and

into this equation, we pass to the characteristic

¯

Í Ô + ÓÔ Í )Ï + ÔÖÔ Í + , ( ÓÏ + Ô )

. Integrating this equation, we find the differential invariant:

;

Ó Í Ô − ÓÔ Í + Ó Í Ô 2 − Ói¯ . Ó 2 ( Ó Ï + Ô ) 2 Ó 2 ( ÓÏ + Ô )2 ÒÍ , one can find Ï Ò˜Í Í Ò and, taking into account the known structure of the function =

ӁÏ

Ï ÒÍ

−

+Ô

Having calculated the derivative ; ê (Ë , Ï ), one obtains the factorization of the original equation: ;˜ÒÍ + Ói; 2 + (ÓÍ % Ó ); = â , ( ÓÏ + Ô ) ÏÒÍ = ( ÓÏ + Ô )2 ; + Ó −2 ( ÓÍ¤Ô − ÓÔÝÍ )( ÓÏ + Ô ) − 12 Ó −2 ( ÓÍ¤Ô 2 − Ói¯ ). Ô vÏ , combined with the corresponding transformation of the independent An equivalence transformation of the form ÓÏ +  variable and changed notation, yields: ;˜ÒÍ + ; 2 = â (Ë ), (13) Ï ÒÍ = ;˜Ï + ã (Ë )Ï −1 . The first equation of system (13) is the Riccati equation. Its general solution can be represented in terms of a fundamental system of solutions of the “truncated” linear equation:

ÏÒ˜Í Í Ò

= â (Ë )Ï ,

(14)

which coincides with (12) for ã ≡ 0. The second equation of system (13) is a Bernoulli equation. It can be integrated by quadrature for an arbitrary function = (Ë , ). Therefore, the general solution of equation (12) can be expressed in terms of a fundamental system of solutions of the linear equation (14). Note that in the general case, equation (12) admits no point groups.

; ;



If an operator admitted by equation (1) has no differential invariants of the first-order, then it is possible to apply the general factorization principle. Theorem 2. An arbitrary B th-order differential equation (1) can be factorized to the system of special structure H (  −  ) = 4   , H , H 
,  , H (  −  −1)‘  , (15)  ) , H = 5   , , 
,  , (Z º ( ¼ ) ≠ 0,

º

œ

 )  is a lowerprovided that equation (7) admits a formal operator (5) for which 5  , , 
,  , (Z order differential invariant on the manifold given by (7). The coordinate ` of this operator satisfies the linear equation with total derivatives: `EN N

H

H

H

+ c  [ ` ] N
+ ))) + c (  ) [ ` ] N ( 

N

N

)

= 0.

(16)

Equation (16) plays a crucial role in both the direct and inverse problems. It can be regarded as an equation for the determination of the coordinate of the canonical operator (if one knows the

© 2003 by Chapman & Hall/CRC

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invariant H ). It can also be regarded as an equation for the determination of an invariant (if one knows the coordinate ` ). In the latter case, this is a first-order partial differential equation. Example 3. The third-order nonlinear equation

ÏÏ Ò˜Í Í Ò˜Í Ò

+

admits two operators X1 =

Ï

€

ë

Ï Ò˜Í Í Òì

,

2

−Ï

X2 =

− â (Ë )Ï

ÒÍ Ï Ò˜Í Í Ò

Ï Ï Ø X

Ë æ Ù

−2

2

€

=0

(17)

,

(18)

which can be found with the help of the direct algorithm, if the structure of the operator is specified by the second expression in (6). The first operator, X1 , is the usual point operator of scaling (the original equation is homogeneous) and provides the usual reduction of order of equation (17) by one. The second operator, X2 , is nonlocal. Let us construct differential invariants of the operator X2 . To this end, we should solve the equations:

€

€ € € Á € è 1 + » Ò [ Á ] € è 1 = 0, Á =€ Ï X Ï −2 Ë , æ Ï Ï ÒÍ Á € è 2 + » Ò [ Á ] € è 2 + » 2Ò [ Á ] € è 2 = 0. Ï Ï ÒÍ Ï Ò˜Í Í Ò Á € € in (19) becomes After differentiation with respect to , the first equation €è €è Ï Ï −2 Ë Ù 1 + … Ï −1 + Ï ÒÍ X Ï −2 Ë Ù † 1Ò = 0.€ æ æ Ø X Ø Ï ÏÍ

(19)

€

Let us show that this equation admits no solutions depending only on Ë , Ï , Ï ÒÍ , and è 1 Ï ÒÍ ≠ 0, i.e., there are no first-order differential invariants. The nonlocal expression  Ï −2 Ë depends on derivatives of arbitrarily high orders and æ can be regarded as an independent quantity. Therefore, the first equation (19) can be split and we obtain the system:

€

€ %

€

€

€

Ï è 1 + Ï ÒÍ è 1Ò Ï ÏÍ

€

Ï

= 0,

−1

€

€

è1 Ï ÒÍ

= 0.

It follows that è 1 Ï ÒÍ = 0. Let us find a second-order differential invariant. After differentiation with respect to becomes

€

Ë

−2

ۏ

€

€è 2 −1 Ï + Ï ÒÍ X Ï −2 Ë † + Ï Ò˜ Í Í Ò X Ï −2 Ë … æ Ù Ï æ Ù Ï ÒÍ Ø æ Ø Ë  −2 , we establish that Splitting this equation with respect to the nonlocal € variable Ï € equation, the nonlocal variable is canceled, €è € è æ Ï Ï Ø X

2

+

Ï

Hence, we find that

è2=;

=

Ï ˜ÒÍ Í Ò % Ï

Ï

2

+ ÏҘ ÍÍÒ

2

Ï Ò˜Í Í Ò

%

Á ,€ the second equation in (19) € è €Ù Ï Ò˜Í€ 2Í Ò = 0. è 2 % Ï ÒÍ = 0. In the remaining

= 0.

, and equation (17) is factorized to the system:

;˜ÒÍ + ; 2 = â (Ë ), ÏÒ˜Í Í Ò − ÏD; = 0.

^_

References for Subsection 0.6.2: R. L. Anderson and N. H. Ibragimov (1979), O. N. Pavlovskii and G. N. Yakovenko (1982), N. H. Ibragimov (1985), P. J. Olver (1986), V. F. Zaitsev (2001).

0.6.3. First Integrals (Conservation Laws)

. = .   , , 
,  , (  A function differential equation

−1)

. if the total derivative of the function if

. c  [ ]≡8

 is called a first integral (conservation law) of the ordinary

 −1)    =    , ,  , (? ( )

(1)

along the trajectories of equation (1) is zero or, equivalently,

 −1)  V (  ) −   , ,  , (?  −1)  W = 0,  , , 
,  , (?

(2)

where 8 is an integrating factor. From this definition it is clear that 8 = ºD(Ân −1) . ºœ The algorithm of finding a first integral is similar to that of finding an admissible operator. It is necessary to prescribe the desired structure of the first integral (or the integrating factor) and

© 2003 by Chapman & Hall/CRC

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substitute it into the determining equation (2). Subsequent splitting with respect to lower derivatives (assumed to be independent variables) leads to the determining system. x3y'z|{}~­³´€ An arbitrary function of first integrals is also a first integral of the same equation. Therefore, having found a first integral depending on ("
 )  , one has to make sure that it is nontrivial; i.e., it cannot be represented as the product of first integrals depending on lower powers of the derivative.

x3y'z|{}~‚µ €

If the equation has , functionally independent first integrals, then its order can be reduced by , by successively excluding higher derivatives (see Example 3). For second-order equations 

 =  ( , , 
), (3) the determining equation (2) can be written in the form

N

.

+ 
N

 N

.

.

+  ( , , 
) N

N




N

.

= 0.

(4)

for the given equation), as well as the In this case, one can solve the direct problem (find inverse problem (find possible  for the given structure of the first integral). Example 1. Let us find all equations of the form

Ï Ò˜Í Í Ò

=

ê (Ë , Ï

)

(5)

admitting a first integral that is quadratic with respect to the first derivative:

4

=

( Ë , Ï )( ÏÒÍ )2 + ( Ë , Ï ) ÏÒÍ +

Ã

Ä

Å

( Ë , Ï ).

Then the left-hand side of the determining equation (4) is a cubic polynomial with respect to with respect to powers of Ï ÒÍ yields the system of four equations:

Ð Ã Ä Å

The solution of this system for

ê (Ë , Ï

where

Æ

=

ê

Ò

= 0, + = 0, + + 2 …ê + ê = 0.

Ä^ Åw Ä

Ò Ò

Ã

Ï ÒÍ

. The procedure of splitting

= 0,

is given by:

à −3  2 Æ (; ) + 21 à −2 … Ø Ã„Ã±ÍÒ˜Í Ò ; = à −1  2 Ï + 21 X ä à −3  2 æ Ë , )=

Æ (; ), Ã = Ã (Ë ), and = ä ä 4 = Ã ( Ï Ò Í )2 − ( Ã Í Ò Ï −

Ã

1 2 ±ÍÒ Ù 2

−

Ï

−

Ã

ÍÒ ä

Ã

1 ±ÍÒ 2

+

ä

†,

( Ë ) are arbitrary functions. The first integral has the form:

ä

)Ï

ÒÍ

+

1 4

Example 2. Consider the equation

Ã

−1

(

Ã

ÏÒ˜Í Í Ò

Í Ò )2 Ï

=



2

−

Ë Ï

1 2

à −1 Ã



.

−1 2

ÍÒ Ï ä

+

1 4

Ã

−1

ä

2

−2X

Æ (; ) ; . æ

Let us find its first integral, which is a cubic polynomial with respect to the first derivative:

4

=

Ã

( Ë , Ï )( ÏÒÍ )3 + ( Ë , Ï )( ÏÒÍ )2 +

Ä

Å

( Ë , Ï ) ÏÒÍ +

Ç

( Ë , Ï ).

In this case, the left-hand side of the determining equation (4) is a fourth-order polynomial in system consists of five equations: = 0, Ò + = 0,

Ð Ã

^Ä  Ä + Åw + 3 Ë Ï −1  2 Ã Å Ò + ÇÈ + 2 Ë Ï −1  2 Ä Ç Ò +  Ë Ï −1  2 Å = 0. Ò

Ï ÒÍ

, and hence the determining

= 0, = 0,

© 2003 by Chapman & Hall/CRC

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Solving this system, we obtain the first integral in the form:

4

= ( ÏÒÍ )3 − 6

Example 3. The equation

1 2

ÏÒ˜Í Í Ò˜Í Í Ò˜Ò

4

admits three first integrals:

4 4

1

=

2

=

3

=

ÏÒÍ ÏÒ˜Í Í Ò˜Í Ò

4

Ë Ë

4

1

−

2

−

Equating these expressions to independent constants equation (see 4.2.1.1).

^_



Ë Ï



 3  2 + 2

ÏÒÍ

=

+ 4 …Ï

…Ï −5  3

Ë

2 3



Ï Ò˜Í Í Ò )2 + 32 …Ï −2 3 , 3 ÏÏ Ò˜Í Í Ò˜Í Ò + 12 Ï ÒÍ Ï Ò˜Í Í Ò , 2 1 Ë 24 3 Í Í Ò − ( Ï Ò Í )2 . 1 + 2 ÏÏ Ò˜ 2 −

1 ( 2

 1,  2, 

3

.

and eliminating

Ï Ò˜Í Í Ò˜Í Ò

and

Ï Ò˜Í Í Ò

, we obtain a first-order

References for Subsection 0.6.3: P. J. Olver (1986), A. D. Polyanin and V. F. Zaitsev (1995).

0.6.4. Discrete-Group Method. Point Transformations Consider transformations of the class of ordinary differential equations

  =    , ,  , (  ( )

−1)

, a ,

(1)

whose elements are uniquely defined by a vector of essential parameters a. Any set of invertible transformations

 =  (a , ‡ ),

= (a , ‡ )

( g



−   ž g ≠ 0),

(2)

mapping each equation of class (1) into some (other) equation of the same class

‡ (g  ) =  a , ‡ ,  , ‡ (g 

−1)

, b ,

(3)

and containing the identical transformation is called a discrete point group of transformations admitted by the class (1). Transformation (2) maps any solution of equation (1) to a solution of equation (3). Therefore, knowing the discrete group of transformations for some class of equations and having a set of solvable equations of this class, one can construct new solvable cases. Point transformations (2) can be found by a direct method — namely, if one substitutes an arbitrary transformation of the form (2) into equation (1) and imposes condition (3), one arrives at a determining equation containing partial derivatives up to order B of the unknown functions  and and having variable coefficients depending on  , , 
,  , (  −1) . Since the functions  and do not depend on the derivatives, the determining equation can be “split” with respect to the “independent” variables 
,  , (  −1) , and we obtain an overdetermined system which is nonlinear, in contrast to that obtained by the Lie method (see Subsection 0.6.1). Example 1. For second-order equations

ÏÒ˜Í Í Ò

ê (Ë , Ï , Ï ÒÍ

=

, a),

(4)

the substitution of (2) into (4) yields (â

8 ãU¥

−ã

8 ⥠)ž¤ Í89Í 8 + ( â¥ãU¥i¥ − ãU¥žâ¥i¥ )(¤žÍ8 )3 + ( â 8 ãU¥i¥ − ã 8 â¥i¥ + 2 â¥ãU¥ 8 − 2ãU¥´â¥ 8 )(¤žÍ8 )2 + ( â¥ã 898 − ãU¥žâ 898 + 2 â 8 ãU¥ 8 − 2ã 8 ⥠8 ) ¤žÍ8 + â 8 ã 898 − ã 8 â 898 = ( â 8 + ⥤žÍ8 )3 ê Ø

â ,ã , ã

8 + ãU¥ ¤ Í8 8â + ⥠¤ Í8

, a . (5)

Ù

Let us require that the transformed equation (5) belong to the class (4), i.e.,

¤žÍ89Í 8

=

ê

: ¤ ¤8

( , , žÍ , b).

(6)

Condition (6) imposed on the determining equation (5), i.e., the replacement of ¤ Í89Í 8 by the right-hand side of equation (6), leads us to the relation ( â 8 ãU¥ − ã 8 ⥠) ê ( : , ¤ , ¤žÍ8 , b) + ( â¥ãU¥i¥ − ãU¥´â¥i¥ )( ¤žÍ8 )3 + ( â 8 ãU¥i¥ − ã 8 â¥i¥ + 2 â¥ãU¥ 8 − 2ãU¥´â¥ 8 )( ¤žÍ8 )2 ã 8 + ãU¥¤ Í8 , a , (7) + ( â¥ã 898 − ãU¥´â 898 + 2 â 8 ãU¥ 8 − 2ã 8 ⥠8 ) ¤žÍ8 + â 8 ã 898 − ã 8 â 898 = ( â 8 + ⥠¤žÍ8 )3 ê â , ã , 8 â + ⥤ Í8 Ù Ø

© 2003 by Chapman & Hall/CRC

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¤ Í8 . Expanding the function ê into a series in powers of ¤ Í8 , we can represent ¼’ 4 Ë , Ï , [ â ], [ ã ] ì ( ¤ Í8 ) õ = 0, (8)

which contains the “independent” variable (7) in the form

õ

õ ë

=0

where the symbols [ â ] and [ ã ] indicate dependence on the functions â , ã and their partial derivatives involved in (7). The sum in (8) is finite if ê is a polynomial with respect to the third variable [for a polynomial of degree ≥ 4, both sides of the equation must be first multiplied by ( â + â Í ) −3 ]. Condition (8) is satisfied if the following equations hold:

5

8 ¥¤ 8 4 E= 0,

ô

õ

ÿ›ÿªÿ

= 0, 1, 2,

Example 2. Consider a special case of equation (4) with the right-hand side independent of the derivative

ÏÒ˜Í Í Ò Relation (7) has the form: (â

8 ãU¥

−ã

8 â¥

=

ê (Ë , Ï

:

, a).

(9)

¥ãU¥i¥ − ãU¥žâ¥i¥ )(¤žÍ8 )3 + ( â 8 ãU¥i¥ − ã 8 â¥i¥ + 2 â¥ãU¥ 8 − 2ãU¥´â¥ 8 )(¤žÍ8 )2 + ( â¥ã 898 − ãU¥žâ 898 + 2 â 8 ãU¥ 8 − 2ã 8 ⥠8 ) ¤žÍ8 + â 8 ã 898 − ã 8 â 898 = ( â 8 + ⥤žÍ8 )3 ê

: ¤

Ï ÒÍ

) ê ( , , b) + ( â

( â , ã , a).

In this case, the sum (8) is finite and the determining system has the form:

â 8 â ¥ã 8 Uã ¥

It can be shown that for Consider the case â

¥

Since

â 8Í

â¥ãU¥i¥ − ãU¥žâ¥i¥ = â ¥ 3 ê ( â , ã , a), â 8 Uã ¥i¥ − ã 8 â¥i¥ + 2 â¥ãU¥ 8 − 2ãU¥žâ¥ 8 = 3 â 8 â ¥ 2 ê ( â , ã , a), â¥ã 898 − ãU¥žâ 898 + 2 â 8 ãU¥ 8 − 2ã 8 ⥠8 = 3 â 8 2 ⥞ê ( â , ã , a), â 8 ã 898 − ã 8 â 898 + ( â 8 ãU¥ − ã 8 ⥠) ê (: , ¤ , b) = â 8 3 ê ( â , ã , a).

(10)

≠ 0, solving system (10) is equivalent to solving the original equation (9). = 0. In this case, the first equation of the system holds identically and the system becomes

â 8 Í Uã ¥i¥ = 0, ãU¥´â 898 − 2 â 8 ãU¥ 8 = 0, â 8 ã 898 − ã 8 â 898 + â 8 ãU¥ê

( : , ¤ , b) = â 8 3 ê

(11) ( â , ã , a).

≠ 0, the first two equations yield

ã

: ¤

( , )=

É (: )¤ + Ê (: ), â8 Í =  [É (: )]2 .

(12)

Substituting (12) into the last equation of system (11) and “splitting” the resulting relation with respect to powers of the “independent” variable , we obtain a new system of (ordinary) differential equations. Solving this system, we find the and , and finally, the desired discrete group of transformations. In order to give calculation details, unknown functions one has to know the specific structure of the function ê ( Ë , Ï ), for in the general case it was only shown that any discrete point group of transformations of equation (9) for â = 0 consists of Kummer–Liouville transformations (12).

¤

É

Ê

N¥

Example 3. Consider the generalized Emden–Fowler equation:

 ËEϱ Here, a = { 5 , ´ , Ì } is the vector of essential parameters, and  ÏÒ˜Í Í Ò

=

by scaling the independent variable and the unknown function).

Ë

( ÏÒÍ ) .

(13)

is an unessential parameter (it can be made equal to unity

O

1 . First, we note that equation (13) admits a discrete group of transformations determined by the hodograph transformation, i.e., by passing to the inverse function:

Ë

¤

= ,

Ï

:

= ,

where

¤ = ¤ (: ).

Í

5 Í

(14)

´

, ÊÏ , This transformation is a consequence of the invariance of equation (13) with respect to the transformation Ë 3− , − (note that the hodograph transformation changes the sign of the unessential parameter ). Denoting the transformation (14) by , let us schematically represent its action on the parameters of the equation as follows:

ÌÍ

Ì  Í



©

Í Î$Î {´ , 5 , 3 − Ì } 5 ´ , Ì} Ï © yields the original equation. Double application of the transformation { ,

transformation

©

.



(15)

© 2003 by Chapman & Hall/CRC

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Ì

O

2 . For

= 0, equation (13) is of the class (8), and the last equation of system (11) becomes

É 8 ÊhÍ8 + „‚ É 2 Ñ: Ði¤^Ò = ƒ 2 (ɐ¤ + Ê )± â E , (16) where Ó , Ô , and ‚ are the parameters of the transformed equation ¤ Í89Í 8 = ‚ƒ: Ð ¤ Ò , and â (: ) =  X [É (: )]2 : . æ ´ Let , Ô ≠ 0, 1, 2. Then relation (16) is possible only if Ê ( : ) ≡ 0. Splitting with respect to powers of ¤ leads us to the system: ÉÉ 89Í 8Í − 2(É 8 Í )2 = 0, (17) ‚ƒ:ÑÐ = ƒ 2 É ± +3 â E . By integration we find that É = : −1 , â = : −1 (to within unessential coefficients). Thus, we arrive at the transformation Ë = : −1 , Ï = : −1 ¤ , where ¤ = ¤ (: ). (18) Denoting the transformation (18) by Õ , let us schematically represent its action on the parameters of the equation: {−5 − ´ − 3, ´ , 0} transformation Õ . (19) { 5 , ´ , 0} Í,Î ÎÎÎ Double application of the transformation Õ yields the original equation. 3 O . Let Ì = 0 and ´ = 2. Then, Ô = 2 and the splitting procedure for equation (16) yields the system of three equations: É…É 89Í 8Í − 2(…É 8 Í )2 = 2ƒ 2 É 6 tÊ â E , ɃhÊ Í89Í 8 − 2…É 8 Í hÊ Í8 = ƒ 2 É 5 Ê 2 â E , ‚ƒ:ÑÐ = ƒ 2 É 5 â E . [

ÉɅ89Í 8Í − 2(Ʌ8 Í )2 ]¤ + ƒÉ ÊhÍ89Í 8

− 2…Í

Its solution gives us the transformation

Ë

:×Ö

= , { , 2, 0}

5

Ï = : õ¤ + ³° : T Í,Î ÎÎÎ {Ó , 2, 0}

where we use the notation:



= (8

5

2

5



+ 40 + 49)−1 2 ,

ô

=



−1 , 2

¤ ¤ :

transformation of the variables, = ( ); transformation of the vector of essential parameters;

Ó

Example 4. Likewise, for the class of equations

= â (Ë )ã (Ï ) (Ï

›

Ï Ò˜Í Í Ò we find two transformations:

©

^_

Õ

›

: {â , ã , } : {â ,

Ï

±

, 1}

ÍÏÎ0Î Í,Î ÎÎÎ

› % Ï ÒÍ )} â (: ), Ï ± , 1}

{ã , â , −( Ï {: ± −

−3

 5

1 [ (2 + 5) − 5], 2

=

ÒÍ

)3 (1

−1

ÒÍ

Ø

 5

= − ( + 2),

°

=

5

5

( + 2)( + 3)



.

)

transformation of the variables; see (14); transformation of the variables; see (18).

References for Subsection 0.6.4: V. F. Zaitsev and A. D. Polyanin (1994), A. D. Polyanin and V. F. Zaitsev (1995).

0.6.5. Discrete-Group Method. The Method of RF-Pairs The direct method (see Subsection 0.6.4) is unsuitable for finding nonpoint transformations of second-order equations (i.e., transformations containing derivatives), since the determining equation cannot be split into equations forming an overdetermined system. Therefore, instead of searching for B¨acklund transformations in the form of arbitrary functions  =  (a , ‡ , ‡ g
), = (a , ‡ , ‡ g
), one uses the superposition of some “standard” transformation containing the derivative and a point transformation which can be found by the direct method. The “standard” dependence on the derivative can be introduced by means of an RF-pair, which amounts to a transformation of successively increasing and decreasing the order of the equation (this transformation is not equivalent to the identity transformation). An additional point-transformation is necessary, since the equation obtained by an RF-pair is usually outside the original class.

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Page 75

1 b . Suppose that any equation of the original class can be solved for the independent variable  :

 ( , 
, 

 ) =  . Termwise differentiation of this equation with respect to  yields the following autonomous equation:



  
+ N


 + N




 = 1,   N N N "
whose order can be reduced with the substitution  = H ( ). This pair of transformations is called N

a first RF-pair.

2 b . Suppose that any equation of the original class can be solved for the dependent variable :

 ( , 
, 

 ) = . Then, termwise differentiation of this equation with respect to  brings us to the following equation which does not explicitly contain :

N N



+ N





N




 

 + N

 N




 = 
.

The order of this equation can be reduced by means of the substitution
 = H ( ). This pair of transformations is called a second RF-pair. Example 1. Consider transformations of the class of generalized Emden–Fowler equations:

Ï Ò˜Í Í Ò

=

 ËEϱ

Ë

ÒÍ

(Ï

).

(1)

This class will be briefly denoted by the vector of essential parameters { 5 , ´ , Ì }. Application of the first RF-pair transforms this equation to ;˜NÍ Í  = (Ì − 1); −1 (;˜Í )2 + ´ Ï −1 ;˜Í + 5³ 1 Ï ± ; Ë −E −1 (;˜Í ) E −1 . (2) E E E E Now we have to find a point transformation that maps class (2) into class (1) (with another vector of parameters): ¤žÍ89Í 8 = ‚ƒ:ÑÐD¤^Ò (¤žÍ ) ” . (3)

Î

Note that in this case, the desired transformation does not map the given class into itself as in Subsection 0.6.4, but is a (1). Nevertheless, the method for finding transformations mapping of the equations classes (2)

Ï

: ¤

= â ( , ),

;

: ¤

=ã (, )

(â

8 ãU¥ − â¥ã 8

≠ 0)

is completely the same and involves solving the determining equation:

8 ⥠) ƒ‚ :ÑÐi¤^Ò (¤žÍ8 ) ” + ( â¥ãU¥i¥ − ãU¥´â¥i¥ )(¤žÍ8 )3 + ( â 8 ãU¥i¥ − ã 8 â¥i¥ + 2 â¥ãU¥ 8 − 2ãU¥´â¥ 8 )(¤žÍ8 )2 + ( â ¥ ã 898 − ãU¥žâ 898 + 2 â 8 ãU¥ 8 − 2ã 8 ⥠8 )¤ Í8 + â 8 ã 898 − ã 8 â 898 = Ì −ã 1 ( â 8 + ⥠¤ Í8 )(ã 8 + ãU¥ ¤ Í8 )2 ´ 8 1 − −1 2 +1 −1 â ± ã Ë E ( â 8 + ⥤žÍ8 ) E (ã 8 + ãU¥ ¤žÍ8 ) E . + ( â + ⥤žÍ8 )2 ( ã 8 + ãU¥ ¤žÍ8 ) + 5³ (4) â E E E E E ¥ ã 8 ãU¥ ≠ 0 and consider transformations Following the procedure set out in Subsection 0.6.4, we omit the general case â 8 â for which at least one of the above partial derivatives is zero. 1 O . Case ⥠= 0, ã 8 = 0. Equation (4) has the form ‚Ñâ 8 ãU¥ :ÑÐi¤^Ò (¤žÍ8 ) ” + â 8 ãU¥i¥ (¤žÍ8 )2 − ãU¥´â 898 ¤žÍ8 = Ì −ã 1 â 8 (ãU¥ )2 (¤ 8 )2 ´ 1 − −1 2 +1 −1 −1 â ± ã Ë E ( â 8 ) E (ãU¥ ) E (¤ 8 ) E . + ( â 8 )2 ãU¥¤ Í8 + 5³ â E E E E E E and for 5 ≠ 0, −1, • ≠ 1, 2 can easily be solved by splitting, â =: Ù , ã =¤ Ú . (â

8 ãU¥

−ã

1 +1

1 −2

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As a result, using an RF-pair, we obtain:

¤ Í8 ) 1 , Ï = : ± 1+1 , Ï ÒÍ = ¤ 2−1 Ë transformation of the variables, ¤ = ¤ ( : ); E ´ 1 5 − 1 Û ÎÎÎÎ ÝÜ − ´ + 1 , Ì − 2 , 5ßÞ transformation of the vector of essential parameters. { 5 , ´ , Ì } Î 2 O . Case â 8 = 0, ãU¥ = 0. Similar calculations bring us to the formulas: Ë = (ž¤ Í8 )− 1 , Ï = ¤ ± 1+1 , ÏÒÍ = : 2−1 transformation of the variables, ¤ = ¤ ( : ); Ë´ E ÎÎÎÎÎ 2 +1 1 5 ´ Ü 1 − Ì , − 5 + 1 , ´ Þ transformation of the vector of essential parameters. {5 , , Ì } Û Ë

=(

(5)

(6)

©

Transformation (6) can be obtained by successive application of transformation (5) and the hodograph transformation (see Subsection 0.6.4). The inverse transformations have a similar structure. For instance, the inverse of transformation (5) can be written (after changing notation) as follows:

Ë

¤

¤ 8 ±1 ,

where ¤ = ¤ ( : ). E Denoting the transformation (7) by à , let us schematically represent its action on the parameters of the equation: ´ Û ÎÎÎÎ Ü 1 1− Ì , − 5 5 + 1 , 2 ´ + 1 Þ transformation à . { 5 , ´ , Ì } Î Applying the transformation à three times, we obtain the original equation. =

1 +1

= ( žÍ )−

Ï

,

ÏÒÍ

=

: 1−1 Ë ,

(7)

(8)

©

It can be shown that all transformations which can be found from equation (4), without additional restrictions on the and parameters of the original and the transformed equations, are obtained by superposition of the transformations (see Subsection 0.6.4, Example 3), which form a group of order 6. The parameters of these equations are given in Figure 1. Example 2. Suppose that

Ì

à

= 0 in equation (1). Then, on the class of Emden–Fowler equations

Õ

Ï Ò˜Í Í Ò

=

5 ´

 ËEϱ

(briefly denoted by { ,

, 0} ),

(9)

(see Subsection 0.6.4, Example 3). Therefore, in this case, the group considered in the one can define the transformation previous example is prolonged to a group of order 12 (see Figure 2). This prolongation takes place each time the third component of the parameter vector becomes equal to zero. This happens, for instance, if = 1 in equation (9). In this case, the order of the group is equal to 24 (see Figure 3).

5

Example 3. The class of second-order equations

= â ( Ë ) ã ( Ï ) ( ÏÒÍ )

›

ÏÒ˜Í Í Ò

(10)

admits a discrete group of transformations similar to that for the generalized Emden–Fowler equation. Most simply, this group can be obtained by inverting the transformation (6). Thus, we seek the parameters of the transformation as functions of a single variable, Ë = (žÍ ), Ï = ( ), ÏÒÍ = ( ).

© Introducing a point generator

¤8

ä

 ¤

á :

(see Subsection 0.6.4), we find a discrete group of transformations relating the equations shown in Figure 4. The functions â 1 ( Ë 1 ), ã 1 ( Ï 1 ), 1 ( Ï ÒÍ 1 ) determine the original equation, while the corresponding functions ( Ï ÒÍ ¼ ) with ô = 2, 3, are determined by the parametric formulas: for the transformed equations, â ( Ë ), ã ( Ï ),

õ

õ



õ

â 2 (Ë

2)

=

ã 2 (Ï

2)

=

2)

=−

› 2 ( and

õ

›

›



õ

Ë

1,

1

â 1 (Ë

ã

1 [ ã 1 ( Ï 1 )]3

â 3 (Ë

3)

=

ã 1 (Ï

ã 3 (Ï

3)

=



3)

=

› 3 (

Ï

,

1)

1

1

æ

1

1)

æÏ æ

1 1



,

Ë

,

Ï

,

â æË1, 1



1 › 1æ ( 1) ,

2

=

X

2

=

X â 1 (Ë

2

=

1

ã 1 (Ï

3

=

X ã 1 (Ï

3

=

X

3

=

â 1 (Ë



1)

› 1 (æ  1

Ë

,

(11)

1)

1)



æ

Ï æ 1

1)

1,

(12)

,

1 ),

where õ = Ï ÒÍ ¼ , ô = 1, 2, 3. The above example allows us to eliminate “singular points” of the group of transformations defined by (7) for = −1, = −1, = 1, 2. For these values of the parameters, the form (10) and the transformations (11), (12) should be used.

´

^_

Ì

5

References for Subsection 0.6.5: V. F. Zaitsev and A. D. Polyanin (1994), A. D. Polyanin and V. F. Zaitsev (1995).

© 2003 by Chapman & Hall/CRC

Page 77

Figure 1

Figure 2

© 2003 by Chapman & Hall/CRC

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Figure 3

Figure 4

© 2003 by Chapman & Hall/CRC

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Chapter 1

First-Order Differential Equations 1.1. Simplest Equations with Arbitrary Functions Integrable in Closed Form* 1.1.1. Equations of the Form â$ä ã = å ( æ ) Solution:**

= X¯ ( )  + L .

M

1.1.2. Equations of the Form

â$äã = å (â )

Solution:  = X

+L . M  ( ) = #  , where #  are roots of the algebraic (transcendental) equation Particular solutions:  ( #  ) = 0.

1.1.3. Separable Equations

â$äã = å ( æ )ç (â )

= Xv ( )  + L . M ( ) M = #  , where #  are roots of the algebraic (transcendental) equation Particular solutions: ( #  ) = 0. x3y'z|{}~/€ The equation of the form  1 ( ) 1 ( )
 =  2 ( ) 2 ( ) is reduced to the form 1.1.3 by dividing both sides by  1 1 . Solution: X

1.1.4. Linear Equation Solution:

=L  Ž + Ž X

ç (æ )â$äã = å 1 ( æ )â + å 0 ( æ )  ( )  ,  −Ž 0 ( ) M

1.1.5. Bernoulli Equation

where

 ( ) = X

ç ( æ )â$äã = å 1 ( æ )â + å„è ( æ )â è

Here, B is an arbitrary number. The substitution ¨ ( ) = ¨2
= (1 − B )  ( )¨ + (1 − B )  ( ). 1  Solution: 1−

 1 ( )  . ( ) M

 = L  Ž + (1 − B )  Ž XZ − Ž   ( )  , ( ) M

where

1−

 leads to a linear equation:

 ( ) = (1 − B ) X

 1 ( )  . ( ) M

* Special cases of equations 1.1.1–1.1.5 for specific functions  ,  0 ,  1 ,   , and are not discussed in this book; such cases can readily be recognized by the appearance of equations investigated, and the solution can be obtained using the general formulas given in Section 1.1. ** Hereinafter we shall often use the term “solution” to mean “general solution.”

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Page 81

1.1.6. Homogeneous Equation

â$äã = å (âêéwæ )

The substitution ‡ ( ) =   leads to a separable equation: ‡
 =  (‡ ) − ‡ . Solution: X

‡

M

= ln | | + L .

 (‡ ) − ‡

= #   , where #  are roots of the algebraic (transcendental) equation

Particular solutions: #  −  ( #  ) = 0.

1.2. Riccati Equation ë ( ì ) í­ï î = ð 2 ( ì ) í

2

+ ð 1(ì )í + ð 0(ì )

1.2.1. Preliminary Remarks For  2 ≡ 0, we obtain a linear equation (see Subsection 1.1.4); and for  0 ≡ 0, we have a Bernoulli equation (see Subsection 1.1.5 with B = 2), whose solutions were given previously. Below we discuss equations with  0  2 1 0. 1 b . Given a particular solution as: =

0 (  ) + ` (  ) ¬ L − X¯` (  )

0

=

0(

 ) of the Riccati equation, the general solution can be written

 2 ( )  ® ( ) M

−1

 , where ` ( ) = exp ™ X V 2  2 ( ) 0 ( ) +  1 ( )W M ¢ . ( )

To the particular solution 0 ( ) there corresponds L = F . Often only particular solutions will be given for the specific equations presented below in Subsections 1.2.2–1.2.8. The general solutions of these equations can be obtained by the above formulas. 2 b . The substitution

‡ ( ) = exp O − X



2

 M P

reduces the Riccati equation to a second-order linear equation:

 2 2 ‡

 + V  2 
− (  2 )
 −  1  2 W‡
 +  0  22 ‡ = 0. The latter often may be easier to solve than the original Riccati equation. Specific second-order linear equations are outlined in Section 2.1.

1.2.2. Equations Containing Power Functions 1.2.2-1. Equations of the form ( )
 =  2 ( ) 1.

ñóò

ôñ

=

2

+

õï

2

+  0 ( ).

+ö .

For  = 0, we have a separable equation of the form 1.1.2. For  ≠ 0, the substitution 'a = ' + ( leads to an equation of the form 1.2.2.4: g
=  2 + 'a . 2.

ñóò

ñ

2

=

–ô

2

ï

2

+ 3ô .

Particular solution: 3.

ñ óò

=

ñ

2

+

ô ï

2

+

õï

0

=  − 

−1

.

+ö .

This is a special case of equation 1.2.2.27 with ƒ = 0 and ý = 0.

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1.2. RICCATI EQUATION ã ( Ë ) Ï

4.

ñóò

ôñ

=

2

õ ïƒ÷

+

ÒÍ

=

â 2 (Ë )Ï

2

+ â 1 (Ë )Ï +

â 0 (Ë

)

.

Special Riccati equation, B is an arbitrary number. 1 ¨2
= − ¨ , where ¨ ( ) = £  L 1 ø Solution:



Q

1 2



1£

O

,

"  P

+L

2

n

1 2



O

1£

,

"  S , P

, = 12 (B + 2); ø + ( H ) and n + ( H ) are the Bessel functions, B ≠ −2. For the case B = −2, see

equation 1.2.2.13. 5.

ñ óò

ñ

2

=

ôù ï ÷

+

–ô

–1

Particular solution: 6.

ñ óò

ôñ

=

2

õ ï 2÷

+

0

ï ÷

+ö

ï 2÷

2

.

=   . –1

. 1

  B +1 B ( . ‰ =  −  leads to an equation of the form 1.2.2.38: ˆ‰ Š
+ ˆ‰ 2 + ‰ = ' ˆ + B +1 B +1

For the case B = −1, see equation 1.2.2.13. For B ≠ −1, the transformation ˆ =

7.

ñ óò

ô ï ÷ ñ

=

2

õ ï –÷

+

–2

.

‡ + L , where ‡ = M 2 ‡ + ý±‡ + 1

Solution: £  ln  = X 8.

ñóò

ô ƒï ÷ ñ

=

2

õ ïƒú

+



2



+

B +1 B +1 2 b . For B = −1 and ¡

+ − ˆ  +1 .

ñóò

=

ñ

2

  

+1

B +1 , ý = £ .  

leads to a Riccati equation of the form 1.2.2.4:

≠ −1, the transformation





=  +

+1

, ¨

= −1 

leads to a Riccati

¨ 2+ −1 . +1 +1 ¡ ¡ = −1, the original equation is a separable equation. In this case we have the

¡

solution: ln | | = X 9.

+1



equation of the form 1.2.2.4: ¨ û
= 3 b . For B =



,

.

1 b . For B ≠ −1, the substitution ˆ =  

Š
=

+1

ï

+ ü (ô

M2 

+ õ )÷ ( ö

+

ï

+L . + ý )– ÷

–4

.

1  +  , ‡ = [( (˜ + )2 + ( ( (˜ + )], where ¢ =  − ˜( , leads (˜ + ¢
M M M M ‡ Š = ‡ 2 + , ¢ −2 ˆ  . to an equation of the form 1.2.2.4: The transformation ˆ =

10.

ñóò

=

ô ƒï ÷ ñ

2

õÿþ ïƒú

+

Particular solution: 11. 12.

13.

ñ óò

= (ô

ï 2÷

+

õï ÷

–1

–1

– ôƒõ

2

 + . 0 = '

)ñ

2

ïƒ÷ +2ú

.

+ö .

¨ 
= ( ¨ The substitution = −1 ¨ leads to an equation of the form 1.2.2.6: I

2

+  2  + ' 

−1

.

( ô 2 ï + õ 2 )(ñ ó ò + ƒñ 2 ) + ô 0 ï + õ 0 = 0. The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.108: (  2  +  2 )‡

 + Œ (  0  +  0 )‡ = 0.

ï 2 ñ óò

=

ô ï 2ñ

Solution:

Œ

2

2

=



Œ

+õ .

+ Π+  = 0.

−  2  O

   2  + L 2 Œ + 1 P

−1

, where Πis a root of the quadratic equation

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14.

ï 2 ñ óò

ï 2ñ

=

–ô

2

2

ï

+ ô (1 – 2 õ )ï

4

Particular solution: 15.

ï 2 ñ óò

=

ï 2 ñ óò

=

ô ï 2ñ

2

+

õ ïƒ÷

−1

=  + '

0

– õ ( õ + 1).

2

.

+ö .

The substitution ¨ = 

+ # , where # is a root of the quadratic equation !# leads to an equation of the form 1.2.2.35:  ¨u
=  ¨ 2 + (1 − 2 !# )¨ + '  . 16.

ï 2ñ

2

ô ï 2ú ( õ ïƒú

+

+ ö )÷ + 41 (1 – ù

The transformation ˆ = ' + + ( , ¨

form 1.2.2.4: ¨3Š
= ¨ 17.

(ö 2 ï

18.

ï 4 ñ óò

+ õ 2ï

2

óò

+ ô 2 )(ñ

Ė

+

) ˆ  . −2

¡ 2

)+

ô

0



1

¡

).

1−



= –ï

ô ï 2 (ï

+

+

1− ¡ 2

¡

 −+

leads to an equation of the

= 0.

ñ

– ô 2. 1  = +

4 2



óò

– 1)2 (ñ

+



2

Ė

2

tan O )+

õï





+L 2

+ö

ï

P

. + s = 0.

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.218:  2 ( − 1)2 ‡

 + Œ ( ' 2 + (˜ + ° )‡ = 0.

ï

20.

(ô

21.

ïƒ÷ +1 ñ óò

2

õï

+

+ ö )2 (ñó ò + ñ 2 ) +




= 0.

The substitution = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.234: (  2 + ' + ( )2 ‡

 + #*‡ = 0. =

ô ï 2÷ ñ

2

+

ö ïƒú

+ý .

The substitution ¨ =  

+ # , where # is a root of the quadratic equation !  # leads to an equation of the form 1.2.2.35:  ¨ 
=  ¨ 2 + (B − 2 !# )¨ + (˜ + . 22.

(ô

ïƒ÷

+ õ )ñó ò =

õÿñ

2

(ô

ïƒ÷

+

õ ïƒú

+

ô ïƒ÷

–2

2

− B±# + = 0,

M

.

0 = −1  .

Particular solution: 23.

− # + ( = 0,

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.179: ( ( 2  2 +  2  +  2 )‡

 + Œ 0 ‡ = 0.

Solution: 19.

+  (

2

=

2

2

+ ö )(ñó ò – ñ 2 ) +

ôù (ù

– 1)ïw÷

B/  −1 +   + ¡ 0 =−   + ' + + (

Particular solution:

–2 −1

+

õÿþ (þ

– 1) ïƒú

–2

= 0.

.

1.2.2-2. Other equations. 24.

ñóò

ôñ

=

2

+

õÿñ

+ö

ï

+ü .

¨2


=− ¨ ¨2

 =  ¨2
−  ( (˜ + , )¨  .

The substitution

25.

+

ô ï ÷

Particular solution:

0

ñ óò

=

ñ

2

+

ô ï ÷ ñ

–1

leads to a second-order linear equation of the form 2.1.2.12:

.

= −1  .

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Page 84

1.2. RICCATI EQUATION ã ( Ë ) Ï

26.

27.

28.

ñóò

=

ñ

2

ñ óò

=

ñ

2

ñ óò

=

ñ

2

ô ƒï ÷ ñ

+

+

õ ïƒ÷

ï

+ (

ô ï ÷ ñ

+

ñóò

= –(ù + 1)ïw÷

ñ

0 2

ñ óò

ô ï ÷ ñ

=

2

+

+

õï ú ñ

ñóò

ô ƒï ÷ ñ

=

2

–ô

ñóò

= – ôù

ïƒ÷ –1 ñ

2

+

ñóò

ô ƒï ÷ ñ

=

2

+

35.

36. 37.

38.

39. 40.

0

2

+

õï

+ö .

ñ

–ô

=.

ô ïƒ÷ +ú =  −

+ 0

+1 −1

õ^ö ï ú

– ô„ö

= −( .

â 0 (Ë

85

)

2

  + +( . 0 = '

+ õ )ñ – ö

   +  )−1 . 0 = ( + ö³ü 0

ï ÷

.

.

õÿþ ïƒú

+ ö )ñ +

–1

ïƒú

ï –1 – õ^ö ïƒú +

= (˜  .

= ôñ 2 + õÿñ + ö ï 2  . The transformation a =  , ¨ = 

ï ñóò

ïƒú

.

ö ïƒú (ô ïƒ÷

õ ïƒú ñ

Particular solution: 34.

+ â 1 (Ë )Ï +

– õ 2.

ïƒ÷ ( õ ƒï ú

Particular solution: 33.

ï

ï ÷

– ôƒõ

Particular solution: 32.

2

The substitution = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.31: ‡

 − ( ƒ… + ý )‡
 + (  2 + ' + ( )‡ = 0.

Particular solution: 31.

â 2 (Ë )Ï

.

+  )ñ + ô

Particular solution: 30.

=

The substitution = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.45: ‡

 −   ‡
 + '   −1 ‡ = 0.

Particular solution: 29.

–1

ÒÍ

−



.

. – ô„ö

2

ïƒ÷ +2

.

¨ g
=  ¨ leads to a separable equation:  3

2

+( .

= ôñ 2 + õÿñ + ö ïƒ÷ . The transformation ˆ =  , ‰ =  −  leads to the special Riccati equation of the form 1.2.2.4:  ( B ‰ Š
= ‰ 2 + ˆ + , where = − 2.

ï ñóò





¡



= ôñ 2 + (ù + õ ïƒ÷ )ñ + ö ï 2÷ . ¨ 
=   The substitution = ¨   leads to a separable equation: 3

ï ñóò

= ï ñ 2 + ôñ + õ ïƒ÷ . The substitution = −‡
 ‡ ‡

 − ‡
 + '  ‡ = 0.

ï ñóò

−1

( ¨

2

+  ¨ + ( ).

leads to a second-order linear equation of the form 2.1.2.67:

ï ñóò

+ ô 3 ï ñ 2 + ô 2 ñ + ô 1 ï + ô 0 = 0. The substitution  3 = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.64: ‡

 +  2 ‡
 +  3 (  1  +  0 )‡ = 0. = ô ïƒ÷ ñ 2 + õÿñ + ö ï –÷ . ¨ 
=  ¨ The substitution ¨ =   leads to a separable equation:  3

ï ñóò

= ô ïƒ÷ ñ 2 + þxñ – ôƒõ 2 ïƒ÷ +2ú . Particular solution: 0 = ' + .

ï ñóò

2

+ (  + B )¨ + ( .

© 2003 by Chapman & Hall/CRC

Page 85

41.

ï ñóò

=

ï 2÷ ñ

= +

Solution: 42.

43.

– ù )ñ + ï

+ (þ

2

ï ñóò

=

ô ƒï ÷ ñ

ï ñ óò

=

ô ï 2÷ ñ

2

−

  ++

 tan O

õÿñ

+

+ (õ

ï ÷

+

2

B +

ö ïƒú

ú

+L

¡

.

. .

P

The transformation ˆ =   −  , ‰ =   leads to a special Riccati equation of the form 1.2.2.4: − B − 2 . (B +  )‰ Š
= ‰ 2 + (˜ˆ  , where , = ¡ B + 2

– ù )ñ + ö .

For B = 0, this is a separable equation. For B ≠ 0, the solution is:

BIX 44.

ï ñ óò

=

ô ï 2÷ +ú ñ

 ¨

ï ÷ +ú

+ (õ

2

¨ =  +L , M ¨ + +(

2

– ù )ñ + ö

ï ú

õ 1 )ñ

ô 0ï

where

¨ =   .

.

¨ 
=   + + The substitution ¨ =   leads to a separable equation: 3

45.

(ô

46.

(ô

2

ï

õ

+

2 )(

ñ óò

+

Ė

2

) + (ô

1

ï

+

+

+

õ

0

−1

( ¨

2

+  ¨ + ( ).

= 0.

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.108: (  2  +  2 )‡

 + (  1  +  1 )‡
 + Œ (  0  +  0 )‡ = 0.

ï

õï

+ ö )ñ„ó ò =

 (ô ñ

+

ñóò

+

ï ñ

– 2ô

)2 +  ( ôñ +

õï

)–

õï

+ .

The substitution a =  + ' leads to a first-order linear equation with respect to  =  (a ):   +( . ( ƒha 2 + ý…a + =/ + ˜( ) g
= 

47.

2ï

2

= 2ñ

2

2ï

2

ñóò

= 2ñ

2

+ 3ï

– 2ô

ñ

Particular solution: 49. 50.

51.

52.

53.

ô ï 2ñ

2

=

ö ï 2ñ

2

ï 2 ñ óò

=

ô ï 2ñ

2

ï 2 ñ óò

=

ô ï 2ñ

2

ï 2 ñ óò

=

ö ï 2ñ

2

ï 2 ñ óò

=

ï 2 ñ óò

+

0

õï ñ

The substitution ¨ =  + (ô

ï

.

£  . 0 = 

Particular solution: 48.

2

ï

2

2

ï

. =  £  − 12  .

+ö .

¨ 
=  ¨ leads to a separable equation:  p

+

õ ï )ñ

+

 ï

+ s.

2

+

ï

2

+ (  + 1)¨ + ( .

+ .

The substitution ( = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.139:  2 ‡

 −  (  +  )‡
 + ( ( ƒ… 2 + ý± + = )‡ = 0. +

õï ñ

+

öï ÷

+

õï ñ

+

ö ï 2÷

The substitution  = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.132:  2 ‡

 − '‡
 +  ( (˜  + ° )‡ = 0. + s ïƒ÷ .

The substitution  = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.133:  2 ‡

 − '‡
 +   ( (˜  + ° )‡ = 0. + (ô

ïƒ÷

+ õ )ï

ñ

+

ï 2÷

+

ïƒ÷

+ . The substitution ( = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.146:  2 ‡

 − (   +  )  ‡
 + ( ( ƒ… 2  + ý±  + = )‡ = 0.

© 2003 by Chapman & Hall/CRC

Page 86

1.2. RICCATI EQUATION ã ( Ë ) Ï

54.

55.

ï 2 ñ óò

ï 2÷

= (

ïƒ÷

+

+ )ñ

ïƒ÷

+ (ô

2

ÒÍ

â 2 (Ë )Ï

=

+ õ )ï

ñ

+ â 1 (Ë )Ï +

2

öï

+

2

â 0 (Ë

87

)

.

¨ 
= (˜ 2 ¨ The substitution = −1 ¨ leads to an equation of the form 1.2.2.53:  2 2 ¨ (   +  ) + ƒ… 2  + ý±  + = . (ï

óò

– 1)ñ

2

+ (ñ

– 2ï

2

−

+ 1) = 0.

2Œ − 1

=

The substitution

ñ

2

Œ

 + 2

(

1−Œ

1

Œ

leads to an equation of the same form:

‡ ( )

− 1)‡
 + ( Œ − 1)(‡

2

− 2‡ + 1) = 0.

If Œ = B is a positive integer, then by using the above substitution, the original equation can be reduced to an equation of the same form in which Œ = 1, i.e., to an equation of the form 1.2.2.58 with  = 1,  = −1. 56.

ï

(ô

2

+ õ )ñó ò +

ñ

2

Particular solution: 57.

ï

(ô

2

+ õ )ñ

óò

+

ñ

0

2

õ (ô

ï ñ

+

=−

 +ý  . ƒ

ï ñ

+ = 0.

+

+



+  ) = 0.

1  +ý  − leads to an equation of the same form: (  ƒ ‡ ( )  +ý  ‡ 2 + (2  + ý )‡ + ƒ = 0. O = − ƒ P =−

The substitution

58.

ï

(ô

2

+ õ )ñó ò + ñ

Solution: 59.

ï

(ô

2

+

õï

2

– 2ï

=  + OX + ö )ñó ò =



ñ

(ô

ï

2

+

õï

+ ö )ñó ò =

Particular solution:

ñ

61.

(ô

ï

2

2

+

õ 2ï

62.

(ô

2

ï

2

+

õ 2ï



+L

+

ï

−1

P

.

+ õ )ñ + ( – ô )ï

0

= − Œ  + # , where # =

2

+ (ô

ï

+  )‡
 +

– õ = 0.

+ (2

+ )ñ –

2

ï

2

2

1 2

+ .

 − ±A ü

 2 − 4“ − 4 Œ(  .

+ (õ – )ï +

wö

.

=Π .

0

+ ö 2 )ñó ò =

Particular solution:

M2

2

2

Particular solutions: 60.

+ (1 – ô ) ï

ñ

2

ñ

2

+ (ô

1

ï

+

õ 1 )ñ

– ( +

1

ï

+

õ 1 )ñ

+ô

– ô 2)ï

ô

1

+

õ 0ï

2

+ (õ

2

–

õ 1 )ï

+

wö

2.

=Π .

0

+ ö 2 )ñó ò =

ñ

2

+ (ô

0

ï

2

+ ö 0.

Let Œ and ý be roots of the system of the quadratic equations

Œ

2

+ Π(

1

−  2) + 

0

= 0,

ý

2

+ ýh 1 + (

0

− Œ(

2

= 0,

where the first equation is solved independently (in the general case there are four roots). If some roots satisfy the condition 2 Œ ý + Œ" 1 + ý… 1 +  0 − Œ" 2 = 0, the original equation possesses a particular solution: 0 = Œ  + ý .

© 2003 by Chapman & Hall/CRC

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88 63.

FIRST-ORDER DIFFERENTIAL EQUATIONS ( ï – ô )( ï – õ )ñ„ó ò + ñ

2

+ ü (ñ +

ï

– ô )(ñ + ï – õ ) = 0.

To the case , = 0 there corresponds a separable equation. To , = −1 there corresponds a linear equation. For , ≠ −1 and , ≠ 0, with the aid of the substitution ,!‡ ( ) = + , ( +  ), we obtain the general solution:

+

õ 2ï

ï 3 ñ óò

=

ô ï 3ñ

2

ï 3 ñ óò

=

ô ï 3ñ

2

ï (ï

+ ô )(ñó ò +

64.

(ö

65.

66.

67.

2

ï

+, ( + − )  −  O =L + , ( +  −  )  − hP 1 1 + =L +, ( + − )  −

2

+ ô 2 )(ñó ò +

69.

70.

71.

72.

73.

2

) + (õ

1

ï

+ ô 1 )ñ + ô

0

if  =  .

= 0.

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.179: ( ( 2  2 +  2  +  2 )‡

 + (  1  +  1 )‡
 + Œ 0 ‡ = 0. + (õ

ï

2

+ ö )ñ + s ï .

The substitution  = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.183:  3 ‡

 − ( ' 2 + ( )‡
 + ! °˜‡ = 0.

ï

+ ï (õ

+ ö )ñ + 

ï

+ .

The substitution  = −‡
 ‡ leads to a second-order linear equation of the form 2.1.2.186:  3 ‡

 −  ( ' + ( )‡
 +  ( ƒ… + ý )‡ = 0. 2

Ė

The substitution Œ  ( 2 +  )‡

 + ( ' 68.

Ė

if  ≠  ,

ï 2(ï

+ ô )(ñó ò +

2

= ‡
 + ( )‡

2

Ė

) + (õ

2

ï

+ ö )ñ + s ï = 0.

2

‡ leads to a second-order linear equation of the form 2.1.2.190:
 + Œ°˜‡ = 0.

ï

) + ï (õ

ï

+ ö )ñ + 

+

= 0.

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.194:  2 ( +  )‡ 

 +  ( ' + ( )‡
 + Œ ( ƒ… + ý )‡ = 0.

ï

õï

+ ö )( ï ñ ó ò – ñ ) – ñ 2 + ï − Solution: ln û û = L + 2 X  û + û

(ô

2

ï 2(ï

+

û

û

+ ô )(ñó ò +

Ė

– 1)2 (ñó ò +

Ė

2

ô ï 2÷ ñ

2

+

2

2

ï

) + ï (õ

2

2

= 0.

2

+ ' + (

M



.

+ ö )ñ + s = 0.

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.219:  2 ( 2 +  )‡

 +  ( ' 2 + ( )‡
 + Œ°˜‡ = 0.

ô (ï

2

õ ï (ï

)+

2

– 1)ñ +

öï

2

+ý

ï

+ s = 0.

The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.227:  ( 2 − 1)2 ‡

 + ' ( 2 − 1)‡
 + Œ ( (˜ 2 +  + ° )‡ = 0. M

ïƒ÷ +1 ñ óò

=

õ ƒï ÷ ñ

+

ö ïƒú

+ý .

 # 2 −(  + B ) # + = 0, The substitution ¨ =   + # , where # is a root of the quadratic equation ! M u ¨
¨ ¨ 2 + (B +  − 2 !# ) + (˜ + . leads to an equation of the form 1.2.2.35:   = 

ï (ô ï

+ õ )ñó ò =

 ƒï ÷ ñ

2

The transformation a =   − , ( ˜H +  ).

+ ( – ôù

ï )ñ

+

ï –÷

. , H =  −  leads to a separable equation: [ƒ…a 2 + (ý + 'B )a + = ]H g
=

© 2003 by Chapman & Hall/CRC

Page 88

1.2. RICCATI EQUATION ã ( Ë ) Ï

74.

75. 76.

ÒÍ

â 2 (Ë )Ï

=

2

+ â 1 (Ë )Ï +

â 0 (Ë

89

)

ï 2 (ô ïƒ÷

– 1)(ñó ò + ƒñ 2 ) + ( ïƒ÷ + ) ï ñ + ïƒ÷ + s = 0. The substitution Œ = ‡
 ‡ leads to a second-order linear equation of the form 2.1.2.254:  2 (   − 1)‡

 + (  +  )‡
 + Œ (&  + ° )‡ = 0.

( ô ïƒ÷ + õ ïƒú + ö )ñó ò = ö^ñ 2 – õ ïƒú Particular solution: 0 = −1  . ( ô ï ÷ + õ ï ú + ö )ñ Particular solution:

óò

=

ô ï ÷ –2 ñ

( ô ïƒ÷ + õ ïƒú + ö )ñó ò =  ï ñ Particular solution: 0 = − Œ .

78.

(ô

+

õï ú

+ ö )( ï

2

+

= .

0

77.

ï ÷

–1

+

2

– ñ ) + sï

ñ óò

ñ

–2

.

õï ú

–1

ñ

+ö .

ï sñ

– 

2

 (ñ

Particular solutions: 0 = A" £ Œ .

ïƒ÷

+ô

2

ï

–

2

ï

+ 

ï

s

.

) = 0.

1.2.3. Equations Containing Exponential Functions 1.2.3-1. Equations with exponential functions. 1.

ñóò

ôñ

=

2

+

õ ó

.



The substitution a = ž 2.

ñ óò

ñ

2

=

+

ô  ó

–ô

Particular solution: 3.

4.

5.

ñóò

=

ñ

2

ñóò

=

ñ

2

ñ óò

=

+

ô

+

ôñ

õ ó

+

+

0

+ 'a .



=   . 2

ó



.

+ö .

The substitution . = −‡
 ‡ leads to a second-order linear equation of the form 2.1.3.10:  ‡

 − ‡
 + . (   + ( )‡ = 0.

ñ

2

+

õÿñ

+ ô ( – õ )

ñóò

ñ

2

=

+

ô ó ñ

0

– ôƒõ

ñóò = ñ 2 + ô 2 ó (  ó

0

1.2.2.4: ¨3Š
= ¨

ñ óò

=

ñ

2

+

2

ô 8 ó

+ Π+

ó





–ô

 

2 2

ó

–

õ

2

2

+ (2 Œ )−2 ( ˆ

=.

+ õ )÷ –

−2



ˆ  .

õ 6 ó 2

.

.

+



1 4

2

.

+ , ¨

=

ö 4 ó

–

The transformation ˆ =  2  , ¨ = 

¨2Š
= ¨

ó

=   .

The transformation ˆ =  

8.

.

+ ö

õ ó

Particular solution: 7.

ó

 

2 2

2

The substitution . = −‡
 ‡ leads to a second-order linear equation of the form 2.1.3.5:   ‡

 + . (  +   + (  2  )‡ = 0.

Particular solution: 6.

leads to an equation of the form 1.2.2.35: Πa 
g = 

+ 'ˆ + ( ).

−2

1

O  −

Œ

2





−

Œ

2

 −

 P

leads to an equation of the form

.

 O

2Œ

−

1 leads to an equation of the form 1.2.2.3: 2P

© 2003 by Chapman & Hall/CRC

Page 89

90 9.

10.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñ óò

ô  ó ñ

=

2

õ só

+

ñ óò

õ ó ñ

=

2

+ ô

ñ óò

ô  ó ñ

=

2

+

ó



ñ óò

ô ó ñ

=

2

+

õÿñ

+

Ė

ñóò

ó

 ñ

=

2



14.

ñóò

óñ

= – 

2

15.

ñóò

ô  ó ñ

=

2

+

ó

0

16.

17.

18.

19.

ñóò =À ô   ó ñ

2

+

õÿñ

ñ óò

ô (2 + )ó ñ

=

2

+ ( + Π)H + ( .

.



ô ( – )ó

.



−

= −Œ   . – ô

 

( – )



=  − . –



ö só

ó

.

õ ó

= −   .

+ [ õ

+ ý

ñ óò

ô  ó ñ

=

2

+

õÿñ

+

 

( + )

 The substitution ¨ = ž

–

ó

.

.

ö  ÷ ó

ó

ó

– ]ñ + ö

.

¨ 
=  (  leads to a separable equation: 3 + ý

 (2÷

+1)

ó

2

  ( ¨

+ )

g
= a 2

2

ñóò

ó (ñ

 

=

–

õ ó

ó

( ô

+

õ  ó

ó ( ô

ó + õ 

)2 + 0

2

+

=− .

ü ó ñ

¡

+ ö )(ñó ò – ñ ) + ô

Particular solution:

+

2

+ 'a

+

.



ñ

+ 'a

.

=   .

+ ö )ñó ò = 0

õ ó

2

+  ¨ + ( ).

 The substitution a =   leads to an equation of the form 1.2.2.52: ,!a 2 
g = a 2 (˜a  +1 + a 2(  +1) . M

Particular solution:

21.

0

+

ó

2

 , a The substitution a =   leads to an equation of the form 1.2.2.51: ! (˜a (  + ) J' + . M

Particular solution: 20.

  

ôƒõ ( + )ó ñ

Particular solution:

.

.

2 ( +2 )

 ñ

Particular solution:

−

=   .

0

+ ô

+ 'a

.

leads to a separable equation: H 
= !H

+

Particular solution:

õ ( +2 )ó 

ö – ó

0

ô  ó ñ

+

2

2

=   .

– ôƒõ

Particular solution: 13.

–ô

0

The substitution H = ž 12.

À

≠ 0.

 The substitution a =   leads to an equation of the form 1.2.2.4: , 
g =  Particular solution:

11.

ü

,

2



2



–þ

ó

2

+

 Œ   + '“     0 =−    +   +(

+

üIþ ó

õ 2   ó

.

= 0.

.

1.2.3-2. Equations with power and exponential functions. 22.

ñ óò

=

ñ

2

+

ô ï óñ

Particular solution:

+ ô 0



ó

.

= −1  .

© 2003 by Chapman & Hall/CRC

Page 90

1.2. RICCATI EQUATION ã ( Ë ) Ï

23.

ñ óò

ô  ó ñ

=

2

ñóò

2

+

õÿù ïƒ÷

Particular solution: 25.

ñóò

ó

 ñ

=

2

ô ƒï ÷ ñ

+

Particular solution: 26.

ñóò

= – 

óñ

ñóò

ô ó ñ

=

– ôƒõ

2

ñóò

ô ƒï ÷ ñ

=

2

0

29.

ñóò

ô ƒï ÷ ñ

=

2

Ė

+

ñóò

ô ƒï ÷ ñ

=

2

– ôƒõ

ñ óò

= –( ü + 1)ï

ñ

34.

2

0

+

0

–

õï ÷

ï ñ óò

=

ô  ó ñ

Solution: 36.

ï ñóò

=

ñóò

=

ñ

2

+

ô ï 2÷   ó ñ 2

ïƒ÷  2 ó

+ 2 ô

 ¨

2

=   .

õ ó

+



+

2

=   . =  −

−1

.

+ ö )ñ +

=  



ó

.

õ  ó

.

– ô

¨

õÿù ï ÷

ôƒõ 2 ï 2   ó −1

ïƒ÷  ó

–ô 0

ï ÷

– )ñ + ö

.

¨ 
=     (  ¨ leads to a separable equation: 3

+ (õ

2



+( .

 

–1

 

ó

2 2

=  





2

2

2

+  ¨ + ( ).

.

.

M

 +L

– ù )ñ + ö

M = XÈ  + ¨ +(

Particular solution:

.

   +( . 0 = '

üIñ

ï  ó

.

.



= '   tan O X  

Solution: X 37.

2

– ö )2 +

Particular solution: 35.



.

=   .

 The substitution ¨ = ž

ô  ó (ñ

–1

ïƒ÷  2 ó

2

ô ï  +1   ó ñ

0

.

.

õÿù ïƒ÷

+

ó 2 + (õ ï ÷  

ñ óò = ô ï ÷  2 ó ñ =



= '  .

ñ óò = ô ï ÷ ñ 2 – ô ï ÷ ( õ  ó

ñ óò

91

)

.

ïƒ÷

–ô

=  − .

ïƒ÷  ó ñ

Particular solution: 33.



– ôƒõ

0

Particular solution: 32.

â 0 (Ë

= −Œ  − .

– ôƒõ

Particular solution: 31.

0

0

Particular solution: 30.

ïƒ÷  – ó

ïƒ÷  ó ñ

Particular solution:

ó  ï 2÷

2

  . 0 = '

ïƒ÷  ó ñ

õ ó

+

– ôƒõ

–1

+ ô

Particular solution: 28.

+ â 1 (Ë )Ï +

.

0

+ô

2

Particular solution: 27.

2

M

2

ô ó ñ

=

õ – ó

â 2 (Ë )Ï

=

H  =  + L , where H = ž . !H + ŒH + 

Solution: X 24.

+

ÒÍ

−1

 



M

P

ó

. .

 + L , where ¨ =   .

.

.

© 2003 by Chapman & Hall/CRC

Page 91

92 38.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñ óò

ô – ó ñ 2

=

ñóò

ô ƒï ÷ ñ

=

2

+

ôƒõ

+ ôƒõ

2

2

.

ï ñ

+

ïƒ÷  ó

2

.

2 2 =    J 2 tan O/XÈ    J 2  + L . M P

Solution: 40.

ï ñ

+

2 2 =    J 2 tan O Z X  − J 2  + L . M P

Solution: 39.

2

– ñ 2 ) = ô + õ exp( ü ï ) + ö exp(2 ü ï ). ˆ = 1  , ¨ = − 2 −  leads to a Riccati equation of the form 1.2.3.3: The transformations ¨ Š
= ¨ 2 +  +    Š + (  2  Š .

ï 4 (ñ ó ò

1.2.4. Equations Containing Hyperbolic Functions 1.2.4-1. Equations with hyperbolic sine and cosine. 1.

ñ óò

=

ñ

2

–ô

2

Particular solution: 2. 3.

= ñ 2 + ô sinh( Particular solution:

2

ñ óò

=

ï )ñ

=  cosh( Π ).

0

+ ôƒõ sinh( = − .

sinh(

ñóò

= [ ô sinh2 (

ï

0 2

sinh3 ( 0

0

ï

= coth( Π ).

6.

[ ô sinh( ï ) + õ ]ñ„ó ò = ñ 2 + ö sinh(

Particular solution: 0 = − .

7.

[ ô sinh(

ï

M

9. 10.

ñ óò

=

ñ

2

ï

)+

ï )ñ

–ý

–ô . 2

+ö

ý

sinh(

ï

).

ô

2

+  + cosh ï .

The transformation  = 2a , ƒ = −‡
 ‡ leads to the modified Mathieu equation 2.1.4.9: ‡ g

g − (  − 2  cosh 2a )‡ = 0, where  = −4ƒ…ý ,  = 2ƒ±= .

ñóò

= ñ 2 + ô cosh( Particular solution:

ñ óò = ñ + ô ï 2

ñóò

ï )ñ

0

+ ôƒõ cosh( = − .

coshú ( õ ï )ñ

Particular solution: 11.

0

).

sinh( ï ) = 0. Œ cosh( Œ  ) =− .  sinh( Œ  ) + 

) + õ ](ñ„ó ò – ñ 2 ) +

Particular solution:

ï

).

– ô sinh2 (

2

).

) – õ 2.

= cosh( Π ).

) – ]ñ

Particular solution:

8.

ï

+ ô sinh ú ( õ = −1  .

–

ï

sinh2 (

ñóò = ñ 2 + ô ï sinhú ( õ ï )ñ

Particular solution: 5.

)–ô

0

ï )ñ

ñ óò

Particular solution: 4.

ï

+ ô sinh(

= [ ô cosh2 (

ï

0

0

)–

+ ô coshú ( õ = −1  .

) – ]ñ

Particular solution:

ï

2

+

ô

+

ï

õ

2

.

).

– ô cosh2 (

ï

).

= tanh( Π ).

© 2003 by Chapman & Hall/CRC

Page 92

1.2. RICCATI EQUATION ã ( Ë ) Ï

12.

2ñó ò = [ ô –

ô

+

Particular solution: 13.

ñóò

ñ

2

=

–

2

+

ï )]ñ

cosh( cosh÷ (

ô

) sinh–÷

ï

â 2 (Ë )Ï

=

–4

ï

(

+ â 1 (Ë )Ï +

2

– ô cosh(

Π .

1 2

= tanh 

0

+ô +

2

ÒÍ

ï

â 0 (Ë

)

93

).

).

1 The transformation ˆ = coth( Œ  ), ¨ = − sinh2 ( Œ  ) −sinh( Œ  ) cosh( Œ  ) leads to an equation

of the form 1.2.2.4: ¨3Š
= ¨ 14.

ñ óò

ô

=

sinh(

ï )ñ

2

+Œ

ˆ  .

Œ

) cosh ÷ (

ï

+ õ sinh(

2

−2

The transformation ˆ = cosh( Œ  ), ¨

¨2Š
= ¨

15.

ñóò

2

ô

=

+ Œ

cosh(

−2

=

ˆ  .

ï )ñ

2

The transformation ˆ = sinh( Œ  ), ¨

¨2Š
= ¨

16.

2

+ Œ

ï

[ ô cosh(

−2

óò

=

Particular solution: 17.

[ ô cosh(

ï

ñ

leads to an equation of the form 1.2.2.4:

).



leads to an equation of the form 1.2.2.4:

–ý

2

+

öý

cosh(

ï

).

=− .

0

) + õ ](ñ„ó ò – ñ

Particular solution:

Œ

ï )ñ

+ ö cosh(

2

).



ï

=

ˆ  .

) + õ ]ñ

Œ

) sinh ÷ (

ï

+ õ cosh(

ï

0

M

cosh( ï ) = 0. Œ sinh( Œ  ) =− .  cosh( Œ  ) +  2

)+

ô

2

1.2.4-2. Equations with hyperbolic tangent and cotangent. 18.

ñóò

ñ

2

=

+

ô

Particular solution: 19.

ñ óò

ñ

2

=

+ 3 ô –

2

ñóò

ñ

2

=

+

ô ï

21.

[ ô tanh(

ï

Particular solution: 22.

ñ óò

ñ

=

2

+

ô

ñóò

ñ

2

=

–

2

ñóò

=

ñ

2

+

ô ï

ñ

2

0

ï )ñ

+ ö tanh(

0

cothú ( õ

Particular solution:

0

). –ý

2

+

öý

tanh(

ï

).

=− .

M

ï

).

=  coth( Π ).

+ 3 ô – ô ( ô + ) coth2 (

Particular solution: 24.

0

ï

= −1  .

– ô ( ô + ) coth2 (

Particular solution: 23.

=

+ ô tanhú ( õ

ï )ñ

0

óò

).

=  tanh( Œ  ) − Œ coth( Œ  ).

tanhú ( õ

) + õ ]ñ

ï

– ô ( ô + ) tanh2 ( 0

Particular solution:

).

=  tanh( Π ).

0

Particular solution: 20.

ï

– ô ( ô + ) tanh2 (

ï

).

=  coth( Œ  ) − Œ tanh( Œ  ).

ï )ñ

+

ô

= −1  .

cothú ( õ

ï

).

© 2003 by Chapman & Hall/CRC

Page 93

94

FIRST-ORDER DIFFERENTIAL EQUATIONS

25.

[ ô coth( ï ) + õ ]ñ„ó ò = ñ 2 + ö coth(

Particular solution: 0 = − .

26.

ñ óò

ñ

2

=

Particular solution: 27.

ñóò

ñ

2

=

ô

+

0

õ

+

M

ï

tanh2 (

2

–2

2

)–2

ï )ñ

–ý

ï

coth2 (

0

ý

coth(

ï

).

).

= Πtanh( Π ) + Πcoth( Π ).

– 2 ôƒõ – ô (ô + ) tanh2 (

Particular solution:

+ö

2

ï

) – õ ( õ + ) coth2 (

ï

).

=  tanh( Π ) +  coth( Π ).

1.2.5. Equations Containing Logarithmic Functions 1.2.5-1. Equations of the form ( )
 =  2 ( ) 1.

= ô (ln ï )÷

ñóò

ñ

2

+

õÿþ ïƒú

Particular solution: 2. 3.

4.

ï ñóò

2

(ln ï )÷ .

= ' + .

+ 'a + ( .

= ï ñ 2 – ô 2 ï ln2 ( ï ) + ô . Particular solution: 0 =  ln(ý± ).

= B − 1:

ï ñóò ï ñóò

=

ï ñ

2

–ô

2

ï

ln2  (

ï

)+

ôƒü

ln 

–1

 0 =  ln ( ý± ).

(

ï

).

+

1 2

leads to an equation of the form 1.2.2.3:

+

1 leads to an equation of the form 1.2.2.4: 2

= ô ïƒ÷ ñ 2 + õ – ôƒõ 2 ïƒ÷ ln2 ï . Particular solution: 0 =  ln  .

ï ñóò

ï 2 ñ óò

= ï 2 ñ 2 + ô ln2 ï + õ ln ï + ö . The transformation ˆ = ln  , ¨ =  ¨2Š
= ¨ 2 + ˆ 2 + 'ˆ + ( − 1 . 4

ï 2 ñ óò

=

ï 2ñ

2

+ ô ( õ ln ï + ö )÷ + 14 .

The transformation ˆ =  ln  + ( , ¨

¨2Š
= ¨

2

ï

ï )(ñ óò

2

ln( ô

+ 

−2

ˆ  .

=

 

– ñ 2 ) = 1.

Particular solution:

0

= −[ ln( )]−1 .

1.2.5-2. Equations of the form ( )
 =  2 ( ) 10.

2

= ôñ 2 + õ ln  ï + ö ln2  +2 ï . The substitution a = ln  leads to an equation of the form 1.2.2.6 with , g
=  2 + 'a  + (˜a 2  +2 .

8.

9.

ï 2ú

ï ñóò

Particular solution:

7.

– ôƒõ

+  0 ( ).

= ôñ 2 + õ ln ï + ö . g The substitution  =  leads to an equation of the form 1.2.2.1: 
g = 

5. 6.

0

–1

2

ñóò

=

ñ

2

+

ô

ln(

ï )ñ

Particular solution:

– ôƒõ ln( 0

ï

2

+  1 ( )
 +  0 ( ).

) – õ 2.

=.

© 2003 by Chapman & Hall/CRC

Page 94

1.2. RICCATI EQUATION ã ( Ë ) Ï

11. 12.

ñóò

=

ñ

2

+

lnú ( õ

ô ï

ï )ñ

Particular solution:

ñóò

+1

ln ï

0

= ' ln  .

=

ô ƒï ÷ ñ

ïƒ÷

– ôƒõ

2

ñóò

= –(ù + 1)ïw÷

ñ

2

+

Particular solution: 14.

ñóò

= ô (ln ï )÷

ñ

2

ñ óò

= ô (ln ï )  (ñ –

ñóò

= ô (ln ï )÷

ñ

2

ï ñ óò

18.

ï ñóò

lnú (

ô

Solution: 19.

ï ñóò

20.

ï ñóò

=

ô ï 2÷

Solution: X 21.

22. 23.

ñ

M2

H

+ õ ln ï + õ .

ñ

–1

.

(ln ï )ú

õ^ö

– ô„ö 2 (ln ï )÷ .

+ L , where H = 

!H + 

+ õ ln ï

95

)

= −( .

0

2

â 0 (Ë

– ô (ln ï )ú .

õÿù ï ÷

+

+

üIñ

+

ôƒõ 2 ï 2

= '  tan /XÈ 

= ô ïƒ÷ (ñ

Solution:

ï )ñ

+1

   +( . 0 = '

= ( ôñ + õ ln ï )2 .

=

+ â 1 (Ë )Ï +

).

ñ

.

– ö )2 +

+ õ (ln ï )ú

Solution: ln  = X

2

= ' ln  .

õï ÷

Particular solution: 17.

−1

(ln ï )÷

0

Particular solution: 16.

ï

(ln ï )ú

+1

=  −

ï

â 2 (Ë )Ï

=

+ õ ln ï + õ .

ñ

ô ïƒ÷

0

– ôƒõ

Particular solution: 15.

ln ú ( õ

ô

0 = −1  .

Particular solution: 13.

+

ÒÍ

Q

−1

+

lnú (

ï

+  ln  . ).

ln ( Π )  + L2S .

M

) –õ . 2

 1 +   =L . +  ln  B

(ln ï )ñ

 ¨

2

+ (õ

ïƒ÷

ln ï – ù )ñ + ö ln ï .

¨ = XÈ  M 2+ ¨ +(

−1

ln 

M

 + L , where ¨ =   .

= ô 2 ï 2 ñ 2 – ï ñ + õ 2 ln ÷ ï . The substitution  2 = −‡
 ‡ leads to a second-order linear equation of the form 2.1.5.27:  2 ‡

 + ‡
 + (  )2 ln *‡ = 0.

ï 2 ñ óò

( ô ln ï + õ )ñ„ó ò = ñ 2 + ö (ln ï )÷ Particular solution: 0 = − Œ .

( ô ln ï + õ )ñ ó ò = (ln ï )÷ ñ 2 + Particular solution: 0 = − Œ .

ñ

–

2

ö^ñ

–

2

+

wö

(ln ï )÷ .

(ln ï )÷ +

ö

.

1.2.6. Equations Containing Trigonometric Functions 1.2.6-1. Equations with sine. 1.

= ñ 2 +  + sin( ï ). The substitution 2a = 2 Œ  + leads to an equation of the form 1.2.6.14: Œ
g = ƒ

ñóò



2

+ ý + = cos a .

© 2003 by Chapman & Hall/CRC

Page 95

96 2.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñóò

ñ

–ô

2

=

2

+ ô sin(

Particular solution: 3.

ñ óò

=

ñóò

=

ñ

2

ñ

2

+ ö sin ÷ (

2

+

0

ï

)+ô

2

sin2 (

ï

).

(

ï

= −  cos( Œ  ). + ô ) sin–÷

ï

–4

+ õ ).

sin( Œ  +  ) ¨ sin2 ( Œ  +  ) , = + cot( Œ  +  )S leads to an equation The transformation ˆ = sin( Œ  +  ) sin(  −  ) Q Œ ¨ 3
¨ of the form 1.2.2.4: Š = 2 + #*ˆ  , where # = ( [ Œ sin (  −  )]−2 . 4.

ô

+

ï )ñ

sin(

Particular solution: 5.

ñóò

ñ

2

=

ô ï

+

0

ñóò

=

ï )ñ

sin(

2

0

ñóò

ô

=[ +

0

ï )]ñ

sin2 (

Particular solution: 9.

ñóò

= –( ü + 1)ï

ñ

2

0

Particular solution: 10.

ñ óò

sin  (

ô

=

ï

0

11.

ï ñóò

=

sinú (

ô ï

[ ô sin(

) + õ ]ñ„ó ò =

Particular solution: 13.

ï

[ ô sin(

).

2

1 2

=  −

−1

õï ÷

+

üIñ

ñ

2 0

0

ï

).

– ô (sin ï )ú .

. – ö )2 +

õÿù ï ÷

–1

ï 2

sinú (

ï

   +( . 0 = '

2

).

.



= − cot( Œ  ).

+ ôƒõ −1

+ ö sin(

=− .

2

.

).

sin + ( Π )  + L2S . M

ï )ñ

–ý

2

+ö

ý

sin(

ï

).

M

sin( ï ) = 0. Œ cos( Œ  ) =− .  sin( Œ  ) + 

) + õ ](ñ„ó ò – ñ 2 ) – ô

Particular solution:

Π +

1 4

– ô + ô sin2 (

+

ï

– ô – ô sin(

+

  X   = '   tan  Q

Solution: 12.

ï )ñ

2

= tan 

+ )(ñ –

Particular solution:

.

).

ô ï +1 (sin ï )ú ñ

+

2

= − cos( Œ  ).

ï )]ñ

2ñó ò = [ + ô – ô sin( Particular solution:

8.

ï

sin3 (

Particular solution: 7.

ï

+ ô sinú ( õ

0 = −1  .

+

õ

)–

= − .

ï )ñ

sinú ( õ

Particular solution: 6.

ï

+ ôƒõ sin(

2

1.2.6-2. Equations with cosine. 14.

15.

ñ óò

=

ñóò

=

ñ

2

+  + cos ï .

The transformation  = 2a , ƒ = −‡
 ‡ leads to a Mathieu equation of the form 2.1.6.29: ‡ g

g + (  − 2  cos 2a )‡ = 0, where  = 4 ƒ…ý ,  = −2ƒ±= .

ñ

2

–ô

2

+ ô cos(

Particular solution:

0

ï

)+ô

2

cos2 (

ï

).

=  sin( Π ).

© 2003 by Chapman & Hall/CRC

Page 96

1.2. RICCATI EQUATION ã ( Ë ) Ï

16.

17. 18. 19.

ÒÍ

=

â 2 (Ë )Ï

= ñ 2 + ô cos( ï )ñ + ôƒõ cos( Particular solution: 0 = −  .

ï

ñóò

= ñ 2 + ô ï cosú ( õ Particular solution:

ï )ñ

ñóò ñ óò

=

ï )ñ

cos(

2

+ ô cosú ( õ = −1  .

0

ï

cos3 (

+

ï

ñóò

ô

ï )]ñ

cos2 (

Particular solution:

ñóò

= –( ü + 1)ï

ñ

2

=

cos  (

ô

ï

=

cosú (

ô

Solution: 25. 26.

[ ô cos(

=  −

−1

ï ÷

ï )ñ

2

+

üIñ

óò

Particular solution:

ñ

õÿù ï ÷

ï 2

cosú (

−1

+ ö cos(

=− .

2 0

M

+

2

ï

).

).

–1

.

ï

).

+ cos ( Π )  + L2S . M

ï )ñ

–ý

2

+ö

ý

cos(

ï

).

cos( ï ) = 0. Œ sin( Œ  ) = .  cos( Œ  ) + 

– ñ ) – ô 2

0

2

– ô (cos ï )ú .

– ö )2 +

+ ôƒõ

 /XÈ  = '   tan  Q

) + õ ](ñ

.

   +( . 0 = '

[ ô cos( ï ) + õ ]ñ„ó ò = Particular solution:

ï

0

ô ï +1 (cos ï )ú ñ

Particular solution:

ï ñ óò

= tan( Π ).

+ )(ñ – õ

ï

– ô + ô cos2 (

+

0

+

Particular solution:

ñ óò

2

+Œ

).

21.

=[ +

2

).

= sin( Π ).

0

97

)

) – õ 2.

2ñó ò = [ + ô + ô cos( ï )]ñ 2 + – ô + ô cos( Particular solution: 0 = tan  12 Œ   .

24.

â 0 (Ë

= ñ 2 + 2 + ö cos÷ ( ï + ô ) cos–÷ –4 ( ï + õ ). The substitution Œ  = ŒH − z 2 leads to an equation of the form 1.2.6.3: 
‘ = ( sin  ( ŒH +  ) sin−  −4 ( ŒH +  ).

20.

23.

+ â 1 (Ë )Ï +

ñóò

Particular solution:

22.

2

2

1.2.6-3. Equations with tangent. 27. 28. 29.

30.

= ñ 2 + ô + ô ( – ô ) tan2 ( ï ). Particular solution: 0 =  tan( Œ  ).

ñóò

= ñ 2 + 2 + 3 ô + ô ( – ô ) tan2 ( ï ). Particular solution: 0 =  tan( Œ  ) − Œ cot( Œ  ).

ñóò

= ôñ 2 + õ tan ï ñ + ö . The substitution  = −‡
 ‡ ‡

 −  tan *‡
 + !(˜‡ = 0.

ñ óò

leads to a second-order linear equation of the form 2.1.6.53:

= ôñ 2 + 2 ôƒõ tan ï ñ + õ ( ôƒõ – 1) tan2 ï . The substitution ‡ = +  tan  leads to a separable equation of the form 1.1.2: ‡
 = ‡

ñ óò

2

+ .

© 2003 by Chapman & Hall/CRC

Page 97

98

FIRST-ORDER DIFFERENTIAL EQUATIONS

31.

= ñ 2 + ô tan( ï )ñ + ôƒõ tan( Particular solution: 0 = −  .

32.

ñóò

=

ñ

2

+

ô ï

ï )ñ

tanú ( õ

ñ óò

= –( ü + 1)ï

ñ

2

ñ óò

tan÷ (

ô

=

ï )ñ

0

Particular solution: 35.

ñóò

ï

tan (

ô

=

0

ï ñóò

Solution: 37.

. +2

2

õ ïƒ÷

+

üIñ

ñ

2

.

).

)+

õ

õÿù ïƒ÷

ï 2

tanú (

+ ôƒõ −1

2

ï

tan2 (

– ö )2 +

–1

ï )ñ

+ ü tan(

=− .

–ý

2

.

.

ï

).

üý

+

õ

)+

tan + ( Π )  + L2S . M

2 0

õ

– ô (tan ï )ú .

=  tan( Π ).

  X   = '   tan  Q

[ ô tan( ï ) + õ ]ñ ó ò = Particular solution:

ï

(

   +( . 0 = '

ï )ñ

tanú (

ô

=

−1

tan÷

2

+ )(ñ –

Particular solution: 36.

=  −

– ôƒõ

2

ï

tanú ( õ

ô

ô ï  +1 (tan ï )ú ñ

+

Particular solution: 34.

+

)–

0 = −1  .

Particular solution: 33.

ï

ñóò

ï

tan(

).

M

1.2.6-4. Equations with cotangent. 38.

= ñ 2 + ô + ô ( – ô ) cot2 ( ï ). Particular solution: 0 = −  cot( Œ  ).

ñ óò

39.

= ñ 2 + 2 + 3 ô + ô ( – ô ) cot2 ( ï ). Particular solution: 0 = Œ tan( Œ  ) −  cot( Œ  ).

40.

ñ óò

41. 42. 43.

ñ óò

= ñ 2 – 2 ô cot( ô Particular solution:

ï )ñ

0

= ñ 2 + ô cot( ï )ñ + ôƒõ cot( Particular solution: 0 = −  . = ñ 2 + ô ï cotú ( õ Particular solution:

ñóò = –( ü + 1)ï  ñ

2

ñ óò

=

cot (

ô

ï

ï ñ óò

=

ô

Solution: 46.

0

+ ô cotú ( õ = −1  .

0

=  −

ï )ñ

−1

õï ÷

. – ö )2 +

   +( . 0 = '

2

+

üIñ

+

 /XÈ  = '   tan  Q

[ ô cot( ï ) + õ ]ñ ó ò = Particular solution:

ï

ô ï +1 (cot ï )ú ñ

+

+ )(ñ –

cotú (

) – õ 2.

ï )ñ

ñ óò

Particular solution: 45.

ï

ñ óò

Particular solution: 44.

+ õ 2 – ô 2. =  cot(  ) −  cot( ' ).

ñ

−1

+ ö cot(

=− .

2 0

ôƒõ 2 ï 2

). – ô (cot ï )ú .

õÿù ï ÷

–1

cotú (

ï

. ).

+ cot ( Π )  + L S . M

ï )ñ

–ý

2

+ö

ý

cot(

ï

).

M

© 2003 by Chapman & Hall/CRC

Page 98

1.2. RICCATI EQUATION ã ( Ë ) Ï

ÒÍ

=

â 2 (Ë )Ï

2

+ â 1 (Ë )Ï +

â 0 (Ë

99

)

1.2.6-5. Equations containing combinations of trigonometric functions. 47.

48.

ñóò

ñ

2

=

2

+

ï

+ ö sin ÷ (

) cos–÷

–4

ï

(

).

This is a special case of equation 1.2.6.3 with  = 0 and  = 2.

ñ óò

ô

=

ï )ñ

sin(

2

) cos÷

ï

+ õ sin(

(

The transformation ˆ = cos( Œ  ), ¨

¨2Š
= ¨

49.

ñóò

2

=

+ Œ

−2

2

ñóò

ô

=

ï )ñ

cos(

2

51.

ñ óò

2

=

+ Œ

sin(

−2

ï

+ õ cos(

ï

) sin ÷ (

2

ô ï ÷

+

Particular solution: 52.

53.

sin ÷

+1

(2 ï )ñó ò =

ôñ

0

ñóò

ñ

2

=

ñóò

ñ

2

=

ñ óò

ñ

2

=

ñóò

ñ

2

=

0

+ þxñ cot ï +

ï

+ õ cos2÷

ñóò

ñ

2

=

–2

+

ñóò

=

ñ

2

–

õ

2

2

tan2 (

wô

+

ˆõ

õ

2

(sin ï )2ú .

=

+ B 2

+1

H + .

1 2

2

–

sin(

ï )ñ

Particular solution:

 +L

2

+

0

= Œ cot( Œ  ) − Œ tan( Œ  ).

cot2 (

ï

+ Πsin(

−2

ï

) + ô cos2 (

ˆ  .

ï )ñ

=

ï

) + õ ( – õ ) cot2 (

) sin ÷ (

Πcos( Π )

– ô tan(

0 = 1 cos( Π ).

.

).

=  tan( Œ  ) −  cot( Œ  ).

tan2 (

2

ô

P

)–2

0

3 4

2

M

.

P

ï

2

ï

 +L M

+ 2 ôƒõ + ô ( – ô ) tan2 (

form 1.2.2.4: ¨3Š
= ¨

ñóò

2

cos2ú'ï .

The transformation ˆ = sin( Œ  ), ¨

59.

.

ï

= −  sin +  cot O/X sin + 

Particular solution: 58.

ï ÷

–ô

= −  cos +  cot O X cos+ 

Particular solution: 57.

leads to an equation of the form 1.2.2.4:

= −  cot  .

– þxñ tan ï +

Solution: 56.



).

– ñ tan ï + ô (1 – ô ) cot2 ï .

Solution: 55.

Œ

ï

(

. The substitution H = tan   leads to a separable equation: 2  sin(2 )H 
= !H

Particular solution: 54.

–1

= 1 cos( Π ).

sin2÷

2

ï )ñ

cos(

).

=

ˆ  .

ï )ñ

leads to an equation of the form 1.2.2.4:

Œ

– ô cos÷

The transformation ˆ = sin( Œ  ), ¨

¨2Š
= ¨



0 = 1 cos( Π ).

Particular solution: 50.

ï )ñ

cos÷ (

ô

+

). = −

ˆ  .

ï )ñ

sin(



ï

ï

+

ï

ï

).

).

sin( Π ) leads to an equation of the 2 cos2 ( Π )

).

© 2003 by Chapman & Hall/CRC

Page 99

100

FIRST-ORDER DIFFERENTIAL EQUATIONS

1.2.7. Equations Containing Inverse Trigonometric Functions 1.2.7-1. Equations containing arcsine. 1.

ñóò

ñ

2

=

+ (arcsin ï ) ÷

Particular solution: 2.

ñóò

ñ

2

=

ï

(arcsin ï ) ÷

+

ñóò

= –( ü + 1)ï

ñ

ñóò

+ (arcsin ï ) ÷ ( ï

2

ñ óò

= (arcsin ï ) ÷

ñ

2

ñ óò

= (arcsin ï ) ÷ = (arcsin ï ) ÷

ñ

2

ñóò

ñ

2

ï ñóò

+

= − .

õ ï ú

–

–

–1

= ý± + .

ïƒú

õ

ñ

2

+

– 

– õ )2 +

üIñ

= ( ô ï 2÷ ñ 2 + õ ïƒ÷ The substitution H =  

ï ñóò

– 1).

−1

2

ñ

+

ï 2ú

õÿþ ï ú

–1

.

(arcsin ï )÷ .

ôþ ïƒú

ˆõ 2 ï 2

+

= '  tan Œ"/XZ 

Q

ñ

(arcsin ï )÷ .

2

  + + . 0 = 

= (arcsin ï ) ÷

+1

(arcsin ï )÷

ï ú

+ þ

= (arcsin ï ) ÷ (ñ – ô

Solution: 9.

ôƒõ

0

Particular solution: 8.

ôñ

+

 + . 0 = '

Particular solution: 7.

.

0

Particular solution: 6.

 0 = 

− −1

Particular solution: 5.

+ (arcsin ï ) ÷ .

ñ

= −1  .

0

Particular solution: 4.

+ ô (arcsin ï ) ÷ .

2

= − .

0

Particular solution: 3.

–ô

ñ

–1

.

(arcsin ï )÷ .

(arcsin  ) 

M

 + L2S .

+ ö )(arcsin ï ) ú – ùñ . leads to a separable equation: H 
=  

ñ

−1

(arcsin  ) + ( !H

2

+ ˜H + ( ).

1.2.7-2. Equations containing arccosine. 10.

ñóò

ñ

2

=

+ (arccos ï )÷

Particular solution: 11.

ñóò

=

ñ

2

+

ï

0

ñóò

= –( ü + 1)ï

ñ

0

ñóò ñ óò

= (arccos ï )÷ = (arccos ï )÷

Particular solution:

= −1  .

 0 = 

.

ôñ

+ ôƒõ –

− −1

ñ

2

Particular solution: 14.

+ (arccos ï )÷ .

ñ

+ (arccos ï )÷ ( ï

2

Particular solution: 13.

+ ô (arccos ï )÷ .

2

= − .

(arccos ï )÷

Particular solution: 12.

–ô

ñ

+

ñ

0 2

= − .

–

õ ï ú

 + . 0 = '

õ

2

+1

ñ

– 1).

(arccos ï )÷ .

(arccos ï )÷

ñ

+

õÿþ ï ú

–1

.

© 2003 by Chapman & Hall/CRC

Page 100

1.2. RICCATI EQUATION ã ( Ë ) Ï

15.

ñóò

= (arccos ï )÷

ñ

Particular solution: 16.

ñóò

0

ï ñóò

ïƒú

ï ñóò

ñ

2

üIñ

+

Q

2

õ ƒï ÷ ñ

+

=

2

ï 2ú

+

ˆõ 2 ï 2

−1

â 2 (Ë )Ï

2

+

â 1 (Ë )Ï

+

â 0 (Ë

–1

.

(arccos ï )÷ .

(arccos  ) 

+ ö )(arccos ï ) ú

M

 + L2S .

– ùñ .

leads to a separable equation: H 
=  

The substitution H =  

101

)

(arccos ï )÷ .

ïƒú

– õ )2 + ôþ

= '  tan Œ"/XZ 

ï 2÷ ñ

= (ô

– 

  + + . 0 = 

= (arccos ï )÷

Solution: 18.

–1

= ý± + .

= (arccos ï )÷ (ñ – ô

Particular solution: 17.

ïƒú

+ þ

2

ÒÍ

−1

(arccos  ) + ( !H

2

+ ˜H + ( ).

(arctan  ) + ( !H

2

+ ˜H + ( ).

1.2.7-3. Equations containing arctangent. 19.

ñóò

ñ

2

=

+ (arctan ï ) ÷

Particular solution: 20.

ñ óò

ñ

2

=

+

0

ñóò

= –( ü + 1)ï

ñ

0

ñóò

0

= (arctan ï ) ÷

ñ

ñóò

0

= (arctan ï ) ÷

ñ

2

ñ óò

= (arctan ï ) ÷

ñ

2

Particular solution: 25.

ñóò ï ñóò

= (arctan ï ) ÷

ï ñóò

= (ô

−1

õ ïƒú

+

õÿþ ï ú

2

+

Q

2

+

–

üIñ

õ ƒï ÷ ñ

õ

– 1).

(arctan ï )÷ .

2

+

+

õÿþ ïƒú

–1

.

(arctan ï )÷ .

ôþ ïƒú

ˆõ 2 ï 2

−1

ñ

ˆõ 2 ï 2ú

– õ )2 +

  + + . 0 = 

ñ

ñ

(arctan ï )÷ –1

= '  tan Œ"/XZ 

ï 2÷ ñ

–

= ' + .

ïƒú

+1

.

ôƒõ

= − .

= (arctan ï ) ÷ (ñ – ô

Solution: 27.

=  −

–

0

Particular solution: 26.

= −1  .

 + . 0 = '

Particular solution: 24.

+ (arctan ï ) ÷ .

ñ

+ ôñ +

2

Particular solution: 23.

(arctan ï ) ÷ .

ô

+

+ (arctan ï ) ÷ ( ï

2

Particular solution: 22.

2

= − .

(arctan ï ) ÷

ï

Particular solution: 21.

–ô

ñ

–1

.

(arctan ï )÷ .

(arctan  ) 

+ ö )(arctan ï ) ú

M

 + L2S .

– ùñ .

leads to a separable equation: H 
=  

The substitution H =  

−1

1.2.7-4. Equations containing arccotangent. 28.

ñóò

=

ñ

2

+ (arccot ï )÷

Particular solution:

0

ñ

–ô

2

+

ô

(arccot ï )÷ .

= − .

© 2003 by Chapman & Hall/CRC

Page 101

102 29.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñóò

ñ

2

=

ï

+

(arccot ï )÷

Particular solution: 30.

ñóò

= –( ü + 1)ï

ñ

ñóò

+ (arccot ï )÷ ( ï

2

 0 = 

− −1

= (arccot ï )÷

ñ

ñóò

= (arccot ï )÷

ñ

2

ñ óò

= (arccot ï )÷

ñ

2

Particular solution: 34.

ñóò ï ñóò

ïƒú

õ

ñ

2

= ( ô ï 2÷ ñ 2 + õ ï ÷ The substitution H =  

ï ñ óò

ˆõ 2 ï 2

+

= '  tan Œ" X  

Q

2

– õ )2 + ôþ

üIñ

+

– 1).

(arccot ï )÷ .

2

– ˆõ

  + + . 0 = 

= (arccot ï )÷

Solution: 36.

–1

= ' + .

= (arccot ï )÷ (ñ – ô

ñ

(arccot ï )÷

õÿþ ï ú

+ 0

Particular solution: 35.

ïƒú

– õ

 + . 0 = '

Particular solution: 33.

= − .

0

+1

.

+ ôñ + ôƒõ –

2

Particular solution: 32.

= −1  .

0

Particular solution: 31.

+ (arccot ï )÷ .

ñ

−1

ñ

ï 2ú

+

õÿþ ïƒú

–1

.

ïƒú

(arccot ï )÷ . –1

.

(arccot ï )÷ .

(arccot  ) 

M

 + L2S .

+ ö )(arccot ï ) ú – ùñ . leads to a separable equation: H 
=  

ñ

−1

(arccot  ) + ( !H

2

+ ˜H + ( ).

1.2.8. Equations with Arbitrary Functions ]

Notation:  =  ( ) and = ( ) are arbitrary functions;  ,  , B , and Πare arbitrary parameters. 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives).

1.

ñóò

ñ

2

=

+

!„ñ

–ô

2

– ô! .

Particular solution: 2.

ñóò

!„ñ

=

2

+ ôñ – ôƒõ – õ

Particular solution: 3.

ñóò

=

ñ

2

+

= .

0

ï !„ñ

!

.

0

=.

0

= −1  .

+! .

Particular solution:

2

4.

= !„ñ 2 – ô ïƒ÷ !„ñ + ôù ïƒ÷ –1 . Particular solution: 0 =   .

5.

ñóò

ñóò

=

!„ñ

2

+ ôù

ïƒ÷

–1

ñóò

= –(ù + 1)ïw÷

ñ

Particular solution:

ï ÷ !

2 2

   . 0 = 

Particular solution: 6.

–ô

2

+

ïƒ÷ +1 „! ñ

 0 = 

− −1

. –! . .

© 2003 by Chapman & Hall/CRC

Page 102

1.2. RICCATI EQUATION ã ( Ë ) Ï

7.

ï ñóò

!„ñ

=

2

=

Solution: 8.

9.

ï ñóò

=

ô ï 2÷ !

+ ùñ +

ï 2÷ „! ñ



2

=

!„ñ

2

ïƒ÷ !

+ (ô

+ "Iñ – ô

2

ñóò

=

!„ñ

2

+ "Iñ + ôù

ñóò

=

!„ñ

2

ïƒ÷ "Iñ

–ô

ñóò

=

ô ó ñ

2

ñóò

=

!„ñ

2

ñóò

=

!„ñ

2

ñóò

=

!„ñ

2

ó !„ñ

– ô

=

Solution: 16.

ñ óò

=

!„ñ

2

Ė



ñóò

=

ó

2

19.

ñóò

=

!„ñ

2

ó

Œ



=

!„ñ

2

.

 ( )( H 2 + !H +  ).

–ô

2

! ï 2÷

ô 2 ï 2 ÷ ("

.

– ! ).

.

.

ô  ó

+ õ )ñ +

ó "Iñ ï  ó

Particular solution:

−1

=   .



=  

0



+ "Iñ + ô

+ 2 ô

if  < 0.

.

 

ó

0 2

0

if  < 0.

.

+ .

õ – ó ! ó"

– ô



.

–ô

=   .

ó

+ô



=   . –ô

if  > 0,

leads to a separable equation: H 
=  ( )( H

+ ô

ñóò

if  > 0,



ó

+ ( ô! – )ñ +

– ô

103

)

.

ó

ñóò

2

+

  !

2 2

0

!„ñ

â 0 (Ë

=   .

0

Particular solution: 20.

–1

Particular solution: =

ïƒ÷ "

#!

+





+

= −  − .

–ô

The substitution H = ž 18.

õ!

£    tan O £ lX      + L M  P  |  |   tanh O − |  | X     + L ü ü P M

– ! ( ô

 !„ñ

ôù ïƒ÷

ô 2 ó !

+

–ô

+ ô

Particular solution: 17.

–1

0

ó

+ ô

+

ïƒ÷

0

Particular solution: 15.

= .

ô ó !„ñ

+

Particular solution: 14.

â 1 (Ë )Ï

leads to a separable equation: H 
=  

   . 0 = 

Particular solution: 13.

– ù )ñ +

0

+

Particular solution: 12.

+

.

   . 0 = 

Particular solution: 11.

2

– ô" .

!

Particular solution: 10.

â 2 (Ë )Ï

=

£ Ñ  tan O £  X   −1   + L M P |  |   tanh O − |  | XÈ  −1   + L ü ü M P

The substitution H =  

ñóò

ÒÍ

2

=  

! 2 

2

 

ó

2 2

2

ó

  !

2 2

ó ("

2

+ !H +  ).

.

– ! ).

.

.

© 2003 by Chapman & Hall/CRC

Page 103

104 21.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñ óò

=

!„ñ

2

+

Solution:

=

ï ñ 

+ ô!



ó

2

.

£     2J 2 tan O £ lX    2J 2  2 |  |   J 2 tanh O − |  | XZ  ü ü

22.

= !„ñ 2 – ô tanh2 ( ï )( ô! + ) + ô . Particular solution: 0 =  tanh( Œ  ).

23.

ñ óò

24. 25. 26.

27. 28.

30.

if  < 0.

ï )(ô !

= !„ñ 2 – ô coth2 ( Particular solution:

0

+ ) + ô . =  coth( Œ  ).

= !„ñ 2 – ô 2 ! + ô sinh( ï ) – ô 2 ! sinh2 ( Particular solution: 0 =  cosh( Œ  ).

ñóò

ï ñóò

= !„ñ 2 + ô – ô Particular solution:

2

ï

).

(ln ï )2 . 0 =  ln  .

!

ï ñ óò

= ! (ñ + ô ln ï )2 – ô .  ( ) 1  =L . +X Solution: +  ln   M

= !„ñ 2 – ô ï ln ï Particular solution:

ñ óò ñóò

= – ô ln ï

ñ

2

+

!„ñ 0

ô! (ï

= sin( ï )ñ 2 + Particular solution:

ñóò

0

+ ô ln ï + ô . =  ln  .

ln ï – ï )ñ – ! . 1 . =  ( ln  −  )

cos( ï )ñ – ! .

0 = 1 cos( Π ).

!

= !„ñ 2 – ô 2 ! + ô sin( ï ) + ô 2 ! sin2 ( Particular solution: 0 = −  cos( Œ  ).

ñóò

31.

= !„ñ 2 – ô 2 ! + ô cos( ï ) + ô 2 ! cos2 ( Particular solution: 0 =  sin( Œ  ).

32.

ñ óò

33.

if  > 0,

ñóò

Particular solution: 29.

 +L M 2J 2 P   +L M P

ñóò

= !„ñ 2 – ô tan2 ( Particular solution:

ñóò

= !„ñ 2 – ô cot2 ( Particular solution:

ï ï

). ).

ï )(ô ! 0

– ) + ô . =  tan( Œ  ).

0

– ) + ô . = −  cot( Œ  ).

ï )(ô !

1.2.8-2. Equations containing arbitrary functions and their derivatives. 34. 35.

ñóò

= ñ 2 – ! 2 + !„ó ò . Particular solution:

ñ óò

= !„ñ 2 – !"Iñ + " Particular solution:

0

óò

= .

. 0

= .

© 2003 by Chapman & Hall/CRC

Page 104

1.2. RICCATI EQUATION ã ( Ë ) Ï

36.

ñóò

= – !„ó ò

ñ

2

!"Iñ

+

ñóò

= " ( ñ – ! )2 +

!„ó ò

ñóò

! óò ñ "

=

2

" óò !

–

! 2 ñ óò

!„óò ñ

–

2

ñóò

!„óò ñ

=

2

+

ô ó !„ñ

Particular solution: 41.

ñ óò

!„ñ

=

2

+"

óò ñ 

ñóò

ñ

2

=

! óò òó !

–

= 1  .

0

= .

0

= −  .

â 1 (Ë )Ï

â 0 (Ë

+

105

)

= .

0

ô ó

+

.

0 = −1  . 2

$

.

£   — tan O £  X   — ü

42.

0

+ ô!

=

Solution:

+

+ " (ñ – ! ) = 0.

Particular solution: 40.

2

.

Particular solution: 39.

â 2 (Ë )Ï

.

Particular solution: 38.

=

–" .

Particular solution: 37.

ÒÍ

|  |  — tanh O − |  | X

ü

M

 +L   —

if  > 0,

M

P

 +L P

if  < 0.

.

Particular solution:


= −   .

0

1.2.9. Some Transformations ]

Notation:  , , and % are arbitrary composite functions of their argument, which is written in parentheses following the function name (the argument is a function of  ). 1.

2.

3.

ñóò

ñ

2

=

ô 2 ! (ô ï

+

+ õ ).

The transformation ˆ =  +  , ‡ =   leads to the equation ‡ Š
= ‡

ñóò

ñ

2

=

+

ï

–4

!

(1 

ï

).

The transformation ˆ = 1  , ¨ = −

ô ï öï

1 !IO ï ( ö + ý )4 The transformation

ñóò

=

ñ

2

+

2

4.

ï 2 ñ óò

=

2

2

+  (ˆ ).

+õ . +ý P

 +  , (˜ + M

ï 4 ! ( ï )ñ

+  (ˆ ).

¨ Š
= ¨ −  leads to the equation 3

¨ = 1 [( (˜ + )2 + ( ( (˜ + )], ¢ M M u ¨
¨ leads to a simpler equation: Š = 2 + ¢ −2  (ˆ ). ˆ =

2

where

¢ = M

− ˜( ,

+ 1.

The substitution ‡ = −



1 2

−



1

leads to the equation ‡
 = ‡

2

+  ( ).

© 2003 by Chapman & Hall/CRC

Page 105

106 5.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ï 2 ñ óò

=

ï 2ñ

2

+

ï 2÷ ! ( ô ï ÷

+ õ ) + 14 (1 – ù

The transformation ˆ =   +  , ¨

¨2Š
= ¨

6.

7.

ñóò

2

2

B



). 1−



+

1 − B −  leads to a simpler equation: 2 B

+ " ( ï )ñ + % ( ï ).

The substitution = −1 ¨

¨ 
= % ( )¨ leads to an equation of the same form: p

ñóò

1 4

ñ

2

=

ó

 2 ! ( 

+

ó

)–



2

ñóò

ñ

=

2

–

2

1

Œ

 −



ó   ô + !IO ó ( ö  + ý )4 ö  ó ó

 2

 2 2 2  2 ¨ = ( (  +  M ) + (   − M , ¢eŒ   2¢   leads to a simpler equation: ¨uŠ
= ¨ 2 + ( ¢eŒ )−2  (ˆ ).

10.

11.

 coth( ï )  . The transformation ˆ = coth( Œ  ), ¨ = − Œ equation: ¨3Š
= ¨ 2 + Œ −2  (ˆ ).

ñ

2

=

–

2

ñ

2

=

ï 2 ñ óò

=

–

ï 2ñ

2

2

ñ óò

ñ

2

=

+

2

+ ! ( ô ln ï + õ ) + 41 .

ï )!

+ sin– 4 (

(cot(

ï

The transformation ˆ = cot( Œ  ), ¨

¨2Š
= ¨

13.

ñ óò

ñ

=

2 2

+Œ

−2

 (ˆ ).

+

2

+ cos– 4 (

14.

−1

sinh2 ( Œ  ) −

ñóò

=

ñ

2

+Œ

−2

 (ˆ ).

2

+

2

+ sin– 4 (

ï

+ õ ) !IO

+

¢ = M

− ˜( ,

1 2

sinh(2 Π ) leads to a simpler

1 2

sinh(2 Π ) leads to a simpler



1 leads to a simpler equation: ¨uŠ
= ¨ 2

2

+

−2

 (ˆ ).

)). = − sin2 ( Œ  )

ï )u !  tan( ï

The transformation ˆ = tan( Œ  ), ¨

¨2Š
= ¨

where

ï )!

+ cosh– 4 (

1 The transformation ˆ =  ln  +  , ¨ =  12.

 (ˆ ).

ï )!

+ sinh– 4 (

 tanh( ï )  . The transformation ˆ = tanh( Œ  ), ¨ = Œ −1 cosh2 ( Œ  ) + equation: ¨ Š
= ¨ 2 + Œ −2  (ˆ ).

ñóò

−2

+õ . +ý P

4 The transformation

ñóò

− ( )¨ +  ( ).

 1 −  −  leads to a simpler equation: ‡ Š
= ‡ 2 + Œ 2

    +  ˆ = ,  (  + M

9.

2

.

The transformation ˆ = ž , ‡ = 8.

1

=

+ ( B )−2  (ˆ ).

= ! ( ï )ñ

2

Q Œ

+ cot( Π )S leads to a simpler equation:

) . = cos2 ( Π )

sin( sin(

ï ï

Q Œ

− tan( Œ  )S leads to a simpler equation:

+ô ) . +õ )P

sin( Œ  +  ) ¨ sin2 ( Œ  +  ) , = + cot( Œ  +  )S leads to a simpler sin( Œ  +  ) sin(  −  ) Q Œ 3 ¨
¨ 2 −2 + [ Œ sin(  −  )]  (ˆ ). equation: Š = The transformation ˆ =

© 2003 by Chapman & Hall/CRC

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107

1.3. ABEL EQUATIONS OF THE SECOND KIND

1.3. Abel Equations of the Second Kind 1.3.1. Equations of the Form â!â$ä ã – â = å ( æ ) 1.3.1-1. Preliminary remarks. Classification tables. For the sake of convenience, listed in Tables 5–8 are all the Abel equations discussed in Section 1.3. Tables 5–7 classify Abel equations in which the functions  are of the same form; Table 8 gives other Abel equations. In Table 5, equations are arranged in accordance with the growth of the parameter . In Table 6, equations are arranged in accordance with the growth of the parameter  . ¡ In Table 7, equations are arranged in accordance with the growth of the parameter ° . The rightmost column of the tables indicates the equation numbers where the corresponding solutions are written out. TABLE 5 Solvable Abel equations of the form !
 − = °˜ + #* + , # is an arbitrary parameter

° ¡

arbitrary −7 −4 −5 2 −2 −2 −5 3 −5 3 −5 3 −7 5

−

Equation

2( + 1) ¡ ( + 3)2 ¡

15 4 6 12 0 2 −3 16 −9 100 63 4 −5 36

¡

°

Equation

1.3.1.10

−1

0

1.3.1.16

1.3.1.56 1.3.1.54 1.3.1.47 1.3.1.33 1.3.1.19 1.3.1.30 1.3.1.23 1.3.1.48 1.3.1.27

−1 2 −1 2 −1 2 −1 2 0 0 1 2 2 2

−2 9 −4 25 0 20 arbitrary 0 −12 49 −6 25 6 25

1.3.1.26 1.3.1.22 1.3.1.32 1.3.1.55 1.3.1.2 1.3.1.1 1.3.1.53 1.3.1.45 1.3.1.46

Solvable Abel equations of the form !
 −



 °

−1

−3

−1

−3

−1 −3 5 −5 11 −1 3 −1 3 −1 3 −1 3 −1 5 0 2

−3 −7 5 −13 11 −5 3 −5 3 −5 3 −5 3 −4 5 −1 2 3

arbitrary 2 +1 ¡ 2 4 ¡0 −5 36 −33 196 −3 16 −3 16 −3 16 15 4 −10 49 −2 9 4 9

TABLE 6 = °˜ + ƒh#*K + ý…#

ƒ

2

'& , # is an arbitrary parameter ý

Equation

1

−1

1.3.1.5

1

−1

1.3.1.13

1 arbitrary 286# 3 arbitrary 3 5 6 13 # 5 arbitrary 2

−1 arbitrary −770 # 9 arbitrary −12 −12 −3 −7 # 20 arbitrary 2

1.3.1.7 1.3.1.62 1.3.1.69 1.3.1.61 1.3.1.40 1.3.1.15 1.3.1.60 1.3.1.68 1.3.1.3 1.3.1.14

© 2003 by Chapman & Hall/CRC

Page 107

108

FIRST-ORDER DIFFERENTIAL EQUATIONS TABLE 7 Solvable Abel equations of the form ! 
− = °˜ + ./# ( ƒ… 1 J # is an arbitrary parameter

°

.

arbitrary ≠ 0 2( − 1) ¡ ( − 3)2 ¡

−1 4 −30 121 −12 49 −12 49 −12 49 −12 49 −12 49 −12 49 −12 49 −12 49 −12 49 −6 25 −6 25 −28 121 −2 9 −2 9 −2 9 −10 49 −4 25 −4 25 0 0 0 0 0 0 0 2 2 20

arbitrary 2 ( − 3)2 ¡

1 4 3 242 arbitrary 1 98 6 49 2 49 4 49 1 49 6 49 2 49 1 2 25 6 25 2 121 arbitrary arbitrary 6 2 49 arbitrary 1 50 arbitrary 1 B +2 B +2 1 2 arbitrary 2 2 arbitrary

ƒ 0

¡

( + 3)

¡

1 21 arbitrary 25 1 5 −10 5 −3 1 3 49 + 3 $ 2 2 5 0 0 0 4 0 7 0 1 1 1 −1 1 0 −10 10 0

4

¡

2

+ ý…# + =/#

J ),

2 −1 2



ý

=

Equation

arbitrary

0

1.3.1.2

3 ( + 3)

1.3.1.12

3 6 0 10 5 15 10 65 12 55 15 196 + 75 $ 16 6 4 15 arbitrary arbitrary 2 12 arbitrary 6 arbitrary arbitrary (B + 1)(B + 3) 2B + 3 0 3 0 30 30 arbitrary

1.3.1.17 1.3.1.29 1.3.1.53 1.3.1.25 1.3.1.38 1.3.1.24 1.3.1.31 1.3.1.52 1.3.1.28 1.3.1.58 1.3.1.64 1.3.1.20 1.3.1.39 1.3.1.51 1.3.1.3 1.3.1.26 1.3.1.11 1.3.1.57 1.3.1.22 1.3.1.59 1.3.1.32 1.3.1.36 1.3.1.34 1.3.1.35 1.3.1.37 1.3.1.4 1.3.1.1 1.3.1.50 1.3.1.49 1.3.1.55

2

+3 +9

¡

5 35 0 41 8 34 27 262 23 166 12 49 − 15$ 2 19 7 106 arbitrary 0 1 61 0 49 0 2 2(B + 2) 2(B + 2) 2 4 arbitrary 19 31 0

¡

¡

Given below in this section are all solvable Abel equations known so far. The equations are arranged into groups, in which all solutions are expressed in terms of the same functions. Notation is given before each group. In most cases the solutions are presented in parametric form:

 =  1 (U , L ),

=

2(

U , L ),

where U is the parameter and L is an arbitrary constant.

© 2003 by Chapman & Hall/CRC

Page 108

109

1.3. ABEL EQUATIONS OF THE SECOND KIND

TABLE 8 Other solvable Abel equations of the form ! 
−

=  ( )

Function  ( )

#* 

− ,$I  + ,$

−1

2

#*

9 625

−

Equation 1.3.1.6 (particular solution)

 

2 2 −1

−1

#

1.3.1.44

27  − #* J + 34 # 2  −1 J 3 − 625 # 4  −5 J 3 4 −5 J 3 6 − 25  + 75 #* 1 J 3 + 313 # 2  −1 J 3 − 100  3 # 6 − 25  +  1 J 3 +  + (˜ −1 J 3 +  −2 J 3 M (coefficients  ,  , ( , and are related by an equality) M − 21  + 7 # 2  123 −1 J 7 + 280 #* −5 J 7 − 400 # 2  −9 J 7  3 4

100

1 3

3 2

1.3.1.66 1.3.1.67 1.3.1.65 1.3.1.70

9

£ #*

,

£ 

1.3.1.63

+ $I + L

2

#

2

1.3.1.18

+ 4#

3 9  2 − 6 2  + £ 2 2 32 64  + 

1.3.1.43

6 2 + 5  2 3  + £ 2 2 8 16  + 

1.3.1.21

6 2 + 9 # 3  + £ 2 8 16  + #

1.3.1.41

30 2 + 33 # 9  + £ 2 32 64  + #

1.3.1.42

−

# + $ exp(−2 # ) # [exp(2 # ) − 1]

 2 Π 2  2 Π 2

 

−  ( Œ + 1)   + Œ   





+  

1.3.1.8 1.3.1.9 +

1.3.1.73 (particular solution)



1.3.1.74 (particular solution) 1.3.1.75 (particular solution)

2  2 Πsin(2 Π ) + 2  sin( Π ) 1.3.1-2. Solvable equations and their solutions. 1. 2.

ññ óò

–ñ =
. Solution:  =

− # ln | + # | + L .

–ñ =
ï +( ,
≠ 0. Solution in parametric form:

ññóò

 = L exp O − X U

2

U

M

U

− U − #P

−

$ #

,

= L3U exp O − X

U

2

U

M

U

− U − #P

.

© 2003 by Chapman & Hall/CRC

Page 109

110 3.

FIRST-ORDER DIFFERENTIAL EQUATIONS – ñ = – 92 ï +
+ ( ï –1 ) 2 . 1 b . Solution in parametric form with # > 0:

ññóò

 = 2¬

2 (2 , − 1) L3U  − ( , − 2)U − , − 1 ® , L3U  + U + 1

where # = 23  ( ,

+1 + , 2 L3U  + U ,  L3U + U + 1

= 23  3 J 2 (2 , − 1)( , − 2)( , + 1).

− , + 1), $

2

( , − 1)2 L3U 

= −6 

2 b . Solution in parametric form with # < 0:

 = ˆ [2 Œ  − + * − ( L 1 Œ − 3 L 2   ) sin  ÑU − (3 L 1   + L 2 Œ ) cos  ÑU ]2 , = 6ˆ , ( L 12 + L 22 )  2 − [ L 1 ( Œ 2 −   2 ) − 2 L 2  ·Œ ]  − + * sin  ÑU − [2 L 1  ·Œ + L 2 ( Œ 2 −   2 )]  − +* cos  ÑU/. , where ˆ =  (  − + * + L

sin  ÑU + L

1

2

cos  ÑU )−2 , # = 19  (3 

− Œ 2 ), $

2

=

2 27

Œ (9 

2

+ 5 Π2 ).

3 b . For the case # = 0, see equation 1.3.1.26. 4.

– ñ = 2
(ï 1) 2 + 4
+ 3
Solution in parametric form:

ññ óò

2

ï –1 ) 2 ).

 = 14  (3 A 2U01 )2 , where

0

5.

=

+

0




X


 X

=X

−

M

 = O ú






Here,

˜

where ¢

=

= 4 # + 1.

P 2





(U

2


























# = − 12  1 J 2 ,

+ U ),

= arctan U − L ,

²

+

= £ 1 + U 2,

+

=£ U

−

= £ 1 − U 2.

− 1,

2

.

,

= (U + 1) O ú

2 + U − # ) exp ” £ −¢ + U − # ) exp ” −

˜

J

−1 2

P

− $ÊO ú

2U + 1 arctan £ −¢ •

2 2U + 1 •

2 13

2U + 1 − £ ¢ (U + U − # ) ” 2U + 1 + £ ¢ • 2U + 1 2 U arctan £ L + 2 $ X exp ” £ −¢ −¢ • M 2 U L + 2 $wX exp ” − 2U + 1 • M 2




ú

J

(U






–3

−1 2





=

˜





2

1 01 2

1 U −1 U = ln −L , ² M U 2 − 1 2 ûû U + 1 ûû û û U 1 1+U = ln û −L , ² M 1−U 2 2 û 1 − U ûû û û

– ñ =
ï + ( ï –1 – ( 2 ï Solution in parametric form:

ññ óò

U

1+U

= A 01 ( ²

2U + 1 − £ ¢ L + 2 $wX¯” 2U + 1 + £ ¢ •

2 13

M

U

˜

J

1 2

P

.

if ¢

< 0,

if ¢

= 0,

if ¢

> 0,

if ¢

< 0,

if ¢

= 0,

if ¢

> 0,

© 2003 by Chapman & Hall/CRC

Page 110

111

1.3. ABEL EQUATIONS OF THE SECOND KIND

6.

ññ óò

–ñ =


ï 

–1

Particular solution: 7.

+

ü/( 2 ï 2 –1 .

# =  − $I  −

0

.

,$

– ñ =
ï –1 –
2 ï –3 . Solution in parametric form:

ññóò

 = U

−1

(U − ln |1 + U | − L )1 J 2 ,

where # =  2 2. 8.

ï 

– ü/(

=

Q

1+U

1 (U − ln |1 + U | − L )1 J 2 − U (U − ln |1 + U | − L )−1 J 2 S , 2

U

ó

– ñ =
+ (4 –2 )65 . Solution in parametric form:

ññóò

£ U 2 + #2$ û # ln U + £ U 2 + #2$ + L û û û ûó û û

û,

 = # ln û 9.

U

 = # ln û

2

U

û

]

û

+1

(arctan U − L ) û ,

# V U + (U U

=

û

+1

1

) - −2 U − L ,

M

² + = £ 1 AIU +

2(þ + 1) ï +
(þ + 3)2 Solution in parametric form:

ññóò

–ñ =–

¡

−1 + +3P

¡

– ñ = – 92 ï + 6
2 (1 + 2
Solution in parametric form:

ññóò

2

 =# where «

=«

²

−1 2, 3 2 ,

J

J

ïƒú

2

−# .

− 1)(arctan U − L )W .

−4

«

−2

² =²

(²

2 −1

+1



 =



(

¡

− 3)2

−2

U

where # = −

2

= !« + +




−3

M

U −L ,

2 −1

O² + « + +

2 U , −1 P

¡

= −12 #

2

²

−4

«

−2

(²

2

« − 2U ),

−1 2 .

J

=

2

) J

+1 −1 2

> 0.

«ÈA 6U 1 J 2 )2 ,

[( −3) ² + « + +3U ]2 ,

¡

= X (1 AIU +

+ .

+3) ï

 1J ¡

1−

,

(þ

=

,0

 + =² + « + −U .

+1 ,

ï –1 ) 2 ),

2(þ – 1) ï 2
Vþ + (þ – 3)2 (þ – 3)2 Solution in parametric form:

ññóò – ñ

« + =« +

.

+3 U « + + −1

+1 O ¡ ± where # = A2¡ 2

12.

û

û

 = ¡

11.

û

In the solutions of equations 10–15, the following notation is used:

« + , - = X (1 AIU + 10.

û

– ñ =
(  2 )65 – 1). Solution in parametric form:

ññóò

# ln U + £ U 2 + #2$ + L û û û £ U 2 + #2$ û

=U

)

1 2

¡



+(4þ

−3

U

−2

2

+3þ

+9)
+3þ

« + [A ( −1)U + ¡

+1

(þ

+3)


ï )

2 –1 2

W.

« + −2 « + +2U ² + ],

.

© 2003 by Chapman & Hall/CRC

Page 111

112 13.

FIRST-ORDER DIFFERENTIAL EQUATIONS

 = 14.

2þ

+1ï +
ï –1 –
4þ 2 Solution in parametric form:

ññóò

–ñ =

«

J

1 2

U 1 J 2 ² 2+

=

,

 = 15.

ï

–3

.

U − V 1 T (2 + 1)U + +1 W ² 2+ « ¡ , 2  U 1 J 2 ² 2+ « 1 J 2 ¡

– ñ = 49 ï + 2
ï 2 + 2
Solution in parametric form:

ññóò

2

1 U 3#

ï

3

−1

 3,

2

1 U 9#

=

ï )

2 –5 3

−2

= −2 # , «

=« +

¡

, 3 2.

J

« 3 (U ²

3

−«

AIU 4 « 3 ).

3

.

 = U 1 J 2 « −3 J 2  3 J 2 , = 14 U 1 J 2 « 1 4J 3  , « = « −5 J 3 ,  =  −5 J 3 . where # = 24 ]

2

.

3 ï – ñ = – 16 + 5
ï –1 ) 3 – 12
Solution in parametric form:

ññóò



where

J 

J (

−3 2

−1 2

2

J «

−2 3

− 2U  − U

2

),

In the solutions of equations 16–18, the following notation is used:

16.

 = X exp(T"U 2 ) U − L , M

– ñ =
ï –1 . Solution in parametric form:

ññ óò

 = 

17.

−1

exp(T"U 2 ),

– ñ = – 41 ï + 14
( ï 1 ) 2 + 5
Solution in parametric form:

ññóò

1 16

2  V 3 A 8U exp(AU 2 )W ,

–ñ =A

2ô

7 ï 2 A

V exp(T"U 2 ) A 2UW ,

−1

= 

ññ óò

 = 18.

= 2U X exp(T U 2 ) U − L2SrA exp(T U 2 ). M Q

+ 3


2

ï –1 ) 2 ).

=  exp(AU 2 ) V (2U

2

# = T 2 2.

A 1)  exp(A"U 2 ) A*U W ,

where # = 14 £  .

2

. 8ô 2 Solution in parametric form:

 = A  (  )−1 ( 2 T 2  2 ), ]

where

=  (  )−1 [exp(T U 2 ) − 2  2 ].

In the solutions of equations 19–21, the following notation is used:

« = £ U (U + 1) − ln L  £ U + £ U + 1  , û û û û 19.

U +1 , U

² =

– ñ = 2 ï +
ï –2 . Solution in parametric form:

 =1−

ññóò

 = 13 ! «

20.

J U ,

−2 3

6 ï 2 – ñ = – 25 + 25
(2ï 1 ) Solution in parametric form:

ññ óò

 = U

−2

(5 ²u« − 3U )2 ,

2

J  U − ²u«  ,

= !«

−2 3 2 3

+ 19


+ 6


= 5 U

−3

2

U +1 ln L  £ U + £ U + 1  . û û U û û

where

ï –1 ) 2 ).

« [(2U + 3)« − 2U 2 ² ],

3 # = − 243 2  .

where

# = −£  .

© 2003 by Chapman & Hall/CRC

Page 112

113

1.3. ABEL EQUATIONS OF THE SECOND KIND

21.

3ï 37 ï 2 + +ô 8 8 Solution in parametric form:

ññ óò

–ñ =

ô

7 16 ï

–

2

2

ô

+

2

2

.

 « 2 − 2U 2   = £ , 2 2 U « £  ]



=

2

4U 

2

−«

U « £ 

4£ 2

.

In the solutions of equations 22–25, the following notation is used:

.

22.

2

2

= A (U

4 ï – ñ = – 25 +
ï –1 ) 2 . Solution in parametric form:

. 2.

.

.

J

J

3 2 −9 8 , 3 4

.

−4 2 (14

.

U

−9

3

12 ï 1 – ñ = – 49 + 98
(25ï Solution in parametric form:

.

. .

−4 3 (21 2 4

4

= A (U

4

2

− 6U

+ 4 L3U − 3).

.

J

−4 3 ( 22 3

.

−U

3 ),

# =A

where

4 5

 £ 5 .

− 16

.

.

J

+ 34


.

)

2 2 3) ,

2 4 ),

2 –1 2

.

.

−4 2 3 (4 3 2

+ 41
= 21 

. .

−

ï )

+ 15


= 28  1 2

.

J

−1 2 −9 8 ( 32 3 4

U

+ 10


.

2

.

# = A 9 2 (10  )2 J 3.

where

).

.

− 3U

ï –1 ) 2 ).

. .

−4 2 4 (9 2 4 3

. T

2 2

2 4

T

. .

where

# = −3 £  .

. .

where

# = −8 £  .

2 3 ),

−8

2 2 3 ),

In the solutions of equations 26–29, the following notation is used: 1 3

8

= exp( £ 3 U ) + L sin U ,

= 2 exp( £ 3 U ) − L sin U + £ 3 L cos U ,

2

8

= 2 exp( £ 3 U ) − L sin U − £ 3 L cos U ,

– ñ = – 92 ï +
ï –1 ) 2 . Solution in parametric form:

=4

4

8 8

1 3

−

8

2 2.

ññóò

 = 3

8

8

−2 2 1 2,

8

= 2

5 ï – ñ = – 36 +
ï –7 ) 5 . Solution in parametric form:

8

−2 2 1 ( 2

−2

8 8

1 3 ),

where

# = 16(3  )3 J 2.

ññóò

 = 48  28.

2

2 2 2) ,

ññ óò

8

27.

.

.

= 9

12 ï 2 – ñ = – 49 + 49
(5ï 1 ) Solution in parametric form:

8

26.

.

= 4

ññóò

 = ]

J

−4 3 , 3

2

9 ï – ñ = – 100 +
ï –5 ) 3 . Solution in parametric form:

 =

25.

.

− 3U + L ,

ññóò

 = 10  24.

3

=U

3

ññóò

 = 5 23.

.

− 1),

8 5J 2 8 1

J

−5 4 , 4

12 ï 6 – ñ = – 49 + 49
(–3ï Solution in parametric form:

ññóò

 =

8

8 8

−4 2 (7 1 3

−2

8

2 2 2) ,

= 5

)

1 2

8

J 8

8

J

−1 2 −5 4 (8 13 1 4

+ 23


= −7 

+ 12


8 8

2

8 8

−

8 8

2 4 ),

where

# = (48 )2 J 5  2 .

ï –1 ) 2 ).

−4 2 1 2 (4 1 2

−4

8 8

2 1 3

+

8 28 2

3 ),

where

# = £  2.

© 2003 by Chapman & Hall/CRC

Page 113

114 29.

FIRST-ORDER DIFFERENTIAL EQUATIONS 30 ï 3 – ñ = – 121 + 242
(21ï Solution in parametric form:

ññóò

8

 = ]

8 8

−6 1 (11 2 4

− 64

8

3 2 1) ,

)

+ 35


1 2

= −11 

ï )

8

+ 6


8

2 –1 2

8

−6 2 4( 4 1

−5

).

8 28

+ 32

4

2

8 38

where # = −32 £  .

2 ),

1

In the solutions of equations 30 and 31, the following notation is used: { = tanh(U + L ) + tan U , { = tanh(U + L ) − tan U , 91 9 2 9 = cosh U − sin( U + L ), 1 2 = sinh U + cos(U + L ), 3 = sinh U − cos(U + L ).

30.

3 ï – ñ = – 16 +
ï –5 ) 3 . 1 b . Solution in parametric form with # < 0:

ññóò

 = 8

{

J

−3 2 , 1

{

= 3

{ {

J

−3 2 (2 − 1

1 2 ),

2 b . Solution in parametric form with # > 0: 31.

 = 4

9 3 J 2 9 −3 J 2 , 1

9 −1 J 2 9 −3 J 2 9 (

= 3

2 4 49

1

2 1

2

12 ï – ñ = – 49 +
(–10 ï 1 ) 2 + 27
+ 10
1 b . Solution in parametric form with # < 0:

ññóò

 =  (10 − 7

{ {

1 2)

2

= 7

,

{ ({ 1

3 1

+3

−

9 9

2 3 ),

ï )

2 –1 2

{ {

2 1 2

2 b . Solution in parametric form with # > 0:

 = ]

9

9 9

−4 1 (7 2 3

9

− 5 12 ),

= −7 

9 −4 9 9 1

3 2( 2

−3

# = −12 8 J 3 .

where

# = 3 2 (4  )2 J 3 .

where

).

{

− 4 2 ),

9 9

2 2 3

+2

where

9 29

1 3 ),

# = −2 £  .

where

# =£  .

In the solutions of equations 32–43, the following notation is used: : L Á (U ) + L 2 n Á (U ) for the upper sign, Á = ™ 1ø L 1 ®Á (U ) + L 2 ; Á (U ) for the lower sign,

: : : : : : : : žÁ = U ( Á )
+ @ Á , = 1 J 3 , ’ 1 = U
+ 13 , ’ 2 = ’ 12 ACU 2 2 , ’ 3 = A 23 U 2 3 − 2 ’ 1 ’ 2 , * * where ø Á (U ) and n Á (U ) are the Bessel functions, and ®Á (U ) and ; Á (U ) are the modified Bessel functions.

: : The solutions of equations 32–43 contain only the ratio
, where the prime :* denotes differentiation with respect to U . Therefore, for symmetry, function is defined in terms of two “arbitrary” constants L 1 and L 2 (instead, we can set, for instance, L 1 = 1 and L 2 = L ). x3y'z|{}~/€

32.

33.

34.

– ñ =
ï –1 ) 2 . Solution in parametric form:

ññóò

J :

−4 3

−2

– ñ =
ï –2 . Solution in parametric form:

ññóò

:  = 2 U 4 J 3

2

’

= U

−4 3

J :

−2

’ 2,

= A 3 U

−2 3

J :

−1

’

’ 12 ,

−1 2 ,

– ñ =
(ù + 2)[ ï 1 ) 2 + 2(ù + 2)
Solution in parametric form:

ññóò

 =

35.

 = U

:

:

Á−2 [ žÁ − ( @ + 1) Á ]2 ,

=

:

– ñ =
(ù + 2)[ ï 1 ) 2 + 2(ù + 2)
Solution in parametric form:

:

2

 =  Á −2 [U 2 ÁrA (2 − @ ) žÁ ] , 1 . where # = T/@ £  , @ = B +2

’ 3,

+ (ù + 1)(ù + 3)


Á−2(  Á 2 −2 @

ññóò

−1 2

:

ÁžÁ!AhU

+ (2ù + 3)


# =T

where

2

:

ï )

2 –1 2

].

where # = @ £  , @ =

Á2 ),

ï )

# = −36 3 .

where

2 –1 2

 J

1 3 2 . 3

1

B +2

.

].

:

= A U 2  Á −2 [  Á 2 + 2(1 − @ ) ÁžžÁÑACU

2

:

Á2 ],

© 2003 by Chapman & Hall/CRC

Page 114

115

1.3. ABEL EQUATIONS OF THE SECOND KIND

36.

– ñ =
ï 1) 2 + 2
2 + ( Solution in parametric form:

where $ 37.

ï –1 ) 2 .

ññóò

 =#

2

= (1 − @ 2 ) #

3

:

:
Á

Á−2 (U

:

− Á )2 ,

2

=#

:

:

:


T U 2 ) Á2 ], Á−2 [U 2 ( Á )2 − ( @ 2 C

and the prime denotes differentiation with respect to U .

– ñ = 2
2 –
ï 1) 2. Solution in parametric form:

ññ óò

: :  =  ( 0
)−2 (U

0

: A 2 0
)2 ,

: : : = A U ( 0
)−2 [U ( 0
)2 + 2

0

:
: 0 AIU

2 0 ],

where # = £  and the prime denotes differentiation with respect to U . 38.

12 ï 6 – ñ = – 49 + 49
(ï 1) 2 + 8
Solution in parametric form:

ññóò

 = 3 ’ where # = 2 £ 3  ; 39.

and ’

−4

:

−6

(’

1

’

2

− 2’

2

− 7U

:

2 2

3)

+ 7


2

2

,

3 ï – ñ = – 16 + 3
ï –1 ) 3 – 12
Solution in parametric form:

+ 4


−4

= 5 U

ï )

2 –5 3

U

2

:

−

’

1

− 3’

2 1 ),

ï –1 ) 2 ).

:

−6

2 2

’ 2(’

−’

1

’ 3 ),

where

# = − £  2.

J

2

:

2 3 2

J ,

J

3 2 3 2

1 J and  3 J 2 are expressed in term of modified Bessel functions; # = 8  4 J 3 .

3 2

2

−1

:

J ’ J ’

2

.

−1 2 2 (2

U

:

−1 2 2 (3 2

’

3

’

1

T 3 ’ 22 ), 2 1

− 12 ’

’

2

A 4U

2

:

3

’ 1 ),

= 32  2 .

9 ï 15 7 ï 2 + T]õ 32 32 Solution in parametric form:

:

2

T

3õ

−1

2

7 64 ï 2 ] T õ

J ’ : = A 14   U −1 −3 J 2 ’

 = − 21 U 2

õ

−1 2 1 −1 2 1

−3 2

–ñ =

where 

:

−4 2 1 (3

’

2 3  3 J 2 − 2 3 J 2  3 J 2 − U = : 4 U 3J 2 3J 2  1J 2

J ’ : = T 18   U −1 −3 J 2 ’

ññ óò

2

.

–ñ =

where 

:

:

 33J J 22  = , : U 3 J 2 33J J 22

 = − 41 U

42.

2

3ï 37 ï 2 3õ 2 + A õ 2A 7 ï 2 8 8 16 A Solution in parametric form:

ññóò

2

= 28 U

),

ññóò

:

ï –1 ) 2 ).

2

are expressed in term of modified Bessel functions.

1

6 ï 6 – ñ = – 25 + 25
(2ï 1 ) Solution in parametric form:

where 41.

:

’

ññóò

 = U

40.

−4 2 1 (5 1

+ 5


−3 2

J ’ J ’

−3 2 2 −3 2 2

2

J

.

:

−1 2 (2 2 3 3 3 −1 2 2 (3 23 3

J

U

’

’

TCU

:

A 3 ’ 23 ), 3

’

3

− 3’

2 3 ),

= 6 2.

© 2003 by Chapman & Hall/CRC

Page 115

116 43.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ññ óò

3 ï 3 7 – 32 32

–ñ =–

ï

+ô

2

2

15 ô

7 64 ï

+

2

+ô

2

Solution in parametric form:

J ’

−3 2 2

 = 12  ’ ]

−1 2 3 ( 3

’

−’

3 2 ),

1 24

=

.

2

J ’

−3 2 2

 ’

−1 2 3 (3 3

’

3 2

− 12 ’

A 4U

2

:

3

’ 3 ).

In the solutions of equations 44–52, the following notation is used:




+  + . = L exp(− #

− $I ).

© 2003 by Chapman & Hall/CRC

Page 154

Ï

1.4. EQUATIONS CONTAINING POLYNOMIAL FUNCTIONS OF

34.

–  2 ( )ñó ò = =]ñ 2 + 2 ( ï ñ + F ï 2 – 2 (
+ = )ñ – 2( F + ( ) ï +  The transformation  = ¨ + ƒ , = ˆ ¨ + ý leads to a linear equation:

( 
ñ

2

– 2


[− ƒh#*ˆ 35.

(


22

ñ

2

ï ñ

+




3

12

+

2

 2


+

ï ñ

F

+

2

(2


− 2ý…#*ˆ + $ )¨ + 2( ƒh$ − ý

2

+




11

=

).

# ).

2

ñ

0.

, ) H 2 + ( $ 2 − # 2 , ) H + $ 0 − # 0 , ]
‘ = ( # 22 , 2 + # 12 , + # 11 ) 2 + [(2 # 22 , + # 12 ) H + # 2 , + # 1 ] + # 22 H 2 + # 2 H + # 0 . ¾ ¾ (
22 ñ 2 +
12 ï ñ +
11 ï 2 +
2 ñ +
1 ï +
0 )ñó ò = ( 22 ñ 2 + ( 12 ï ñ + ( 11 ï 2 + ( 2 ñ + ( 1 ï + ( 0 . Here, #2© , $3© , and # 1 are arbitrary parameters, and the other parameters are defined by the [( $

22

−#

22

relations:

#

ƒ − 2 # 22 ý , 2 + # 22 ý 2 − # 1 ƒ , 0 = − # 11 ƒ 2 = (2 # 11 − $ 12 ) ƒ + ( # 12 − 2 $ 22 ) ý + # 1 , 1 = −2 $ 11 ƒ − $ 12 ý , 2 + ( $ 12 − 2 # 11 ) ƒ…ý + ( $ 22 − # 12 )ý 0 = $ 11 ƒ 2

# $ $ $

= −#

12

( ƒ , ý are arbitrary parameters). The transformation  = ¨ + ƒ , [− #

22

ˆ 3 + ($

where , = 2 #

11

22

ƒ +#

−# 12

12 )

ˆ 2 + ($

=ˆ ¨ +ý 12

−#

11 )

2

−#

1

ý

leads to a linear equation:

ˆ +$

11 ]

¨ Š
= (#

22

ˆ 2+#

12

ˆ +#

11 )

¨ +, ,

ý + # 1.

1.4.5. Equations of the Form ( M 3 â = N 3 â 3 + N 2 æ0â 2 + N 1 æ 2 â + N 1.

2

2

+
2 ñ +
1 ï +
0 )ñó ò = ( 22 ñ 2 + ü (2
22 ü +
12 – 2 ( 22 ) ï + ü (–
22 ü 2 + ( 22 ü +
11 ) ï 2 + ( 2 ñ + ü (
2 ü +
1 – ( 2 ) ï + ( = H + ,! leads to a Riccati equation with respect to  =  ( H ):

+

ï

+ $Iˆ + c ]¨ Š
= ( ƒh#*ˆ

2

+ (2ý…# + L )ˆ

The substitution

36.

( ï

155

æ0â 2 + M 1 æ 2 â + M 3 + Z â + Z æ 0æ 1 0 ï ï ï ï ï 3 2 2 3 (ñ – ñ +  ô ñ + õ )ñ óò = ñ – + õÿñ + ô . 3+M

2

0

æ 3 + Y 1 â + Y 0 æ )âäã

Solution in parametric form (  ≠ 0): −1

 =L

2.

(ñ 3 – ï ñ 2 – ï 2 ñ + ï 3 + ôñ )ñ Solution in parametric form:

 =L

3.

− − a |a | 2    −g + 12  L |a |− 2    g ,

−1

sign a 

−1

óò

= –ñ

3

+

ï ñ

2

J g + 1  L |a |  1 J g , 8

(ñ 3 + ï ñ 2 – ï 2 ñ – ï 3 + ôñ + õ ï )ñó ò = –ñ Solution in parametric form (  ≠ −  ):

3

− 4a 2  = a + L |a |  +  exp O − ,  +  P

4.

(ñ 3 + ï ñ 2 – 2 ï 2 ñ + 2 ôñ + ô Solution in parametric form:

 =L

−1 −1 −1

a



ï )ñ óò

= –ñ

3

=L

+

J g + 1  L3a 2  1 J g , 3

+

–ï

ï 2ñ

=L

ñ

2

− − a |a | 2    −g − 12  L |a |− 2    g .

−1

–ï

3

−1

+ï

ï

+ô

sign a 

2

ñ

+ï

3



−1

+

J g − 1  L |a |  1 J g . 8

õÿñ

+

= a − L |a |  +  exp O − −

ï ñ

.

2

+ 4ï =L

2

– 4ï

ñ

−1 −1 −1

a



3

ô ï

.

4a 2 .  +  P

– ôñ + 4 ô

ï

.

J g − 2  L3a 2  1 J g . 3

© 2003 by Chapman & Hall/CRC

Page 155

156 5.

FIRST-ORDER DIFFERENTIAL EQUATIONS (ñ 3 + ï ñ 2 – 5 ï 2 ñ + 3 ï 3 + ôñ + Solution in parametric form: −1

 =L 6.

Jg +

−1

a |a | 

−1

1 32

−1

−1

sign a 

= –3ñ

– 3ï

3

L |a |  1 J g ,

ô ï )ñ óò

ï )ñ óò

= 2ï

= –2ñ

3

ñ

– 6ï

3

2

ñ

2

−2  27 a 2  = a + L |a |   +  exp − S , Q 2(  +  )

9.

10.

−1

  

3 − − 2( − ) −

a |a |

g  

−

1 16 (

ô ï )ñ óò

3

= –ñ

12.

3

ï )ñ óò

= 3ñ

 = a + L3a 2 exp O − 13.

(ñ 3 – 4 ï ñ 2 + 5 ï 2 ñ – 2 ï 3 + 2 ôñ – 3 ô Solution in parametric form: −1

 =L

14.

sign a 

−1

3

−1

sign a 

−1



3

3

2a P

ï )ñ óò

ñ

3 32

L |a |  1 J g .

ï

– ôñ + 4 ô

−1

ï

.

.

ï

.

J g − 2  L |a |  1 J g . 9



ï

.

27 a 2 S . Q 2(  +  )

ñ

+ 9ï

ñ

2

a |a |

2

– 3ï

L

    g +

2

– 3ï

2

ñ

3

= 3ñ

– 8ï

=L 3

−1

– 15 ï

=L

ñ

2

ñ

−1

2

– ôñ +

3

.

3 −   −  ) L |a | 2(  −  )  g .

ô ï

.

–ï

3

+

õÿñ

+

ô ï

.

− 4a 3 O | a |  +  + . 2  +  P

– 8ï

3

+ 10 ï

sign a 

ñ

3 16 (

ï

£ 2a 4 + 1.

2

3

–ï

ñ

2

£ 

+ 20 ï

2

– (2 ô – õ )ñ + 3 ô

3

3 − − 2( − ) −

−1

2

– 9ï

ñ

= a + 2 L3a 2 exp O − = 2ñ

J g + 1  L |a |  1 J g , 8

– ôñ + 3 ô

+ ( õ – ô )ñ + 2ô

= L3a − L

,

ï )ñ óò

2

– 3ï

– 14 ï

2

ï ñ

– 3ï

J g + L |a |  1 J g ,

(ñ 3 – 5 ï ñ 2 + 7 ï 2 ñ – 3 ï 3 + ôñ – 2 ô Solution in parametric form:

 =L

+

− 4a 3 O | a |  +  + , 2  +  P

(ñ 3 – 4 ï ñ 2 + 4 ï 2 ñ + ôñ – ô Solution in parametric form:

3

3

J g + 2  L |a |−1  1 J g . 3

sign a 

= L3a T

(ñ 3 + 3 ï ñ 2 + 3 ï 2 ñ + ï 3 + ôñ + õ ï )ñ ó ò = –ñ Solution in parametric form (  ≠ −2  ):

 = L3a + L

+ 8ï

=L

L 2  = L3aA £ £ 2a 4 + 1, 

11.

−1

Jg −

– ôñ + 4 ô

3

=L

−1

3

−2

3 −   −  ) L |a | 2(  −  )  g ,

(ñ 3 + 3 ï ñ 2 + 3 ï 2 ñ + ï 3 – ôñ + Solution in parametric form:

a |a | 

– 4ï

ñ

– 4ï

ñ

– 9ï

ñ

−1

sign a 

2

−1

2

= a − 2 L |a |   +  exp −

(ñ 3 + 3 ï ñ 2 – ï 2 ñ – 3 ï 3 + ôñ + õ ï )ñó ò = –ñ Solution in parametric form (  ≠  ):

 =L

−1

=L

+ 6ï

+ 15 ï

2

+ 2ï

2

J g + 1  L |a |  1 J g , 9

(ñ 3 + 3 ï ñ 2 – 4 ï 3 + ôñ + õ ï )ñó ò = –2ñ Solution in parametric form (  ≠ −  ):

ñ

=L

J g − 1  L |a |−1  1 J g , 3

(ñ 3 – 3 ï 2 ñ + 2 ï 3 + 2 ôñ + ô Solution in parametric form:

 =L 8.

−1

(ñ 3 + 2 ï ñ 2 – ï 2 ñ – 2 ï 3 + 2 ôñ + Solution in parametric form:

 =L 7.

sign a 

ô ï )ñ óò

−1

+ 21 ï

sign a 

+ 2 ôñ – 2 ô



2a 2 P

– 4ï

ñ

2

ï

.

. 3

J g + 2 L |a |  1 J g .

−1

2

ñ

– 9ï

3

ï

.

ï

.

+ 3 ôñ – 4 ô

+ 2 ôñ – 3 ô

J g + 3  L |a |  1 J g . 8

© 2003 by Chapman & Hall/CRC

Page 156

Ï

1.4. EQUATIONS CONTAINING POLYNOMIAL FUNCTIONS OF

15.

(ñ 3 – 5 ï ñ 2 + 8 ï 2 ñ – 4 ï 3 + ôñ + õ ï )ñó ò = 2ñ Solution in parametric form (  ≠ −  ):

– 10 ï

3

2 + a2  = a + L |a |  +   exp S , Q 2(  +  )

16.

(ñ 3 + 5 ï ñ 2 + 3 ï 2 ñ – 9 ï 3 + ôñ + õ ï )ñ Solution in parametric form (  ≠ −  ):

óò

18.

−1

= –3ñ

– 15 ï

3

 = L3a"A 19.

−1

−1

−1

  −g   −

24.

1 27 (

ñ

=L

−1

3

ñ

−1 −1 −1

a



3

a |a | 

−1

ñ

3

– 12 ï

ñ ñ

2

2

J g − 1  L |a |−1  1 J g , 8

( ï ñ 2 – 2 ü ï 2 ñ + ü 2 ï 3 + ôñ – ô ï )ñ ó ò = Solution in parametric form ( , ≠ 1):

ñ

3

  = a + L |a | exp − S , Q 2( , − 1)a 2

a2

Q 2(  +  )

+ 27 ï

ñ

+ 18 ï

2

– 9ï

ñ

ï

.

ï

.

S .

+ ( õ – 2 ô )ñ + 3ô

3



.

32 a 2 .  +  P + (4 ô + õ )ñ – 3 ô

3

3 + 3 + a |a |− 4  +2   −g − 3(  + 12  ) L |a | 4  +2   g .

ñ

+ 24 ï

2

L

2

2

– 16 ï

ñ

3

ï

– ôñ + ô

.

£ 2a 4 + 1.

£ 

ñ

=L

−1

– 2ü

2

+ 6ï

2

2

3

ï ñ

– 3ï

a

2

– 3ï −1

+

2

– 8ï

ï

.

  g   .

2 − −2

 − 2  ) L |a | 

+ (3 ô + õ )ñ – 2 ô

ï

.

2 + 2 + a |a |− 3  +2   −g − 2(3  + 2  ) L |a | 3  +2   g .

−1 −1 −1

=L

– ( ô – õ )ñ + 2ô

ñ

+ 18 ï

=L

ñ

3

2 27 (

2

2

– 8ï

ñ

  −g   +

+ï

ñ

a |a |−  +2   −g .

2 − −2

a |a |− 

ñ

−1

–ï

−1

=L

+ 3ï

=L

+ 3ï

+ 3ï

2

– 13 ï

3

J g + 1  L3a 2  1 J g , 8

(3 ñ 3 + ï ñ 2 – 3 ï 2 ñ – ï 3 + ôñ )ñó ò = Solution in parametric form: −1

3

2

ï

+ (3 ô + õ )ñ – 2 ô

3

+ ( ô + õ )ñ .

2

  g   ,

(3 ñ 3 – ï ñ 2 – 3 ï 2 ñ + ï 3 + ôñ )ñó ò = –ñ Solution in parametric form:

– 9ï

= L3a"A

2 − −2

 − 2  ) L |a | 

2

– 11 ï

£ 2a 4 + 1,

2 + 2 + a |a |− 3  +2   −g − (3  + 2  ) L |a | 3  +2   g ,

 =L

ñ

a |a |−  +2   −g + (  + 2  ) L |a |  +2   g ,

2 − −2

a |a |− 

 =L

23.

2£ 

= 2ñ

(2 ñ 3 – 9 ï ñ 2 + 13 ï 2 ñ – 6 ï 3 + ôñ + õ ï )ñó ò = 3ñ Solution in parametric form (  ≠ − 32  ):

 =L

22.

3

(2 ñ 3 + 3 ï ñ 2 – 3 ï 2 ñ – 2 ï 3 + ôñ + õ ï )ñó ò = –ñ Solution in parametric form (  ≠ 2  ):

 =L

21.

2

   exp

−3

(2 ñ 3 – 3 ï ñ 2 + ï 2 ñ + ôñ + õ ï )ñó ò = ñ 3 – ï Solution in parametric form (  ≠ −2  ):

 =L

20.

L

óò

– 8ï

ñ

= a − 3 L |a |   +  exp O −

3 + 3 + a |a |− 4  +2   −g − (  + 12  ) L |a | 4  +2   g ,

(ñ 3 – 6 ï ñ 2 + 12 ï 2 ñ – 8 ï 3 – ôñ + ô ï )ñ Solution in parametric form (  > 0):

2

2 + +

(ñ 3 – 6 ï ñ 2 + 11 ï 2 ñ – 6 ï 3 + ôñ + õ ï )ñó ò = 2ñ Solution in parametric form (  ≠ − 21  ):

 =L

+ 16 ï

2

= a + 2 L |a | 

−3  32 a 2  = a + L |a |   +  exp O − ,  +  P

17.

ñ

157

a |a | 

+ô

3



3

ü 2ï 2ñ

.

J g − 1  L3a 2  1 J g . 8

+

−1

ï

ô ï

.

J g + 1  L |a |−1  1 J g . 8 +

üôñ

– üô

ï

.

 S . Q 2( , − 1)a 2

= a + , L |a | exp −

© 2003 by Chapman & Hall/CRC

Page 157

158 25.

26.

FIRST-ORDER DIFFERENTIAL EQUATIONS (ñ 3 – 3 ü ï ñ 2 + 3 ü 2 ï 2 ñ – ü 3 ï 3 – ôñ + ô ï )ñó ò = Solution in parametric form (  > 0, , ≠ 1):

(ñ

3

– 3ü

ï ñ

2

ï 2ñ

3

+ 3ü

2

, −1£ 2 2a + 1, 2£ 

 = L3a"AeL 2

–ü

3

27.

3

+

1 2

ï ñ

( , − 1)3 a 3S , ( , − 3)  − 2 

3

3

– ( ü + 2) ï

ñ

ï

4

3

sign a 

−1

J g +  L |a |  1 J g , ( , − 1)2

=L

−1

  + + + 

Q

sign a 

ü 2ï

−1

3

– ( ü + 2) ï

ñ

3

– (2 ü + 1)ï

ñ

ï

– ôñ + ô

.

ï

.

3

ï

.

üô ï

.

ï

.

+ ( ü + 1) ôñ – 2 üô

2

ñ

–

ü 3ï

3

+

üôñ

–

 S . Q 2( , − 1)2 a 2

= a + , L3a 2 exp −

2

(
ñ 3 + ï ñ 2 –
ï 2 ñ – ï 3 + ôñ + õ ï )ñ ó ò = 1 b . Solution in parametric form with  ≠ 0:

3

( , − 1)3 a 3S . ( , − 3)  − 2 

2

+ ü ( ü + 2) ï 2 ñ – ü 2 ï 3 + ôñ + õ ï ]ñó ò = üIñ 3 – ü (2 ü + 1)ï ñ 2 + ü 2 ( ü + 2) ï 2 ñ – ü Solution in parametric form (  ≠ −  , , ≠ 1):

[ñ

ï

J g +  , L |a |  1 J g . ( , − 1)2

– ü ( ü – 4) ï 2 ñ + ü 2 ( ü – 2) ï 3 + ôñ – ô ï ]ñó ò = (2 ü – 1)ñ 3 – ü (4 ü – 1)ï ñ 2 + ü 2 (2 ü + 1) ï Solution in parametric form ( , ≠ −1):

[ñ

4

+ [( ü + 1)ô + õ ]ñ – üô

|a | 

2

−1

–ü

3 2

, −1£ 2 2a + 1. 2£ 

2

= L3a + , L

ï ñ

+ 3ü

2

–ü

 ( , − 3), , ≠ 1):

+ ( , − 1)3 2  = a + L |a |  +   exp a S , Q 2(  +  )

30.

2

= L3aAe, L

  = a + L3a 2 exp − S , Q 2( , − 1)2 a 2

29.

– 3ü

+ (2 ü + 1)ï 2 ñ – ü ï 3 + 2 ôñ – ( ü + 1) ô ï ]ñ„ó ò = üIñ 3 – ü ( ü + 2) ï ñ 2 + ü (2 ü + 1)ï 2 ñ – Solution in parametric form ( , ≠ 1):

[ñ

 =L 28.

Q

  + + + 

|a | 

3

ô ñ + õ ï )ñóò = üIñ 3 – 3 ü 2 ï ñ 2 + 3 ü 3 ï 2 ñ

ï

Solution in parametric form (  ≠

 = L3a + L

üIñ

ï

3

3

+ [( ü + 1)ô + õ ]ñ – üô

  + ( , − 1)3 2 + exp a S .  Q 2(  +  )

ñ

= a + , L |a |  3

+


ï ñ

2

–ï

2

ñ

–


ï

3

+

õÿñ

+ô

ï

.

− − − − a |a | 2   |a + 1|  O 2  − 14 L |a |  2  |a + 1| 2  O , − − − − = L −1 a |a | 2   |a + 1|  O 2  + 14  L |a |  2  |a + 1| 2  O .

 =L

−1

2 b . Solution in parametric form with  = 0:

 =L 31.

−1

a |a | O

−1 −1 2



J g − 1  L |a | 1−2O  1 J g , 8

=L

−1

a |a | O

−1 −1 2



J g + 1  L |a | 1−2O  1 J g . 8

(
3 ñ 3 +
2 ï ñ 2 +
1 ï 2 ñ +
0 ï 3 + ô ï )ñó ò = ( 3 ñ 3 + ( 2 ï ñ 2 + ( 1 ï 2 ñ + ( 0 ï 3 + ôñ . This is a special case of equation 1.7.1.13 with ² + ( , ) =  . The transformation a =   , ‡ =  2 leads to a linear equation:

. .
P (a ) − a O (a )]‡ g = 2 O (a )‡ + 2  , . (a ) = # a 3 + # a 2 + # a + # and . (a ) = $ a 3 + $ a 2 + $ a 3 2 1 0 3 2 1 P O [

where

.

+$

0.

© 2003 by Chapman & Hall/CRC

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1.5. EQUATIONS OF THE FORM

32.

33.

– 4) ï 2 ñ – (
– 2) ï 3 + ôñ – ô = –(
– 2)ñ 3 – (
– 4) ï ñ Solution in parametric form:

[
ñ

[
ñ

+ 2) ï

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

+ (


3

+ 1)ï

+ 3(


3

ñ

2

– (




 = a + L |a |1− O exp O

ñ

2

2

– 4(
– 3)ï 3 + ôñ – ô ï ]ñ = –(2
– 3)ñ 3 – 6(
– 2)ï ñ

+ 12 ï

2

+ 2) ï

+ (


ñ

óò

2

+ 12 ï

ñ

2

8a 2 P

ñ

2


ï

+



= a − L |a |1− O exp O

,

8a 2 P

ï ]ñ óò

159

3

ô ï

.

ï

.

– ôñ +

.

ï

+ 8


3

– 2 ôñ + 2 ô

Solution in parametric form:

 = a + L |a |1− O exp O



18a P 2

= a − 2 L |a |1− O exp O

,



18a 2 P

.

1.5. Equations of the Form ð ( ì , í ) í ï î = ë ( ì , í ) Containing Arbitrary Parameters 1.5.1. Equations Containing Power Functions 1.5.1-1. Equations of the form 
 =  ( , ). 1.

2.

7

=
ñ + ( ï –1 ) 2 . The substitution ¨ = 2 # ¨ + 2 $C# −2  −1 J 2 .

ñóò ñóò

=

7

ñ




( ï

+

Let # = A 2 

−1

–1

4.

5.

=  [2U2Aç (U )]2 ,

 (U ) = exp(T"U 2 ) X exp(T"U 2 ) U + L2S Q M

where

7

−1

£

leads to an Abel equation of the form 1.3.1.33:

ñóò

−1

.

¨*¨p
=

7

=ô ñ + õ ï + ö ïƒú . leads to the Abel equation ¨*¨ 
= ¨ + 2  The substitution ¨ = 2  −1 £ is discussed in Subsection 1.3.1 (see Table 5).

ñóò ñóò

¨*¨p
=

.

=
ñ + ( ï –2 . The substitution ¨ = 2 # ¨ + 2 $C# −2  −2 .

ñ óò

=

ôñ ÷

÷ õ ï 1–÷

+

Solution: X 6.

leads to an Abel equation of the form 1.3.1.32:

£  , $ = T 4  (  > 0). Solution in parametric form:

 =  (U ), 3.

£

−1

=


ñ

s

–

(

−2

( ' + (˜ + ), which

.

¨ ¨ ¨  + M 1 ¨ +  = ln | | + L , where =   1−  À ï .

‘ The transformation  = (¨
)1 J' , Emden–Fowler equation:

1 −1

.

À

À Œ = ($ # )1 J , leads to the generalized = Œ (¨u H )1 J , where

¨ '‘

‘ = − Œ", H °$

−

À

À

1

1−

¨

 −1 (¨ ‘
)  ,

which is discussed in Section 2.5 (in the classification table, one should search for the equations satisfying the condition B + + 1 = 0).

¡

© 2003 by Chapman & Hall/CRC

Page 159

160 7. 8.

FIRST-ORDER DIFFERENTIAL EQUATIONS = ( ô ï + õÿñ + ö )÷ . This is a special case of equation 1.7.1.1 with  (ˆ ) = ˆ  .

ñ óò

= ô ïƒú Solution:

ñóò X

9. 10. 11. 12.

÷ ¿÷ ú ñ ÷

– –

õ ïƒú

+

.

¨  1 J  ln | |, ¨  −M Œ ¨ + 1 + L = O hP

ñ óò = ô ï ÷ –1 ñ ú

+1

where

¨ =O 

+ õ ï ÷' –1 ñ úI +1 .

1

J

hP

 −+

−1

, Œ = ¡



+1

O

 1 J  .  P ·

This is a generalized homogeneous equation of the form 1.7.1.3 with  (ˆ ) = ˆ + 'ˆ  .

7

À

7

= ô ï ñ ñ + õ ïƒú ñ + ö ï s ñ . This is a special case of equation 1.7.1.4 with  ( ) =   , ( ) = ' + , % ( ) = (˜ , and B = 1 2.

ñóò

À

= ô ï ñ 1+÷ + õ ïƒú ñ + ö ï s ñ 1–÷ . This is a special case of equation 1.7.1.4 with  ( ) =   , ( ) = ' + , and % ( ) = (˜ .

ñóò

= ï ÷ –1 ñ 1–ú ( ô ï ÷ + õÿñ ú )  . This is a special case of equation 1.7.1.7 with  (ˆ ) = ˆ  .

ñ óò

1.5.1-2. Other equations. 13.

14. 15.

ï ñ óò

=

ôñ

+

õü ñ

2

+ö

ï

2.

The substitution ¨ =   leads to a separable equation:  ¨ 
= (  − 1)¨ +  £ ¨

= ñ + ô ïƒ÷ –ú ñ ú + õ ïƒ÷ –  ñ  . ¨ 
=   The substitution =  ¨ leads to a separable equation: p

ï ñóò

( ôñ

÷

+

õ ï )ñ ó ò

17. 18.

19.

20.

+( .

(  ¨2+ +  ¨  ).

= 1.

Solution:  =    OL + lX 16.

−2

2

ï (ï ñ ÷

  − 

M P

.

+ ô )ñó ò + õÿñ = 0. Solution: B… −  = lÝL  J  +   .

ï (ô ñ ú

+ þ )ñó ò = ñ V õ ïƒ÷ (  –1) ñ ú  – ù·W . This is a special case of equation 1.7.1.16 with  (ˆ ) = ˆ , (ˆ ) = 1, % (ˆ ) = 'ˆ  , and , = B .

( ô ï ÷ + õ ï 2 + ö ï ñ )ñ ó ò = ü ï ÷ + õ ï ñ + ö^ñ 2 . The transformation a =   , H =   −2 leads to a linear equation with respect to H = H (a ): ( , − a ) H g
= (B − 2)(!H +  + (˜a ). ( ôñ ÷ + õ ï 2 + ö ï ñ )ñó ò = üIñ ÷ + õ ï ñ + ö^ñ 2 . The transformation a =   , H =   −2 leads to a linear equation with respect to H = H (a ): a  ( , − a ) H g
= (B − 2)(a  H +  + (˜a ).

( ô ïƒ÷ + õÿñ ÷ + ï )ñó ò =  ï ñ ÷ –  +  ïƒú ñ ÷ –ú + ñ . The transformation a =   , H =   −1 leads to a linear equation: ( ƒ…a  −  + ý±a  − + − 'a  (B − 1)( 'a  +  ) H + B − 1.

+1

− a ) H g
=

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1.5. EQUATIONS OF THE FORM

21.

22. 23. 24. 25. 26.

( ô ï ÷ + õÿñ ÷ +
ï 2 + ( ï ñ )ñ ó ò =  ï  ñ ÷ –  +  ï ú ñ ÷ –ú +
ï ñ + (­ñ 2 . ý a  − + − 'a  The transformation a =   , H =   −2 leads to a linear equation: ( ƒ…a  −  + ±  (B − 2)( 'a +  ) H + (B − 2)( $Ia + # ).

28.

30. 31.

33. 34.

− a ) H g
=

(  ï + ñ + ) ÷ ñó ò = ( ô ï + õÿñ + ö )÷ . This is a special case of equation 1.7.1.6 with  (ˆ ) = ˆ  .

( ô ï ÷ + õÿñ ú )ñ ó ò = ï ÷ –1 ñ 1–ú . This is a special case of equation 1.7.1.7 with  (ˆ ) = 1 ˆ .

( ôñ ú + õ Solution:

ï ÷

+ s)ñ



óò

+

+

 ï 

+

õÿù ï ÷ –1 ñ

+  = 0.

!Y ( ) + ƒ…j ( ) + '  +1

+1 ¡ | | ln

≠ −1,

if if

¡ ¡

= −1,

j ( ) =

+°



 

+ ý± = L , +1

, +1 ln | |

if , ≠ −1, if , = −1.

( ô ï 2 ñ ÷ + õ ï ñ ú + ö^ñ  )ñó ò = ñ[ + ñ\ + . This is a Riccati equation with respect to  =  ( ). ( ô ïƒ÷ ñ ú + ï )ñó ò = õ ï ñ ÷ +ú –  + ñ . The transformation a =   , H =   + + (B + − 1)(a + H + 1).

ï (ô ï ÷ ñ ú

+  )ñ

óò

+ ñ (õ

ï ÷ ñ ú

leads to a linear equation: a + ( 'a  −  − a ) H g
=

−1

+  ) = 0.

( 9  : )P ý − B±ƒ  − B± (    )O + = L , where # = ¡ , $ = ¡ . Solution: # $  ý − ˜ ƒ  ý − ˜ ƒ

ï (ô ù ï  ñ ÷ +

+ s)ñó ò + ñ ( õÿþ Solution: ,  + ,! + − ° (

ïƒú + ñ 

)−  = L .

( ô ï ÷ ñ ú +
ï 2 + ( ï ñ )ñ ó ò = õ ï  ñ ÷ +ú The transformation a =   , H =   + + (B + − 2)(a + H + $Ia + # ).

¡

32.

+1

[( ô ï + õÿñ ) ÷ + õÿñ ]ñó ò = ö ( ô ï + õÿñ )ú – ôñ . This is a special case of equation 1.7.1.15 with  (ˆ ) = ˆ  , (ˆ ) = 1, and % (ˆ ) = (˜ˆ + .

¡

29.

161

[( ô ï + õÿñ ) ÷ + õ ï ]ñó ò = ö ( ô ï + õÿñ )ú – ô ï . This is a special case of equation 1.7.1.14 with  (ˆ ) = ˆ  , (ˆ ) = 1, and % (ˆ ) = (˜ˆ + .

where Y ( ) = 27.

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

+ s) = 0. – −2



+
ï ñ + (­ñ 2 . leads to a linear equation: a + ( 'a  −  − a ) H g
=

À

( ôþ ïƒ÷ ñ ú –1 + õÿñ  )ñó ò + ôù ïƒ÷ –1 ñ ú + ö ï s = 0. This is a special case of equation 1.7.1.19 with  ( ) =   and ( ) = (˜ .

( ô ïƒ÷ ñ ú + õ ï ñ  )ñó ò = ñ s +  . This is a Bernoulli equation with respect to  =  ( ) (see Subsection 1.1.5).

ï (ô ïƒ÷ – ñ ú

+ þ )ñó ò = ñ ( õ ï  ÷ –  ñ ú – ù ). This is a special case of equation 1.7.1.16 with  (ˆ ) = ˆ , (ˆ ) = 1, and % (ˆ ) = 'ˆ  .

© 2003 by Chapman & Hall/CRC

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162 35. 36.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ï (ô ï ÷ ñ ú –

+ þ )ñ ó ò = ñ ( õ ï  ÷ ñ  ú –  – ù ). This is a special case of equation 1.7.1.17 with  (ˆ ) = ˆ , (ˆ ) = 1, and % (ˆ ) = 'ˆ  .

À

( ô ïƒ÷ +1 ñ ú –1 + õ ïƒ÷' +1 ñ úI –1 )ñó ò = ö ïƒ÷ s ñ ú s . This is a special case of equation 1.7.1.3 with  (ˆ ) = (˜ˆ ( ˆ + 'ˆ  )−1 .

37.

( ô ï ÷ + õÿñ ú )  ñ ó ò = ö ï ÷ –1 ñ 1–ú . This is a special case of equation 1.7.1.7 with  (ˆ ) = (˜ˆ −  .

38.

”

ï

1

+ô

+

3 1



ï

2

–ô

ñóò

•

3 2

– ñç”



1



+

3 1

2

23 •

= 0,

where 12 = ( ï + ô )2 + ñ 2 , 22 = ( ï – ô )2 + ñ 2 . This is the equation of force lines corresponding to the Coulomb law in electricity.  +  − + 2 =L . Solution:  1 39. 40. 41.

&

– ñ = ( ô ï

ï ñóò

&

1

2

+ ï ). This is a special case of equation 1.7.1.24 with  (‡ ) = ‡ J + ÿõ ñ  )( ñ ñóò

– ñ = ( ô ï  + õÿñ  )(ññ ó ò – ï ). This is a special case of equation 1.7.1.25 with  (‡ ) = ‡ J

ï ñ óò

+ ï = ( ô ï + õÿñ  )( ï ñó ò – ñ ). This is a special case of equation 1.7.1.24 with  (‡ ) = ‡

ññóò

−

2

and ( , ¨ ) =   +  ¨  .

2

and ( , ¨ ) =   +  ¨  .

J 2 and ( , ¨ ) = (  +  ¨  )−1 .

1.5.2. Equations Containing Exponential Functions 1.5.2-1. Equations with exponential functions. 1.

ñóò

=

ô]

+õ . =−

Solution: 2.

ñóò

=

ô]

+

Solution:

ó ñ óò = B
 ] +D

õ ó = 

1

Œ

ln OL 

−





−

.





hP

.



− ln L −  X exp(   )  S .

Q

M

3.

–ô . Š This is a special case of equation 1.7.1.2 with  (ˆ ) = #  , B = 1, and  = 0.

4.

ñóò

= ô + ] + õ  . Á  and ( ) =     . This is a special case of equation 1.7.2.5 with  ( ) =  

ó

ó

5.

ñ óò

ó

ó

= ô  + ] + õ – ] .  Á This is a special case of equation 1.7.2.8 with  ( ) =   , ( ) = 0, and % ( ) =    .

ó

ó

ó

6.

= ô 2 ^ –_ ] + õ ^ + ö ^ –_ ] . This is a special case of equation 1.7.2.9 with  (ˆ ) = ˆ + ( .

7.

= ô ^ + ] + õ _ + ö` – ] . ¨ 
= Œ  9  ¨ The substitution ¨ = ž leads to a Riccati equation: 3

ñóò ñ óò

ó

ó

ó

2

  + Œ  : ¨ + (Œ  ; .

© 2003 by Chapman & Hall/CRC

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1.5. EQUATIONS OF THE FORM

8.

( ô

]

õ ó )ñ óò

+

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

= 1.

Solution:  =    − ln L −  X exp(    ) 9.

( õ

^ ]

( ô

ó ] +_

+ õ )ñ

( ô

^ ó

õ _ ] )ñ óò

óò

+ ö )ñ

163

 D

=

ó + ]

Q

^ ]

– ô

S. M

.

Š

This is a special case of equation 1.7.2.13 with  (ˆ ) = ( , (ˆ ) = 1, and % (ˆ ) =  .

10. 11.

óò

=ö .

¨ 
= ý + ( (   The substitution ¨ ( ) = + ý± leads to a separable equation: p +

ó  ^ –_ ]

=

(

^ ó +` ]

+ ô )ñó ò +

õ ó +_ ]

+  )−1 .

.

This is a special case of equation 1.7.2.9 with  (ˆ ) = ˆ

12.

?

+ ô

−1

.

= 0.

This is a special case of equation 1.7.2.15 with  ( ) =  ;  and ( ) =  

Á .

1.5.2-2. Equations with power and exponential functions. 13.

ñóò

=

ô]

+

õ ïƒ÷

.

1 b . Solution in parametric form with B ≠ −1: 1 +1

 =U 

=

,

 ' U   U!S . U − ln L − X U −  +1 exp O B +1 Q B +1 B + 1 P3M

2 b . Solution in parametric form with B = −1,  ≠ −1:

 = * ,

= − ln OL 

−

*

−

  * .  +1 P

3 b . Solution in parametric form with B = −1,  = −1:

14.

ñóò

=

ôñ

–1

+

 = * ,

õ ó

= −U − ln( L − U ).

.

Solution in parametric form:

 = ln( #2« where  = T 2 $ 15.

2

,  = A"#

ô ó + ]

+

=

ô ƒï ÷  ]

ñóò

=

ñóò

=

ñóò

=

ñóò

−1

õ ïƒ÷

.

+

õ ó

.

ô ƒï ÷  ]

+

õ ïƒú

.

ô ƒï ÷  ]

+

õ ïƒú  – ]

−1

) TçU 2 ,

= $ [2 A exp(T"U 2 ) «

17. 18.

],

$ , « = X exp(T"U 2 ) U + L . M

This is a special case of equation 1.7.2.5 with  ( ) =   16.

−1

Á  and ( ) = '  .

Á This is a special case of equation 1.7.2.5 with  ( ) =   and ( ) =   . This is a special case of equation 1.7.2.5 with  ( ) =   and ( ) = ' + . . This is a special case of equation 1.7.2.8 with  ( ) =   , ( ) = 0, and % ( ) = ' + .

© 2003 by Chapman & Hall/CRC

Page 163

164

FIRST-ORDER DIFFERENTIAL EQUATIONS

ó

ó

19.

= ö (ñ + ô  + õ )÷ – ô  . This is a special case of equation 1.7.2.10 with  (ˆ ) = (˜ˆ  .

20.

ñóò

ñ óò

= ( ô] +

õ ï –  )1 ) 

.

Solution in parametric form:

 = exp ¦"U − 21.

,

23. 24.

where

 (U ) = X

, U . M , (  +    * )−1 J' + 1

1V  (U ) + L2W!§ ,

where

 (U ) = X

, U . M , (  +   −  * )1 'J  − 1

ó

= ( ôñ  + õ )1 )  . Solution in parametric form:

ñ óò

 =  (U ) + L , 22.

=  (U ) + L ,

1V  (U ) + L W § ,

= exp ¦U +

,

= ô ï ÷ –1   ÷ ] + õ ï ú –1   ú ] . This is a special case of equation 1.7.2.2 with  (ˆ ) = ˆ 

ñ óò ñóò

=

ô ïƒ÷ –1  ^ ]

+

õ ïƒ÷¿ú

–1

 ^ ú ]

−1

+ 'ˆ +

+1

.

This is a special case of equation 1.7.2.4 with  (ˆ ) = ˆ + 'ˆ + .

ó

ó

= ô  ÷ ñ ÷ +1 + õ –  . This is a special case of equation 1.7.2.1 with  (ˆ ) = ˆ 

ñ óò

ó

+1

ó

+ .

= ô ^ ñ ú +1 + õ ^ ÷ ñ ÷¿ú +1 . This is a special case of equation 1.7.2.3 with  (ˆ ) = ˆ + 'ˆ  .

26.

ñóò

28.

.

.

25.

27.

−1

ñóò

=

ô ÷ ó ñ ÷

+1

+

õ ú ó ñ ú

+1

. This is a special case of equation 1.7.2.1 with  (ˆ ) = ˆ 

+1

ó

+ 'ˆ +

= ô ïƒ÷ ñ  + õ ïƒ÷  ^ ñ  +1 – ñ . This is a special case of equation 1.7.2.7 with  ( ) =   , (ˆ ) =  + 'ˆ , and

ñóò

ñ óò = ô ó ñ 1+÷

ó + õ  ñ

ó + ö ñ 1–÷

.



This is a special case of equation 1.7.1.4 with  ( ) =   , ( ) =  

ó

ó



¡

= 1.

, and % ( ) = ( 

Á .

29.

= ô ñ 1+÷ + õ  ñ + ö ïƒú ñ 1–÷ .   This is a special case of equation 1.7.1.4 with  ( ) =   , ( ) =    , and % ( ) = (˜ + .

30.

ñóò

ñóò

=

ô ï  ñ 1+÷

+

õ ó ñ

+

ö ïƒú ñ 1–÷

.

 This is a special case of equation 1.7.1.4 with  ( ) =   , ( ) =   , and % ( ) = (˜ + .

ó

ó

31.

= ô  ñ 1+÷ + õ ï ú ñ + ö ñ 1–÷ .   This is a special case of equation 1.7.1.4 with  ( ) =   , ( ) = ' + , and % ( ) = (   .

32.

ñóò

33.

ñ óò

=

ô ó ñ 1+÷

+

õ ïƒú ñ

+

ö ï  ñ 1–÷

.

 This is a special case of equation 1.7.1.4 with  ( ) =    , ( ) = ' + , and % ( ) = (˜  . = ô ï ÷ +   ] + õ ï ÷¿ú +   ú ] – ù . This is a special case of equation 1.7.2.6 with  ( ) =  

ï ñ óò

−1

and (ˆ ) = ˆ + 'ˆ + .

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1.5. EQUATIONS OF THE FORM

34. 35. 36. 37.

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

165

ó

( õÿñ + )ñ ó ò = ö D +  ] – ôñ . Š This is a special case of equation 1.7.1.15 with  (ˆ ) = Œ , (ˆ ) = 1, and % (ˆ ) = (  . = ô ïƒ÷ ] – ùñ . This is a special case of equation 1.7.2.11 with  (ˆ ) = ˆ and ƒ = 1.

ï ññóò

= ô ïƒ÷ ] – ùñ 2 . This is a special case of equation 1.7.2.12 with  (ˆ ) = ˆ and ƒ = 1.

ï ñ 2 ñ óò

ó

( ôñ ÷ + õ )ñ ó ò = 1. 1 b . Solution in parametric form with B ≠ −1:

  U  U!S , U − ln L − X U −  +1 exp O B +1 Q B +1 B + 1 PuM 2 b . Solution in parametric form with B = −1,  ≠ −1: 

 =

 = − ln OL  −  * −



 +1

 * , P

=U 

1 +1

.

= * .

3 b . Solution in parametric form with B = −1,  = −1:

38.

( ô] +

õ ï )ñ óò

= 1.

a

Solution in implicit form:  = 39.

= * .

 = −U − ln( L − 'U ),

( ô] + õ ï 2 )ñó ò = 1. Solutions in parametric form:

 =−

: 1 U (ln )
, * 2

 L   +   1−   (L +  )

= ln O

U

2

4 "P

,

if  ≠ 1, if  = 1.

:

=L

ø

1 0(

U ) + L 2 n 0 (U )

and

: : 1 U 2 U (ln )
, = ln O − , = L 1 ® 0 (U ) + L 2 ; 0 (U ), * 2 4 "P where ø 0 (U ) and n 0 (U ) are the Bessel functions, and ® 0 (U ) and ; 0 (U ) are the modified Bessel functions.  =−

40.

( ô] + õ ï –1 )ñó ò = 1. A # $ ,  = T 2# Let  =

2

. Solution in parametric form:

 = # V 2UÑA exp(T U 2 )  (U )W ,

ó

ó

= ln V $ç (U )W T‹U 2 ,

where

 (U ) = X exp(T U 2 ) U + L S Q M

41.

( D +  ] + õ ï )ñó ò = öD +  ] – ô ï . Š Š This is a special case of equation 1.7.1.14 with  (ˆ ) =  , (ˆ ) = 1, and % (ˆ ) = (  .

42.

( D +  ] + õÿñ )ñó ò = öD +  ] – ôñ . Š Š This is a special case of equation 1.7.1.15 with  (ˆ ) =  , (ˆ ) = 1, and % (ˆ ) = (  .

43.

ó

−1

.

ó

ó

( ô ^ ñ ú + õ )ñ ó ò = ñ . This is a special case of equation 1.7.2.3 with  (ˆ ) = ( ˆ +  ) −1 .

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166 44. 45. 46.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ó

ó

(  ^ ñ ÷ + ô )ñ ó ò + õ  +_ ] + ô = 0. Á This is a special case of equation 1.7.2.15 with  ( ) =  and ( ) =   .

ó

(  ^ ñ ÷ + ô )ñó ò + õ ïƒú  _ ] + ô = 0. This is a special case of equation 1.7.2.15 with  ( ) =  and ( ) = ' + .

ó

ó

(  ^ ñ ú + þ ï )ñ ó ò = ñ ( õ ^ ÷ ñ ÷¿ú –  ï ). This is a special case of equation 1.7.2.17 with  (ˆ ) = ˆ , (ˆ ) = 1, and % (ˆ ) = 'ˆ  .

47.

+ ñ )ñó ò = õ ïƒ÷¿ú  ^ ú ] – ùñ . This is a special case of equation 1.7.2.16 with  (ˆ ) = ˆ , (ˆ ) = 1, and % (ˆ ) = 'ˆ + .

48.

(ô

ï (ƒï ÷  ^ ] ïƒ÷ ]

+

õ ï   ] )ñ óò

=

]

.

This is a Bernoulli equation with respect to  =  ( ) (see Subsection 1.1.5).

49. 50. 51. 52. 53. 54. 55. 56.

( ô ïƒ÷ ] + õ ï ñ ú )ñó ò =   ] . This is a Bernoulli equation with respect to  =  ( ). ( ô ïƒ÷ ñ ú + õ ï ] )ñó ò = ñ  . This is a Bernoulli equation with respect to  =  ( ). ( ô ïƒ÷ ñ ú + õ ï ñ  )ñó ò = ] . This is a Bernoulli equation with respect to  =  ( ).

ó

( ôþ ïƒ÷ ñ ú –1 + õ )ñó ò + ôù ïƒ÷ –1 ñ ú + ö = 0.  This is a special case of equation 1.7.1.19 with  ( ) =  and ( ) = (  .

( ôþ ï ÷ ñ ú –1 + õ ] )ñ ó ò + ôù ï ÷ –1 ñ ú + ö = 0. This is a special case of equation 1.7.1.19 with  ( ) =   and ( ) = ( .

ó

( ôþ ïƒ÷ ñ ú –1 + õÿñ  )ñó ò + ôù ïƒ÷ –1 ñ ú + ö = 0.  This is a special case of equation 1.7.1.19 with  ( ) =   and ( ) = (  .

ó

ó

[( ô ï + õÿñ ) ÷ + õ ^ ]ñ ó ò = ö ( ô ï + õÿñ )ú – ô ^ . This is a special case of equation 1.7.2.14 with  (ˆ ) = ˆ  , ( ) = 1, and % (ˆ ) = (˜ˆ + .

[( ô ï + õÿñ ) ÷ + õ ^ ] ]ñó ò = ö ( ô ï + õÿñ )ú – ô ^ ] . This is a special case of equation 1.7.2.13 with  (ˆ ) = ˆ  , ( ) = 1, and % (ˆ ) = (˜ˆ + .

1.5.3. Equations Containing Hyperbolic Functions 1. 2. 3.

= ô cosh( ƒñ ) + õ cosh( b ï ). This is a special case of equation 1.7.2.18 with  ( ) = 0, ( ) =  , and % ( ) =  cosh( @! ).

ñóò

= ô sinh( ƒñ ) + õ sinh( b ï ). This is a special case of equation 1.7.2.18 with  ( ) =  , ( ) = 0, and % ( ) =  sinh( @! ).

ñ óò

= ô ïƒ÷ cosh( ƒñ ) + õ ïwú . This is a special case of equation 1.7.2.18 with  ( ) = 0, ( ) =   , and % ( ) = ' + .

ñóò

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1.5. EQUATIONS OF THE FORM

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

= ô ï ÷ sinh( ƒñ ) + õ ï ú . This is a special case of equation 1.7.2.18 with  ( ) =   , ( ) = 0, and % ( ) = ' + .

ñ óò

= ôñ 1+÷ + õÿñ + ö sinh( ï )ñ 1–÷ . This is a special case of equation 1.7.1.4 with  ( ) =  , ( ) =  , and % ( ) = ( sinh( Œ  ).

ñóò

= ôñ 1+÷ + õ sinh( ï )ñ + ö^ñ 1–÷ . This is a special case of equation 1.7.1.4 with  ( ) =  , ( ) =  sinh( Œ  ), and % ( ) = ( .

ñóò

= ñ cosh ï ( ôñ ÷¿ú sinh÷ –1 ï + õÿñ ú ). This is a special case of equation 1.7.2.22 with  (ˆ ) = ˆ  + 'ˆ .

ñóò

= ñ sinh ï (ôñ ÷¿ú cosh÷ –1 ï + õÿñ ú ). This is a special case of equation 1.7.2.24 with  (ˆ ) = ˆ  + 'ˆ .

ñóò

= ( ô ï ÷ cosh ñ + õ ) coth ñ . This is a special case of equation 1.7.2.25 with  (ˆ ) = ˆ +  .

ï ñ óò

= ( ô ïƒ÷ sinh ñ + õ ) tanh ñ . This is a special case of equation 1.7.2.23 with  (ˆ ) = ˆ +  .

ï ñóò

( ôñ ú cosh ï + õ )ñó ò = ñ ú +1 sinh ï . This is a special case of equation 1.7.2.24 with  (ˆ ) = ˆ ( ˆ +  ) −1 .

( ôñ ú sinh ï + õ )ñ„ó ò = ñ ú +1 cosh ï . This is a special case of equation 1.7.2.22 with  (ˆ ) = ˆ ( ˆ +  ) −1 .

( ô ïƒ÷ + õ ï coshú ñ )ñó ò = ñ  . This is a Bernoulli equation with respect to  =  ( ) (see Subsection 1.1.5). ( ô ï ÷ + õ ï tanhú ñ )ñ ó ò = ñ  . This is a Bernoulli equation with respect to  =  ( ). ( ô ïƒ÷ + õ ï coshú ñ )ñó ò = cosh  ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

16.

( ô ïƒ÷ + õ ï tanhú ñ )ñó ò = tanh ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

17.

( ôþ

ïƒ÷ ñ ú

–1

18.

( ôþ

ïƒ÷ ñ ú

–1

19.

( ô ï ÷ ñ ú + õ ï )ñ ó ò = cosh ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

20.

167

+ õ )ñó ò + ôù

ïƒ÷ –1 ñ ú

+ ö sinh  (

+ õ )ñó ò + ôù

ïƒ÷ –1 ñ ú

+ ö tanh (

ï

) = 0.

This is a special case of equation 1.7.1.19 with  ( ) =  and ( ) = ( sinh  ( Π ).

ï

) = 0.

This is a special case of equation 1.7.1.19 with  ( ) =  and ( ) = ( tanh  ( Π ).

( ô ïƒ÷ ñ ú + õ ï )ñó ò = tanh ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

© 2003 by Chapman & Hall/CRC

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168 21.

FIRST-ORDER DIFFERENTIAL EQUATIONS ( ô ïƒ÷ coshú ñ + õ ï )ñó ò = sinh  ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

22.

( ô ïƒ÷ tanhú ñ + õ ï )ñó ò = ñ  . This is a Bernoulli equation with respect to  =  ( ).

23.

( ôþ

ïƒ÷ ñ ú

–1

( ôþ

ïƒ÷ ñ ú

–1

24.

+ õ sinh 

ñ )ñ óò

+ ôù

ïƒ÷ –1 ñ ú

+ ö = 0.

+ õ tanh

ñ )ñ óò

+ ôù

ïƒ÷ –1 ñ ú

+ ö = 0.

This is a special case of equation 1.7.1.19 with  ( ) =  sinh 

and ( ) = ( .

This is a special case of equation 1.7.1.19 with  ( ) =  tanh 

and ( ) = ( .

1.5.4. Equations Containing Logarithmic Functions 1. 2. 3. 4. 5. 6.

= ñ (  ï + þ ln ñ +  ). This is a special case of equation 1.7.2.3 with  (ˆ ) = ln ˆ + ý .

ñóò

= ô ï ÷ –1 ñ  ú +1 (ù ln ï + þ ln ñ ). This is a special case of equation 1.7.1.3 with  (ˆ ) = ˆ  ln ˆ .

ñóò

= ô ïƒ÷ ñ ln2 ñ + õ ïƒú ñ ln ñ + ö ï ñ . This is a special case of equation 1.7.3.1 with  ( ) =   , ( ) = ' + , and % ( ) = (˜  .

ñóò

ï ñóò

= ( ñ + ù ln ï )ú +  . + This is a special case of equation 1.7.2.4 with  (ˆ ) = ln ˆ + ý .

ï ñóò

= ñ (ù ln ï + þ ln ñ ). This is a special case of equation 1.7.1.3 with  (ˆ ) = ln ˆ .

þ ï ñóò

=

ô ï s ñ  (ù

ln ï + þ

ln ñ ) – ùñ .

This is a special case of equation 1.7.1.5 with  ( ) = 7. 8. 9.

 ¡



À −1

and (ˆ ) = ln ˆ .

( ï D + õ )ñ ó ò = ñ ï D –1 + ö (ln ñ – ln ï ). This is a special case of equation 1.7.1.12 with  (ˆ ) =  , (ˆ ) = ( ln ˆ , and % (ˆ ) = 1.

ï ( ñ

+  )ñ ó ò = ù ln ï + (  – ù )ñ . This is a special case of equation 1.7.2.16 with  (ˆ ) = ý , (ˆ ) = 1, and % (ˆ ) = ln ˆ .

+ þ ï  )ñ ó ò = ñ ( õÿù ln ï + õÿþ ln ñ – ù ï  ). This is a special case of equation 1.7.1.16 with  (ˆ ) =  , (ˆ ) = 1, and % (ˆ ) =  ln ˆ .

ï (ô

10.

ï (ô

+ þxñ  )ñ ó ò = ñ ( õÿù ln ï + õÿþ ln ñ – ùñ  ). This is a special case of equation 1.7.1.17 with  (ˆ ) =  , (ˆ ) = 1, and % (ˆ ) =  ln ˆ .

11.

( ôþ

12.

( ô ln ñ + õ

ï ÷ ñ ú

–1

+ õ )ñ

óò

+ ôù

ï ÷ –1 ñ ú

+ ö ln  (

ï

) = 0.

This is a special case of equation 1.7.1.19 with  ( ) =  and ( ) = ( ln  ( Π ).

ï )ñ ó ò

= 1.

Solution:  =    ”

 

X

 −  M

+L

•

−

 

ln .

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1.5. EQUATIONS OF THE FORM

13. 14.

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

(ln ñ )ñ ó ò = ñ ( ô ï ÷' ñ  + õ ï ÷ ñ ) – ùñ ln ñ . This is a special case of equation 1.7.3.7 with  (ˆ ) = ˆ  + 'ˆ and

169

ï

ï (ô

+þ

ln ñ )ñ„ó ò = ñ ( õ

ïƒ÷ ñ ú

– ù ln ñ + ö ).

¡

= 1.

This is a special case of equation 1.7.3.9 with  (ˆ ) =  , (ˆ ) = 1, and % (ˆ ) = 'ˆ + ( .

15. 16. 17. 18.

19.

( ô ïƒ÷ + õ ï lnú ñ )ñó ò = ln  ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

ï (ô ï ÷ ñ ú

+ þ ln ï )ñ ó ò = ñ ( õ ï ÷' ñ úI – ù ln ï ). This is a special case of equation 1.7.3.10 with  (ˆ ) = ˆ , (ˆ ) = 1, and % (ˆ ) = 'ˆ  .

ï (ô ƒï ÷ ñ ú

+þ

ïƒ÷' ñ I ú 

ln ñ )ñó ò = ñ ( õ

– ù ln ñ ).

This is a special case of equation 1.7.3.9 with  (ˆ ) = ˆ , (ˆ ) = 1, and % (ˆ ) = 'ˆ  .

ï ÷ ñ ú

( ôþ

–1

+ õ ln 

ñ )ñ ó ò

+

ôù ï ÷ –1 ñ ú

+

ö

= 0.

This is a special case of equation 1.7.1.19 with  ( ) =  ln 

(ô

lnú

ïƒ÷

ñ

+

õ ï )ñ óò

and ( ) = ( .

= ln  ( Ė ).

This is a Bernoulli equation with respect to  =  ( ).

20.

( ô ïƒ÷ lnú ñ + õ ï ln  ñ )ñó ò = ñ s . This is a Bernoulli equation with respect to  =  ( ).

1.5.5. Equations Containing Trigonometric Functions 1.

ñóò

=

ñóò

= sin( ô



cos( ôñ ) +  cos( õ

ï

).

This is a special case of equation 1.7.4.11 with  ( ) = ƒ , ( ) = 0, and % ( ) = ý cos( ' ). 2.

ï

) cos( õÿñ ) + cos( ô

ï

) sin( õÿñ ).

This is a special case of equation 1.7.1.1 with  (ˆ ) = sin ˆ and ( = 0. 3.

= ô tan( õ ï ñ ). The solution is given by the relation:

ñ óò

?

X

0

exp  a  cos  £   a  1 2 2

M

a = L exp  '  , 1 2

2

where ¨ =





.

4.

= õ ï ÷ cos( ôñ ) + ö ï ú . This is a special case of equation 1.7.4.11 with  ( ) = '  , ( ) = 0, and % ( ) = (˜ + .

5.

ñóò

6. 7.

ñ óò

=

õ ïƒ÷

sin( ôñ ) + ö

ïwú

.

This is a special case of equation 1.7.4.11 with  ( ) = 0, ( ) = '  , and % ( ) = (˜ + .

= ñ cos ï ( ôñ ÷¿ú sin÷ –1 ï + õÿñ ú ). This is a special case of equation 1.7.4.4 with  (ˆ ) = ˆ  + 'ˆ .

ñóò

= ñ sin ï ( ôñ ÷¿ú cos÷ –1 ï + õÿñ ú ). This is a special case of equation 1.7.4.3 with  (ˆ ) = ˆ  + 'ˆ .

ñ óò

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Page 169

170 8.

9. 10. 11.

FIRST-ORDER DIFFERENTIAL EQUATIONS sin2 ñ cos2 ñ + õ . cos2 ï sin2 ï This is a special case of equation 1.7.4.14 with  (ˆ ) = ˆ + 'ˆ

ñ óò

=

ô

= ôñ 1+÷ + õÿñ + ö sin( ï )ñ 1–÷ . This is a special case of equation 1.7.1.4 with  ( ) =  , ( ) =  , and % ( ) = ( sin( Œ  ). = ôñ 1+÷ + õ sin( ï )ñ + ö^ñ 1–÷ . This is a special case of equation 1.7.1.4 with  ( ) =  , ( ) =  sin( Œ  ), and % ( ) = ( .

ñóò

ï ñóò

+

ô

sin( õ

ï

+ ö^ñ ) = 0.

B = 2: 2 ¨2
−  ¨

13.

14.

15. 16. 17. 18. 19. 20. 21. 22. 23.

.

ñóò

The substitution ¨

12.

−1

=  tan

2

' + (

2 + 2( !  ( − 1)¨ − '

leads to a Riccati equation of the form 1.2.2.35 with 2

= 0.

= ô ï 2 tan( õÿñ ) + ñ . ¨ 
=  tan( ' ¨ ). The substitution =  ¨ leads to an equation of the form 1.5.5.3: I

ï ñóò

= ô ïƒ÷ cos2 ñ + õ cos ñ sin ñ . This is a special case of equation 1.7.4.8 with  (ˆ ) = 12 ( ˆ +  ).

ï ñóò

= ô ï ÷ sin2 ñ + õ cos ñ sin ñ . This is a special case of equation 1.7.4.7 with  (ˆ ) = 12 ( ˆ +  ).

ï ñ óò

= ô ï ú sin  ñ cos2–  ñ – ù sin 2ñ . This is a special case of equation 1.7.4.18 with  ( ) =  +

ï ñ óò

−2

´

−1

and (ˆ ) = ˆ  .

(1 + tan2 ñ )ñ ó ò = ô tanú +1 ñ + õ tan ñ + ö ï ÷ tan1–ú ñ . This is a special case of equation 1.7.4.19 with  ( ) =  , ( ) =  , and % ( ) = (˜  .

( ôþ ï ÷ ñ ú –1 + õ )ñ ó ò + ôù ï ÷ –1 ñ ú + ö sin  ( ï ) = 0. This is a special case of equation 1.7.1.19 with  ( ) =  and ( ) = ( sin  ( Œ  ).

( ôþ ï ÷ ñ ú –1 + õ )ñ ó ò + ôù ï ÷ –1 ñ ú + ö tan ( ï ) = 0. This is a special case of equation 1.7.1.19 with  ( ) =  and ( ) = ( tan  ( Œ  ).

( ô ïƒ÷ ñ ú + õ ï )ñó ò = cos ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ) (see Subsection 1.1.5). ( ô ïƒ÷ ñ ú + õ ï )ñó ò = tan ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

( ôñ ú cos ï + õ )ñó ò = ñ ú +1 sin ï . This is a special case of equation 1.7.4.3 with  (ˆ ) = ˆ ( ˆ +  ) −1 .

( ôñ ú sin ï + õ )ñó ò = ñ ú +1 cos ï . This is a special case of equation 1.7.4.4 with  (ˆ ) = ˆ ( ˆ +  ) −1 . ( ô ïƒ÷ + õ ï cosú ñ )ñó ò = ñ  . This is a Bernoulli equation with respect to  =  ( ).

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1.5. EQUATIONS OF THE FORM

24. 25. 26. 27. 28. 29. 30.

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY PARAMETERS

171

( ô ï ÷ + õ ï cosú ñ )ñ ó ò = cos  ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

( ôþ ï ÷ ñ ú –1 + õ cos ñ )ñ ó ò + ôù ï ÷ –1 ñ ú + ö = 0. This is a special case of equation 1.7.1.19 with  ( ) =  cos 

( ô ïƒ÷ cosú ñ + õ ï )ñó ò = cos  ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

and ( ) = ( .

( ô ïƒ÷ + õ ï tanú ñ )ñó ò = ñ  . This is a Bernoulli equation with respect to  =  ( ). ( ô ïƒ÷ + õ ï tanú ñ )ñó ò = tan ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

( ôþ ïƒ÷ ñ ú –1 + õ tan ñ )ñó ò + ôù ïƒ÷ –1 ñ ú + ö = 0. This is a special case of equation 1.7.1.19 with  ( ) =  tan 

( ô ïƒ÷ tanú ñ + õ ï )ñó ò = tan ( ƒñ ). This is a Bernoulli equation with respect to  =  ( ).

and ( ) = ( .

1.5.6. Equations Containing Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions 1. 2. 3.

= ô ïƒ÷ ] + õ lnú ï . + This is a special case of equation 1.7.2.5 with  ( ) =   and ( ) =  ln  .

ñóò

= ô ln÷ ( b ï )  ] + õ ï ú . This is a special case of equation 1.7.2.5 with  ( ) =  ln  ( @! ) and ( ) = ' + .

ñ óò

= ô ] ( ƒñ + ln ï )ú . + This is a special case of equation 1.7.2.2 with  (ˆ ) =  ln ˆ .

ñ óò

ó

4.

= ô –  ( ï + ln ñ )ú . + This is a special case of equation 1.7.2.1 with  (ˆ ) =  ln ˆ .

5.

ñóò

= ôñ ln2 ñ + õÿñ ln ñ + ö ñ .  This is a special case of equation 1.7.3.1 with  ( ) =  , ( ) =  , and % ( ) = (   .

6.

ñóò

ñóò

ó

ó

= ôñ ln2 ñ + õ ñ ln ñ + ö^ñ .  This is a special case of equation 1.7.3.1 with  ( ) =  , ( ) =   , and % ( ) = ( .

7.

= ô] sin ï + õ tan ï . This is a special case of equation 1.7.5.6 with  (ˆ ) = ˆ +  .

8.

ñóò

= ( ô sin ñ + õ ) tan ñ . This is a special case of equation 1.7.5.4 with  (ˆ ) = ˆ +  .

9.

ñóò

ñóò

ó

ó

ó

= ô sin2 ñ + õ – cos2 ñ . This is a special case of equation 1.7.5.8 with  (ˆ ) = 12 ( ˆ +  ˆ ).

© 2003 by Chapman & Hall/CRC

Page 171

172 10.

11.

12.

13.

14.

15.

FIRST-ORDER DIFFERENTIAL EQUATIONS cos÷ (

ï ) ]

õï ú

.

ï

).

ñ óò

=

ô

ñóò

=

ô ƒï ÷  ]

+ õ cosú (

ñóò

=

ô ƒï ÷  ]

+ õ tanú (

ñóò

=

ô

ñóò

=

ñóò

=

+

This is a special case of equation 1.7.2.5 with  ( ) =  cos  (“/ ) and ( ) = ' + . This is a special case of equation 1.7.2.5 with  ( ) =   and ( ) =  cos + (“/ ).

ï

).

This is a special case of equation 1.7.2.5 with  ( ) =   and ( ) =  tan + (“/ ).

ï ) ]

tan÷ (

+

õ ïƒú

.

This is a special case of equation 1.7.2.5 with  ( ) =  tan  (“/ ) and ( ) = ' + .


B

ó

cos( ôñ ) +

ó

(4 

sin( ôñ ) +

ó


B-

.

  The substitution ¨ = tan( 12  ) leads to a linear equation: ¨ 
= !$   ¨ + !#   .

ô

sin(

ï

ï

) sinh( ƒñ ) + õ cos(

) cosh( Ė ).

This is a special case of equation 1.7.2.18 with  ( ) =  sin(“/ ), ( ) =  cos(“/ ), and % ( ) = 0. 16.

17.

ñóò

=

ôñ

ln2 ñ +

õÿñ

ln ñ + ö sin ÷ (

óò

ô

tan1+ú

ï )ñ

.

This is a special case of equation 1.7.3.1 with  ( ) =  , ( ) =  , and % ( ) = ( sin  ( Œ  ). (1 + tan2 ñ )ñ

=

ñ

+ õ tan ñ + ö



ó

tan1–ú

ñ

.



This is a special case of equation 1.7.4.19 with  ( ) =  , ( ) =  , and % ( ) = ( ! . 18.

ó

( ô

cos ñ + õ )ñó ò = cot ñ .

This is a special case of equation 1.7.5.5 with  (ˆ ) = ( ˆ +  ) −1 . 19.

ó

( ô

sin ñ + õ )ñ

óò

= tan ñ .

This is a special case of equation 1.7.5.4 with  (ˆ ) = ( ˆ +  ) −1 . 20.

( ô] cos ï + õ )ñó ò = tan ï . This is a special case of equation 1.7.5.6 with  (ˆ ) = ( ˆ +  ) −1 .

21.

( ô

]

sin ï + õ )ñ

óò

= cot ï .

This is a special case of equation 1.7.5.7 with  (ˆ ) = ( ˆ +  ) −1 . 22.

(

^ ó ñ ÷

+

ô )ñ óò

+

õ _ ]

lnú

23.

(

^ ó ñ ÷

+

ô )ñ óò

+

õ _ ]

cosú

24.

(

^ ó

ï

+ ô

= 0.

+ This is a special case of equation 1.7.2.15 with  ( ) =  and ( ) =  ln  .

ï

+

ô

= 0.

This is a special case of equation 1.7.2.15 with  ( ) =  and ( ) =  cos +  . cos÷

ñ

+ ô )ñó ò +

õ _ ]

cosú (

ï

) + ô

= 0.

This is a special case of equation 1.7.2.15 with  ( ) = cos 

and ( ) =  cos + ( Π ).

© 2003 by Chapman & Hall/CRC

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ê ( Ë , Ï , Ï ÒÍ

1.6. EQUATIONS OF THE FORM

) = 0 CONTAINING ARBITRARY PARAMETERS

173

1.6. Equations of the Form c ( ì , í , í ï î ) = 0 Containing Arbitrary Parameters 1.6.1. Equations of the Second Degree in â0ä ã 1.6.1-1. Equations of the form  ( , )(
 )2 = ( , ). 1.

(ñó ò )2 =

ôñ

õï

+

2

.

See equation 1.6.3.43. 2.

3.

(ñó ò )2 =

ñ

ï

+ô

2

õï

+

+ö .

The substitution ¨ = 2 ¨*¨2
− ¨ = 4  + 2  . ü

(ñ

óò

ôñ

)2 =

3

+

õÿñ

+ 

+ö . 3

Solution:  = L¥A\X (  4.

(ñó ò )2 =

ôñ

+

7 õ ï

2

+ ' + ( leads to an Abel equation of the form 1.3.1.2:

+ ( )−1 J

+

2

.

M

.

See equation 1.6.3.26. 5.

6.

(ñó ò )2 =

ôñ

+

7 õ ï

+ö ,

ïƒú

–

ô

The substitution  ¨ = 2  ¨*¨2
− ¨ = ˜ −2  −1 J 2 . ü

(ñó ò )2 =

ñ

+ô

+1

The substitution ¨

þ

2(þ

=2

Q

≠ 0. +  £  + ( leads to an Abel equation of the form 1.3.1.32:

+1 ï + 3)2 +  +

+1

2

+õ .

−

¡

+1  + 3)2

2( 2( + 1) ¡
* ¨ ¨ ¨  − =− ¡  + 2 ( form 1.3.1.10: ¡ ( + 3)2 7.

(ñó ò )

8.

ï (ñ óò

2

= ƒñ + ô ï

2

+ õ ïƒú

¡

+1

2

+ S

+ 1)

J

1 2

leads to an Abel equation of the

+ .

+ö .

For Œ ≠ 0, the substitution Œ ¨ = 2( Œ +  2 + ' + +1 + ( )1 J 2 leads to the Abel equation ¨*¨2
− ¨ = 4 Œ −2  + 2 Œ −2 ( + 1) + , which is outlined in Subsection 1.3.1 (see Table 5). ¡ Special cases of the original equation are equations 1.6.1.1–1.6.1.6. )2 =

ô ï ñ

+õ .

See equation 1.6.3.32. 9.

10.

ï (ñ óò

)2 =

ô ï ñ

+

õï

The substitution  ¨ ¨*¨2
− ¨ = −2 ( −2 

ñ 2 (ñ ó ò

)2 =

ô ï 2ñ

2

+ö , =2  . ü

−2

ô

≠ 0. +  + (˜

−1

leads to an Abel equation of the form 1.3.1.33:

+õ .

See equation 1.6.3.34. 11.

ñ 2 (ñ ó ò

)2 =

ô ï –2 ) 5 ñ

2

+õ .

See equation 1.6.3.28.

© 2003 by Chapman & Hall/CRC

Page 173

174 12.

FIRST-ORDER DIFFERENTIAL EQUATIONS ( ôñ

2

+

õ ï )(ñ óò

)2 = 1.

See equation 1.6.3.44. 13.

ï

(ô

2

+

õÿñ )(ñ óò

)2 =

ï 2ñ

.

See equation 1.6.3.46. 14.

ï ñ

(ô

+ õ )(ñ„ó ò )2 = ñ .

See equation 1.6.3.33. 15.

ï ñ 2 (ñ ó ò

ôñ

)2 =

2

+

õï

.

See equation 1.6.3.45. 16.

ï 2ñ

(ô

2

+ õ )(ñó ò )2 =

ï

2

.

See equation 1.6.3.35. 17.

7

(ô

ñ

õ ï )(ñ óò

+

)2 = 1.

See equation 1.6.3.27. 18.

ï 2 ñ 3 ) 5 + õÿñ )(ñ óò

(ô

)2 =

ï 2ñ

.

See equation 1.6.3.29. 19.

(ñ

óò

ô ]

+õ .

(ñó ò )2 =

ô

õ ó

(ñó ò )2 =

ôñ

)2 =

See equation 1.6.3.3 with , = 2.

20.

+

.

See equation 1.6.3.4 with , = 2.

21.

2

õ ó

+

.

See equation 1.6.3.8 with , = 2.

22.

ï 2 (ñ óò

)2 =

ô ï 2  ]

+õ .

See equation 1.6.3.9 with , = 2. 23.

( ô] +

õï

2

)(ñó ò )2 = 1.

See equation 1.6.3.9 with , = −2.

24.

( ô

óñ

2

óò

+ õ )(ñ

)2 =

ñ

2

.

See equation 1.6.3.8 with , = −2.

25.

(ñó ò )2 =

ôñ

+ õ ln ï .

See equation 1.6.3.13. 26.

27.

(ñ

óò

)2 =

Ė

+

ô

ln ï + õ ,

The substitution Œ ¨ = 2 £ Œ ¨*¨2
− ¨ = 2 Œ −2  −1 .

( ô ln ñ + õ

ï )(„ñ óò

≠ 0. +  ln  +  leads to an Abel equation of the form 1.3.1.16:

)2 = 1.

See equation 1.6.3.14.

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1.6. EQUATIONS OF THE FORM

ê ( Ë , Ï , Ï ÒÍ

) = 0 CONTAINING ARBITRARY PARAMETERS

175

1.6.1-2. Equations of the form  ( , )(
 )2 = ( , )
 + % ( , ). 28.

(ñó ò )2 + ôñó ò + õÿñ = 0. Solution in parametric form:

' = −2a −  ln a + L ,

29.

30.



= −a 2 − a .

(ñó ò )2 + ôññó ò = õ ï + ö . We differentiate the equation with respect to  , take as the independent variable, and assume ˆ = 
 to obtain a linear equation with respect to = (ˆ ): (ˆ 2 −  )
Š + ˆ + 2ˆ 2 = 0. (ñ

óò

)2 +

ô ï ñ óò

+

õÿñ

+

öï

2

= 0.

2 g The transformation  =  , =  2 ‡ leads to an autonomous equation:  ‡ g
+ 2‡ + 12   = 1 2 1 4  − ( − '‡ . Having extracted the root and carried over the terms 2 ‡ + 2  from the left-hand

side to the right-hand side, we obtain a separable equation of the form 1.1.2.

31.

32. 33.

34.

35.

36.

= ï ñó ò + ô ï 2 + õ (ñó ò )2 + ö^ñó ò + ý , ô ≠ 0. Differentiating with respect to  and changing to new variables a = "
 and ¨ (a ) = −2  , we arrive at an Abel equation of the form 1.3.1.2: ¨*¨ g
− ¨ = −4 'a − 2!( .

ñ

(ñ ó ò )2 + ( ô ï + õ )ñ ó ò – ôñ + ö = 0,  L 2 + ( Solutions: = (  +  ) L + 

ô

≠ 0. and 4 

(ñ ó ò )2 + ( ôñ + õ ï )ñ ó ò + ôƒõ ï ñ = 0. This equation can be factorized: (
 +   = L  −  and = − 21 ' 2 + L .

= 4 ( − (  +  )2 .

)(
 + ' ) = 0. Therefore, the solutions are:

(ñó ò )2 + ô ï 2 ñó ò + õ ï ñ = 0. The transformation H = ln  , ‡ =  −3 leads to an equation independent implicitly of H : (‡
‘ )2 + (  + 6‡ )‡
‘ + (3  +  + 9‡ )‡ = 0. Rewriting the latter equation to solve for ‡
‘ , we obtain a separable equation of the form 1.1.2. )2 – ññó ò – ï = 0. Solution in parametric form:

ô (ñ óò

a V L +  ln  a + £ a 2 + 1  W ,  = £ 2 a +1

ï (ñ óò

= a −

 a

.

)2 – ôññó ò + õ = 0. 1 b . For  ≠ 1, the solution in parametric form is written as:

 = L3a  +



2 − 1

2 b . For  = 1, the solution is: 37.

−1

a 2,

a = a 2 +  ,

where

, =

= L3 +  L .

1 .  −1

)2 + ôññó ò + õ ï = 0. 1 b . For  ≠ −1, the solution in parametric form is written as:

ï (ñ óò

 = L3a |(  + 1)a 2 +  | There are two singular solutions:

= A"

ü



+2 − 2( +1)



,

−  (  + 1).

=−

 2 (a +  ).  a

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176

FIRST-ORDER DIFFERENTIAL EQUATIONS 2 b . For  = −1, the solution in parametric form is written as:

 = L3a exp O − 38.

ï (ñ óò

)2 – ññ

óò

a2 , 2 P

= O a +



a P

.

+ ôñ = 0.

Solution in parametric form:

 = L (a −  ) exp(−a  ), = 0.

There is a singular solution: 39.

40.

ï (ñ óò

)2 – ññ

ñ (ñ óò

)2 +

óò

+ô

ï 2 ñ óò

+

= L3a 2 exp(−a  ).

õÿñ óò

+

ö

ô

= 0,

≠ 0.

We divide the equation by 
and differentiate with respect to  . Passing to the new variables a = 
 and ¨ (a ) = −2 , we arrive at an Abel equation of the form 1.3.1.33: ¨*¨pg
− ¨ = !(˜a −2 .

ô ï ñóò

+

õÿñ

= 0.

Solution in parametric form:

a + (  + a 2 ) = 0,

(a 2 +  +  ) + = a  J ( 

L



+ )

,

where

¡

=

 + 2 . 2(  +  )

There two singular solutions = A" £ −  −  corresponding to the limit LZD = 0 is also a singular solution. 41.

ï (ñ óò

)2 + ( ô – ñ ) ñ

óò

õ

+

F . In addition,

= 0.

Solutions: L ( L3 − +  ) +  = 0 and ( −  )2 = 4 ' . 42.

ô ï (ñ óò

)2 + ( õ

ï

– ôñ + ü )ñó ò –

, L

õÿñ

= 0.

. In addition, there is a singular solution which can be written in Solution: = L3 + L +  parametric form as:

 =− 43.

ô ï (ñ óò

)2 – ( ôñ +

õï

– ô – õ )ñ„ó ò +

45.

ï (ñ óò

)2 + ôññó ò +

,!a . a + 

= 0.

õ ƒï ÷ ñ ú

L ( +  ) L − 

+ ' −  −  )2 − 4 '

and ( 

= 0.

= 0.

g The substitution  =  leads to an equation of the form 1.6.1.68: ("
g )2 +  !
g +   ( 

ï 2 (ñ ó ò

)2 – (2 ï

ô ï 2 (ñ óò

= L

2

A ü

óò

+ñ

 + L

ï ññóò

)2 – 2 ô

Solutions:

+ ô )ñ

ñ

Solutions: 46.

õÿñ

= a +

,

Differentiating with respect to  and factorizing, we obtain (2  "
 −  − ' +  +  )

 = 0. Equating both factors to zero and integrating, we arrive at the solutions: = L3 +

44.

,

( a +  )2

2

2

+1)

g + = 0.

= 0. = − 41 

and

+ñ

2

– ô ( ô – 1) ï

+ 

2

= L3

1+

2

−1

.

= 0.

 , where , = ü

(  − 1)  .

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ê ( Ë , Ï , Ï ÒÍ

1.6. EQUATIONS OF THE FORM

47.

(ô

2

– 1) ï

2

(ñó ò )2 + 2 ï

ññóò

–ñ

+ô

2

2

ï

) = 0 CONTAINING ARBITRARY PARAMETERS

2

177

= 0.

Solution in parametric form:

 = L (a 2 + 1)−1 J 2  a + £ a 2 + 1  48.

ï 2 (ñ ó ò

)2 + ( ô

ï 2ñ

3

+ õ )ñ

óò

ññóò

–ï

+ ôƒõÿñ

3

−1

J

= a +  £ a 2 + 1.

,

= 0.

The equation can be factorized: (
 +  3 )( 2 
 +  ) = 0. Equating each of the factors to zero, we obtain the solutions: = A (2  + L )−1 J 2 and =   + L .

49.

(ï

2

50.

(ï

2

51.

(ï

2

– ô )(ñó ò )2 – 2 ï

2

= 0.

Solving for , differentiating with respect to  , and setting ¨ ( ) =
 , we obtain a factorized equation: ( ¨3
− ¨ )( 2 ¨ 2 +  2 −  ¨ 2 ) = 0. Equating each of the factors to zero, we arrive at the solutions: 1 2 2 = ( −  − L 2 ) and +  2 =  ( ≠ 0). 2L – ô 2 )(ñó ò )2 + 2 ï

+ñ

ññóò

2

= 0.

The equation can be factorized: ( 
 +  
 + )( 
 −  
 + ) = 0. Equating each of the factors to zero, we obtain the solutions: ( +  ) = L and ( −  ) = L .

óò

+ ô )(ñ

)2 – 2 ï

ññ óò

+ñ

2

õ

+

= 0.

Differentiating with respect to  , we obtain a factorized equation: [( Therefore, the solutions of the original equation are: =L 52.

ï 3 (ñ óò

)2 +

ï 2 ñ ñóò

ô ï ñ (ñ óò

=L

)2 – ( ôñ

2 1

where L

+L

2 2

+  = 0;

'

2

+

2

+  )
 − 

]

 = 0.

+  = 0.

+ ô = 0.

Solutions: L3 53.

 + L 2,

1

2

2

2

õï

+

 + 2

and 

2

+ ü )ñó ò +

õï ñ

= 4 . = 0.

This differential equation represents an equation of curvature lines of a surface defined by the relation #* 2 + $ 2 + LuH 2 = 1, where  = #2$ ( L − $ ),  = #2$ ( # − L ), , = L ($ − # ). Solutions: ( L −  ) 54.

ñ 2 (ñ óò

)2 + 2 ô

ï ññóò

Solutions:

55.

( ô – õ )ñ

2

= L ( L −  )

+ (1 – ô )ñ 2

(ñó ò )2 – 2 õ

2

+ 

ï ññóò

2

2

+

ô ï

2

−, L

2

+ ( ô – 1) õ = 0.

and

−  = (  − 1)( + L )2 + ôñ

2

–

õï

2



2

and

= '

2

2

+ 

A 2 £ − , − , .

2

−  = 0.

– ôƒõ = 0.

Solutions:

 56.

( ôñ – ï

2

2

+

)(ñó ò )2 + 2 ï

Solution: ( L

2

= L3 +  −

ññóò

–ñ

+  )2 = 4 

.

2

 − L 4

2

and (  −  )

2

− '

2

= ( −  )  .

= 0.

© 2003 by Chapman & Hall/CRC

Page 177

178 57.

FIRST-ORDER DIFFERENTIAL EQUATIONS (ñ 2 – ô 2 ï 2 )(ñó ò )2 + 2 ï ññó ò + (1 – ô 2 ) ï Solution in parametric form:

2

= 0.

L3a ,  = £ 2 a +1

58.

( ôñ – õ ï )2 [ ô 2 (ñó ò )2 + õ 2 ] – ü 2 ( ôñó ò + õ )2 = 0. We solve the equation for  − ' and differentiate with respect to  . Setting ¨ ( ) = 
, we obtain a factorized equation with respect to ¨ ( ): (  ¨ −  )[(  2 ¨ 2 +  2 )3 J 2 A\, ¨2
] = 0. Equating each of the factors to zero and integrating, we arrive at the solutions: ( ' − L )2 + ( 

59.

60.

(ï

ñóò

(ï

ñóò

63.

64.

ïƒ÷ +1 ñ óò

+ ùñ )2 + ô =

L

B

2 2

B

ï 2÷

+ ùñ )2 – ô

+

+2

õ

+

67.

− ' = A/, £ 2.

(ñó ò )2 – õ = 0. 2

+ 'B

−2

.

(ññó ò + ï )2 = ô 2 ( ï 2 + ñ 2 )  [(ñó ò )2 + 1]. This equation splits into two equations of the form 1.8.1.4 with  (‡ ) = A"‡ J 2 :

! 
+  = A  (

ô ï 2÷ (ï ñ óò

2

+

2

) J

2


 2 ü ( ) + 1.

+ þxñ )2 + õ ï 2ú ( ï ñó ò + ùñ )2 = 1. = L 1  −  + L 2  − + . Here, the constants L Solution: 2 2 ( L 1 + L 2 )(B − )2 = 1.

( ï ñó ò

1

and L

2

are related by the constraint

¡

– ñ ) = ô ( ï 2 + ñ 2 )  (ññó ò + ï )2 . This equation splits into two equations of the form 1.7.1.20 with  (‡ ) = A ‡ J 2 : 2

2

2

2

+


) J 2 (!  +  ).

( ï ñó ò – ñ )2 = ô 2 ( ï 2 + ñ 2 )  [(ñó ò )2 + 1]. This equation splits into two equations of the form 1.8.1.3 with  (‡ ) = A"‡ J 2 :

 
− = A  (

66.



and

= 0.

 
− = A  (

65.

2

 − + L .

£ L = L3 −  A

Solutions:

62.

− L )2 = ,

ô ≠ 0. ( ï ñó ò + ô )2 – 2 ôñ + ï 2 = 0, 2 2 The substitution 2  −  = ‡ leads to the equation ‡"‡
 −  (‡ −  )+  2 = 0. Further assuming ‡ −  =  ¨ ( ), we obtain ( ¨ +  )¨3
+ ¨ 2 + 1 = 0. Taking ¨ to be the independent variable, we arrive at a linear equation whose solution is:  = (¨ 2 + 1)−1 J 2 V L −  ln  ¨ + £ ¨ 2 + 1  W . Solution:

61.

L

. = L − £ a 2+1

2

+

2

) J

2


 2 ü ( ) + 1.

( ï ñó ò + ñ + 2 ô ï )2 = 4( ï ñ + ô ï 2 + õ ). The substitution ‡ =  +  2 +  leads to a separable equation: ‡
 = A 2 £ ‡ .

( ô 2 ï + õ 2 ñ + ö 2 )(ñó ò )2 + ( ô 1 ï + õ 1 ñ + ö 1 )ñó ò + ô 0 ï + õ 0 ñ + ö 0 = 0. The Legendre transformation  = ‡ g
, = aև g
− ‡ (
 = a ) leads to a linear equation: [  (a ) + aÝ (a )]‡ g
= (a )‡ + % (a ),

where  (a ) =  2 a 2 +  1 a +  0 , (a ) =  2 a 2 +  1 a +  0 , and % (a ) = − ( 2 a 2 − ( 1 a − ( 0 .

© 2003 by Chapman & Hall/CRC

Page 178

ê ( Ë , Ï , Ï ÒÍ

1.6. EQUATIONS OF THE FORM

68.

(ñ

óò

)2 +

ôññ óò

= 0.

ñóò

+ñ =

¡

ô 3ó (ñ óò

ñóò

– ñ )2 .

= 12   L

Solution: 70.

179

 ≠ 2, solving for 
and performing the substitution ¨ = ´ + −2 , we arrive at a 2− ¡  ‹A £  2 − 4  ¨  ¨ (see Subsection 1.1.2). separable equation: ¨ 
= Œ ¨ + 2 = 2, solving the original equation for "
 , we obtain a separable equation: 2 b . With ¡ £ 2 
 − 4    . 2  =  − ‹A 1 b . With

69.

õ  ó ñ ú

+

) = 0 CONTAINING ARBITRARY PARAMETERS

2

= (ñó ò 2 – ñ 2 )( ô

ó





− 12 L 

+

õ –ó

−



.

).

This is a special case of equation 1.8.1.56 with  (‡ ) = ‡ . 71.

72.

ô 2 ó (ñ óò

+ ñ )2 +

õ 2_ ó (ñ óò

+



Ė

)2 = 1.

 = L 1  −  + L 2  −: . Here, the constants L Solution: constraint ( L 12 + L 22 )(ý − Œ )2 = 1.

ñóò

= ô (ñó ò 2 – ñ 2 ) cosh ï .

ñóò

= ô (ñó ò 2 – ñ 2 ) sinh ï .

1

and L

are related by the

2

This is a special case of equation 1.8.1.58 with  (‡ ) = ‡ . 73.

This is a special case of equation 1.8.1.59 with  (‡ ) = ‡ . 74.

ô (ñ óò

cosh ï – ñ sinh ï )2 + õ (ñó ò sinh ï – ñ cosh ï )2 = 1.

= L 1 sinh  + L Solution: constraint L 12 + L 22 = 1.

75.

(ñó ò )2 – ï

ññóò

Solutions: 

76.

+ñ

2

cosh  . Here, the constants L

ln( ôñ ) = 0.

= exp( L3 − L

2

1

and L

2

are related by the

= exp( 41  2 ).

) and 

) cos ï .

ñóò

= ô (ñó ò 2 + ñ

ñóò

= õ (ñó ò 2 + ñ 2 ) sin ï .

2

2

This is a special case of equation 1.8.1.64 with  (‡ ) = ‡ . 77.

This is a special case of equation 1.8.1.65 with  (‡ ) = ‡ and  = 0. 78.

ô (ñ óò

cos ï + ñ sin ï )2 + õ (ñó ò sin ï – ñ cos ï )2 = 1.

Solution: = L 1 sin  + L L 12 + L 22 = 1. 79.

ô (ñ óò

2

cos  . Here, the constants L

cosh ï – ñ sinh ï )2 + õ [(ñ

óò

)2 – ñ 2 ] +

ö

1

and L

ô (ñ óò

cos ï + ñ sin ï )2 + õ [(ñó ò )2 + ñ

Solution: L 12 +  ( L

2 1

2

are related by the constraint

1

and L

2

are related by the constraint

= 0.

= L 1 sinh  + L 2 cosh  . Here, the constants L Solution: constraint L 12 +  ( L 12 − L 22 ) + ( = 0. 80.

2

] + ö = 0.

= L 1 sin  + L 2 cos  . Here, the constants L + L 22 ) + ( = 0.

1

and L

2

are related by the

© 2003 by Chapman & Hall/CRC

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180

FIRST-ORDER DIFFERENTIAL EQUATIONS

1.6.2. Equations of the Third Degree in

â0äã

1.6.2-1. Equations of the form  ( , )( 
)3 = ( , ) 
+ % ( , ). 1. 2.

3. 4. 5. 6. 7.

(ñó ò )3 + ô ï + õÿñ + ö = 0. This is a special case of equation 1.8.1.9 with  (¨ ) = ¨

( ô ï + õÿñ + ö )(ñ„ó ò )3 =  ï Dividing both sides by  of the form 1.7.1.6 with 

9. 10. 11. 12.

.

+ ñ + . +  + ( and raising to the power 1/3, we finally arrive at an equation (¨ ) = ¨ −1 J 3 .

)3 + õÿñó ò = ï . This is a special case of equation 1.8.1.7 with  (¨ ) =  ¨

ô (ñ óò ô (ñ óò

)3 + õÿñó ò = ñ . This is a special case of equation 1.8.1.8 with  (¨ ) =  ¨

3

+ ¨ .

3

+ ¨ .

)3 + ï ñó ò = ñ . This is a special case of equation 1.8.1.10 with  (¨ ) =  ¨

ô (ñ óò

3

.

)3 + õ ï ñó ò = ñ . This is a special case of equation 1.8.1.11 with  (¨ ) =  ¨ and ( ) =  ¨

ô (ñ óò

(ñó ò )3 – ô ï ñó ò + ï 3 = 0, Solution in parametric form:

ô

3

.

≠ 0.

 = 8.

3

a

a 3+1

,

=L +



4a 3 + 1 . 6 (a 3 + 1)2 2

(ñó ò )3 – ô ï ññó ò + 2 ôñ 2 = 0. Differentiating with respect to  and eliminating , we obtain a factorized equation with respect to ¨ ( ) = 
: [2(¨2
)2 −  ¨2
+  ¨ ](9¨ −  2 ) = 0. Equating each of the factors to 1  3 . zero and integrating, we find the solutions: = 14 L ( − L )2 and = 27

ô ï (ñ óò

)3 + õÿñó ò = ñ . This is a special case of equation 1.8.1.11 with  (¨ ) =  ¨

ô ï 3 ) 2 (ñ óò

)3 + 2 ï ñó ò = ñ . = 2 L £  + L Solution:

ô ïƒ÷ (ñ óò

3

.

)3 + ï ñó ò = ñ . This is a special case of equation 1.8.1.15 with  (¨ ) =  ¨

ô 3ó (ñ óò

and (¨ ) =  ¨ .

3

)3 + õ (ñó ò + ñ ) + ö = 0.  = L  − + ( L 3 − ( )  Solution:

−1

3

.

.

1.6.2-2. Equations of the form  ( , )(
 )3 = ( , )(
 )2 + % ( , )
 + & ( , ). 13.

)3 + õ (ñó ò )2 = ï . This is a special case of equation 1.8.1.7 with  (¨ ) =  ¨

ô (ñ óò

3

+ ¨

2

.

© 2003 by Chapman & Hall/CRC

Page 180

ê ( Ë , Ï , Ï ÒÍ

1.6. EQUATIONS OF THE FORM

14. 15.

ô (ñ óò

)3 + õ (ñó ò )2 = ñ . This is a special case of equation 1.8.1.8 with  (¨ ) =  ¨

(ñó ò )3 + ô (ñó ò )2 + õÿñ + ôƒõ ï + Solution in parametric form:

ý

2

ln(a +  ) + L ,

In addition, there is a singular solution: 16.

17. 18. 19.

20.

21. 22. 23. 24.

25.

26. 27.

2

.

= − ' − a 3 − a 2 − .



M

M

2

 = L + 32  a 2 + 2 'a + ( ln |a |,

= a 3 + 'a 2 + (˜a − .

M

) =ñ . This is a special case of equation 1.8.1.11 with  (¨ ) =  ¨ 3

+ ¨

= −  −  .

ô (ñ óò ) + õ (ñ óò ) + ö^ñ óò = ñ + ý . Solution in parametric form: ô (ñ ó ò ) + õ ï (ñ ó ò

3

= 0.

2 ' = −3a 2 + 2 a − 2 

3

181

) = 0 CONTAINING ARBITRARY PARAMETERS

2

ô ï (ñ óò

)3 + õ (ñó ò )2 = ñ . This is a special case of equation 1.8.1.11 with  (¨ ) =  ¨

2

and (¨ ) =  ¨

3

.

3

and (¨ ) =  ¨

2

.

( ï 2 – ô 2 )(ñó ò )3 + õ ï ( ï 2 – ô 2 )(ñó ò )2 + ñó ò + õ ï = 0. The equation can be factorized: (
 + ' )[(
 )2 ( 2 −  2 )+1] = 0, whence we find the solutions: = − 21 ' 2 + L and = A arcsin(  ) + L .

ô (ñ óò

ó

+ ñ )3 + õ 3 (ñ ó ò )3 + ö = 0.  = L 1 − L 2  − . Here, the constants L Solution: L 13 + L 23 + ( = 0.

1

and L

2

are related by the constraint

( ï ñ ó ò – ñ )3 + ôñ + õ ï = 0. This is a special case of equation 1.8.1.16 with  (¨ ) = 1, (¨ ) =  , % (¨ ) =  , and B = 3.

( ï ñó ò – ñ )3 + ôññó ò + õ ï = 0. This is a special case of equation 1.8.1.16 with  (¨ ) = 1, (¨ ) =  ¨ , % (¨ ) =  , and B = 3.

( ï ñó ò – ñ )3 + ô ï ñó ò + õÿñ = 0. This is a special case of equation 1.8.1.16 with  (¨ ) = 1, (¨ ) =  , % (¨ ) =  ¨ , and B = 3. cosh ï – ñ sinh ï )3 + õ (ñ ó ò sinh ï – ñ cosh ï )3 + ö = 0. = L 1 sinh  − L 2 cosh  . Here, the constants L 1 and L Solution: constraint L 13 + L 23 + ( = 0.

ô (ñ óò

cosh ï – ñ sinh ï )3 + õ (ñ ó ò 2 – ñ 2 ) + ö = 0. = L 1 sinh  + L 2 cosh  . Here, the constants L Solution: constraint L 13 +  ( L 12 − L 22 ) + ( = 0.

ô (ñ óò

cosh ï – ñ sinh ï )3 = õ (ñó ò 2 – ñ 2 ) cosh ï . This is a special case of equation 1.6.4.21 with B = 3 and

ñóò (ñ óò

sinh ï – ñ cosh ï )3 = õ (ñó ò 2 – ñ 2 ) sinh ï . This is a special case of equation 1.6.4.22 with B = 3 and

1

and L

2

are related by the

2

are related by the

= 1.

ñóò (ñ óò

¡ ¡

= 1.

© 2003 by Chapman & Hall/CRC

Page 181

182 28.

29. 30. 31.

FIRST-ORDER DIFFERENTIAL EQUATIONS cos ï + ñ sin ï )3 + õ (ñó ò sin ï – ñ cos ï )3 + ö = 0. Solution: = L 1 sin  − L 2 cos  . Here, the constants L 1 and L L 13 + L 23 + ( = 0.

ô (ñ óò

2

are related by the constraint

cos ï + ñ sin ï )3 + õ (ñó ò 2 + ñ 2 ) + ö = 0. This is a special case of equation 1.6.4.24 with B = 3 and , = 1.

ô (ñ óò

cos ï + ñ sin ï )3 = õ (ñó ò 2 + ñ 2 ) cos ï . This is a special case of equation 1.6.4.25 with B = 3 and

ñóò (ñ óò

= 1.

sin ï – ñ cos ï )3 = õ (ñ ó ò 2 + ñ 2 ) sin ï . This is a special case of equation 1.6.4.26 with B = 3 and

¡

ñ ó ò (ñ ó ò

= 1.

¡

1.6.3. Equations of the Form (â$ä ã ) d = å ( â ) + ç ( æ ) 1.6.3-1. Some transformations. 1 b . In the general case, the equation ( 
)  =  ( ) + ( )

(1)

can be reduced with the aid of the transformation a = X [ ( )]1 J'  , ‡ = X [  ( )]−1 J' M M same form (‡ g
)  =  (‡ ) + 4 (a ), where functions 

=  (‡ ) and 4

to the

= 4 (a ) are defined parametrically by the following formulas:

 (‡ ) = 4 (a ) =

1

 ( )

,

1 , ( )

‡ = X [  ( )]−1 J' a = X [ ( )]1 J'

,

M M

 .

2 b . Taking as the independent variable, we obtain from equation (1) an equation of the same class for  =  ( ): (
)−  = ( ) +  ( ). 3 b . The equation




=  £

+ ( )

 , = 1,  =  ¨ can be reduced with the aid of the substitution ( ) = 2  −1 £ 2  −2 ( ), which is outlined in Subsection 1.3.1. 4 b . The equation 
= −1 + ( ) ( , = 1,  = is an alternative form of representation of the Abel equation !"
 Subsection 1.3.2. À 5 b . The equation 
=  + ( )  , = 1,  = 

£



to the Abel equation ¨*¨3
− ¨

−1

=

)

= ( ) + 1, which is outlined in

À 

can be reduced, with the aid of the substitution  ¨

= − ù ( )  followed by raising both sides M

À of the equation to the power of 1 ° , to an equation of the class in question: ( ¨ 
)1 J =  ¨ + X (  )  .

M

© 2003 by Chapman & Hall/CRC

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ê ( Ë , Ï , Ï ÒÍ

1.6. EQUATIONS OF THE FORM

6 b . The equation

( 
)2 = 

183

) = 0 CONTAINING ARBITRARY PARAMETERS

( , = 2,  =  ,  ≠ 0) ¨ can be reduced with the aid of the substitution  = 2 £  + ( ) to an Abel equation of the second kind:

+ ( )

¨*¨ 
= ¨ + Y ( ),

where

Y = 2

−2


( ),

which is outlined in Subsection 1.3.1. 7 b . The equation

( 
)1 J

2

=

+ ( )

( , = 1 2,  = 

)

can be reduced by squaring both sides and performing the substitution H =  equation: H 
= !H 2 + 
.

+ ( ) to the Riccati

For some specific functions = ( ), the solutions of the latter equation are given in Section 1.2. 8 b . The equation

( 
)1 J

2

J + ( )

1 2

=

( , = 1 2,  = 

J )

1 2

can be reduced by squaring both sides and performing the substitution equation of the second kind:

ˆˆ 
=  exp(− 21  2  ) !ˆ + 12 exp(−  2  )

= exp(  2  )ˆ

2

to an Abel

2

(see Subsection 1.3.3). 9 b . The equation

( 
)−1 J

2

=  ( ) + 

( , = −1 2, =  )

can be reduced by squaring both sides and performing the substitution  =  ( ) +  to a Riccati equation: 
=  2 + 
.





For some specific functions  =  ( ), the solutions of the latter equation are given in Section 1.2. 1.6.3-2. Classification tables and exact solutions. For the sake of convenience, Tables 9–13 given below list all the equations outlined in Subsection 1.6.3. The five tables classify the equations in which functions  and are of the same form. The right-most column of a table indicates the numbers of the equations where the corresponding solutions are given. After the tables follow the equations—they are arranged into groups so that the solutions of the equations within each group are expressed in terms of the functions indicated before the groups as a notation list. 1.

(ñó ò )  =


ñ

s

+

(

.

Solution:  = X ( # 2.

(ñó ò )  =




+

4.

(ñó ò )  =

+ $ )−1 J'

.


B]

+

(

Ÿ




Solution:

+

(4

 +L . M

.

Solution:  = X ( #   + $ )−1 J'

(ñó ò )  =

+L .

M

= X ( # + $I )1 J'

Solution: 3.

( ïe

À

ó

.



= X ( # + $  )1 J'

M

M

+L .

 +L .

© 2003 by Chapman & Hall/CRC

Page 183

184

FIRST-ORDER DIFFERENTIAL EQUATIONS

À

TABLE 9 Solvable equations of the form ( 
)  = #

°

&

Equation

,

°

&

Equation

arbitrary

arbitrary (° ≠ , )

,° , −°

1.6.3.7

−1

1

−1

1.6.3.23

arbitrary

arbitrary

0

1.6.3.1

−1

1 2

1.6.3.42

1.6.3.6

−1 2

1

1.6.3.17

1.6.3.2

−1 2

1 arbitrary ( ° ≠ −1, 0)

1 arbitrary (& ≠ −1, 0)

1.6.3.38

,

,

−

arbitrary

1− , 0

1+ , arbitrary

arbitrary

1

1

1.6.3.5

1 2

1

−2

−1

−2

1.6.3.46

1 2

1

−1

1.6.3.37

−2

−1

1

1.6.3.33

1

−1

−2

1.6.3.20

−2

1.6.3.29

1

−1

−1 2

1

1.6.3.27

1

−1

1

1.6.3.22

−2

1.6.3.30

−1

1.6.3.11

−1

1.6.3.16

−2

−2 5 1 2

−2

2

−2

1.6.3.35

1

−2

2 arbitrary ( ° ≠ 0) arbitrary ( ° ≠ −2, 0)

1

1.6.3.44

1

1 2 1 2

1

1.6.3.10

1

1 2

−1 2

1.6.3.24

2

1.6.3.15

1

1 2

1

1.6.3.41

−1

−2

−1

1.6.3.21

2

−2

−1

−2

1.6.3.31

2

−2

−2 5

1.6.3.45

−1

1 2

−1

−2

2

1.6.3.36

2

−2

2

1.6.3.34

−1

−1

1.6.3.12

2

1

−1

1.6.3.32

1.6.3.40

2

1

1 2

1.6.3.25

2

1

2

−2

−1 −1

−1 −1

1 2

−1 2 −1 2

−1

1 2

1.6.3.39

1.6.3.28

1.6.3.26 1.6.3.43

(ñó ò )  =
ñ + ( ï . Solution in parametric form:

 = X ( #*U 1 'J  + $ )−1 U + L , M 6.

Ÿ

,

arbitrary ( , ≠ −1, 1)

5.

+ $I





(ñó ò )  =
ñ 1–  + ( ï – 1+  , Solution in parametric form:

 = l” X where # = 



− 1+

 

 

+L M • = U 1 'J  + ý



#

1

¬ U − $ÈX ( #*U 1 'J  + $ )−1 U − $çL2® . M

| ü | ≠ 1.

U

− 1−

=

 ý…$ , $ = 



1+

+1

,

= /”U − ý X

U

− ýhL M • = U 1 'J  + ý

 

−1

,

  ¬  ( , − 1) ="® .  ( , + 1)

© 2003 by Chapman & Hall/CRC

Page 184

1.6. EQUATIONS OF THE FORM

ê ( Ë , Ï , Ï ÒÍ

185

) = 0 CONTAINING ARBITRARY PARAMETERS

TABLE 10 Solvable equations of the form Ÿ (
 )  = #  + $I

TABLEÀ 11 Solvable equations of the form  +$  (
 )  = #

,

&

Equation

,

arbitrary

−,

1.6.3.9

arbitrary

,

1.6.3.8

arbitrary

0

1.6.3.3

arbitrary

0

1.6.3.4

−1

−1

1.5.2.40

−1

arbitrary

1.5.2.37

−1

1

1.5.2.38

1 2

1

1.6.3.18

−1

−1 2

2

1.5.2.39

1

−1

1.5.2.14

1

1.6.3.19

1

arbitrary

1.5.2.13

7.

8.

Equation

−1

1.5.2.8

1

1.5.2.2



(ñó ò )  =
ñ s + ( ï  –s , À Solution in parametric form:

ñ óò ) (

s

 =L 

−

=L 

™

ó +4 ( 

=
ñ 

Equation

(
 )−2 = # ln + $I

1.6.3.14

(
 )−1 = # ln + $I (
 )2 = # + $ ln 

ü

≠ s.

À

1.5.4.12 1.6.3.13

À

1  , −° O# + $IU −À   − U!® ° À P 1  , −° O# + $IU −   − !U ® U exp X¬ ° P

−1

M −1

M

Uƒ¢ , À  − ƒU ¢ .

. Solution in parametric form: 1 ( # + $  −  * )1 J' − S

Q

1

 = exp ¦ U −

(ñó ò )–1 =


ñ

s

+

Solution:  = 

( ï

=X .

P  O# X

Q

X

,

(ñó ò )  =
B] + ( ï –  . Solution in parametric form:

,

X

Q

−1

1 ( $ + #   * )−1 J' + S

−1

Q

,

+L

P

−1

,

M

M

L

U +

,

À

M

U +L , M

1 ( # + $  −  * )1 J' − S

1 ( $ + #   * )−1 J' + S

 −P 

−1

,

1

= exp ¦U +

10.

Form of equation

exp ™ X¬

 =X

9.

Equation

TABLE 13 Solvable equations containing logarithmic functions

TABLE 12 Solvable equations of the form  (
 )  = #  + $ 

,

°

M

U −

§ .

L

,

,

§ ,

U +L .

.

© 2003 by Chapman & Hall/CRC

Page 185

186

]

FIRST-ORDER DIFFERENTIAL EQUATIONS

In the solutions of equations 11–14, the following notation is used:

11.

=
ñ 1 ) 2 + ( ï –1 . Solution in parametric form:

2

=  [2U2AC exp(T"U 2 )] ,

(ñó ò )–1 =
ñ –1 + ( ï 1 ) 2 . Solution in parametric form:

2

 =  [2U2AC exp(T"U 2 )] ,

13.

= ˜ exp(T U 2 ),

(ñó ò )2 =
ñ + ( ln ï . Solution in parametric form:

where # = 4 

−2

# = A 2

where

# = T 4 , $ = A 2 1 J 2 

2

2

.

−1

.

,

 , $ = T 4 ˜# .

(ñ ó ò )–2 =
ln ñ + ( ï . Solution in parametric form:

2

 = , [2U2AC exp(T"U 2 )] A 4 ln( ˜ ) − 4U 2 . , where # = T 4 !$ , $

]

 J , $ = T 4 .

−1 1 2

where

= f, [2U2Aç exp(T U 2 )] A 4 ln( ! ) − 4U

 = ! exp(T U 2 ), 14.

.

ñóò

 = ! exp(T"U 2 ),

12.

−1

 = X exp(T"U 2 ) U + L2S Q M

= 4 

−2

= ˜ exp(T U 2 ),

.

In the solutions of equations 15–19, the following notation is used:

:

=

™

L

ø

n Á (U ) for the upper sign, Á (U ) + L 2 L 1 ®Á (U ) + L 2 ; Á (U ) for the lower sign, 1

where ø Á (U ) and n Á (U ) are the Bessel functions, and ®Á (U ) and ; Á (U ) are the modified Bessel functions. x3y'z|{}~/€ : : : The solutions of equations 15–19 contain only the ratio
= (ln )
. Therefore, * * for the sake of symmetric appearance, two “arbitrary” constants L 1 and L 2 are indicated in the : definition of function (instead, we can set, for instance, L 1 = 1 and L 2 = L ). 15.

s ≠ –2, s ≠ 0. (ñó ò )–1 =
ñ s + ( ï 2 , Solution in parametric form:

 = U where @ = 16.

1 ° +2 , # =T  ° +2 2

−2 −1−

≠ –1, (ñ ó ò )1 ) 2 =
ñ + ( ï e , Solution in parametric form:

Á V U (ln : )
+ @W , À

= 'U

*

, $

=−

° +2 2



−1 −1



2

Á ,

.

≠ 0.

Á U (ln : )
+ @2A & + 1 U 2 S , * 2& Q 1 J 2 1 (& + 1)  & + 1 −Ÿ , # =  −1 − S , $ =T  ˜# . where @ = & +1 Q 2 2&  = U 2 Á ,

= 'U

2

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Page 186

1.6. EQUATIONS OF THE FORM

17.

ê ( Ë , Ï , Ï ÒÍ

s ≠ –1, s ≠ 0. (ñó ò )–1 ) 2 =
ñ s + ( ï , Solution in parametric form:

:  = U 2 Á U (ln À Q 1 ° +1 − , # =T  $ , where @ = ° +1 2°

18.

° +1 2 Á U S , = 'U 2 , * 2° ( ° + 1)  1 J 2 S . $ =  −1 − Q 2

)
+ @*A

ó

(ñó ò )1 ) 2 =
ñ + (4 . Solution in parametric form:

: =  V U (ln )
A

 = ln( U 2 ), where @ = 0, # =  19.

1J 2  − 12   , $ = T

−1

1 −1 2



(ñó ò )–1 ) 2 =
B] + ( ï . Solution in parametric form:

*

where @ = 0, # = T

1 2



−1

$ , $ =

−1

1 2 2

U W,

*

:  =  V U (ln )
A

]

187

) = 0 CONTAINING ARBITRARY PARAMETERS

˜# .

1 2

U 2W ,

= ln( 'U 2 ),

1J 2  − 21   .

In the solutions of equations 20–35, the following notation is used:

:

=

™ L

L

ø

J U ) + L 2 n 1 J 3 (U ) for the upper sign, 1 J 3 ( U ) + L 2 ; 1 J 3 ( U ) for the lower sign,

1 1 3( 1

®

where ø 1 J 3 (U ) and n 1 J 3 (U ) are the Bessel functions, and ® 1 J 3 (U ) and ; 1 J 3 (U ) are the modified Bessel functions; : : : : ’ 1 = U
+ 13 , ’ 2 = ’ 12 ACU 2 2 , ’ 3 = A 23 U 2 3 − 2 ’ 1 ’ 2 .

*

x3y'z|{}~/€

:

:

:

The solutions of equations 20–35 contain only the ratio
= (ln )
. Therefore, * * for the sake of symmetric appearance, two “arbitrary” constants L 1 and L 2 are indicated in the : definition of function (instead, we can set, for instance, L 1 = 1 and L 2 = L ). 20.

21.

22.

23.

=
ñ –1 + ( ï –2 . Solution in parametric form:

ñ óò

J :

 = U

−4 3

 = U

−2 3

−2

’ 2,

(ñó ò )–1 =
ñ –2 + ( ï –1 . Solution in parametric form:

J :

−1

J :

−2 3

= 'U

−1

’

−1 2

’ 3,

where

# = 2

J :

−2

’ 2,

where

# =T

’

−1 2

−1

’ 1,

= 'U

−4 3

J :

−2

’ 2,

where

# =T

’ 2,

= 'U

−2 3

J :

−1

’ 1,

where

# = 2

’ 3,

=
ñ –1 + ( ï . Solution in parametric form:

= 'U

−4 3

−1 2

2 3

 , $ =T

2 3

 .

 , $ = 2  2 

−1

.

ñóò

J :

 = U

−2 3

 = U

−4 3

(ñ ó ò )–1 =
ñ + ( ï –1 . Solution in parametric form:

J :

−2

2 −1 2 , 3



−2



$ = 2

 , $ =T

−2

.

2 2 −1 . 3

 

© 2003 by Chapman & Hall/CRC

Page 187

188 24.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñóò


ñ 1 )

=

2

( ï

+

)

–1 2

.

Solution in parametric form:

25.

(ñó ò )–1 =

J :

−2

)

+

−4 3

 = U


ñ

–1 2

’ 12 ,

= 'U

( ï 1)

2

−8 3

J :

−4

’ 22 ,

where

# = 2

J :

−8 3

26.

(ñó ò )2 =


ñ

−4

J :

−2

’ 12 ,

where

# =T

’ 22 ,

( ï 1)

+

2

27.

(ñ

óò

J :

−4 3

)–2 =

−2

2

+

( ï

J :

−4

J :

−4

−8 3

= 'U

’

2 2

(’

2 2

4 2 3

A

U

:

−8 3

 = U 4 3

28.

(ñó ò )2 =


ñ

–2

J $ , $ = 4 



−1 2

( ï

+

2 3

J , $ = 2 1J 2 

−1 2



−1

.

3

’ 1 ,

# = 4

where

−2

 , $ =T

J ˜# .

4 −1 2 3



.

Solution in parametric form:

where # = T



.

’ 12,


ñ 1 )

−4 3

= 'U

Solution in parametric form:

 = U

J .

2 −1 2 3

.

Solution in parametric form:

 = U

 J , $ =T

−1 1 2

)

–2 5

−2

4 3

A

2

U

:

3

’ 1 ),

= 'U

−4 3

= 'U

J :

−4 3

−2

’ 12 ,

.

.

Solution in parametric form:

 = U

(ñ

óò

)–2 =

J ’ 5J 2, 1

−5 2

J  $ , $ =

4 −2 5 2 3

where # = T 29.

J :

−5 3



)


ñ

–2 5

( ï

+

–2

J :

−2

(’

A

2 2

4 3

U

2

:

3

’ 1 )1 J 2 ,

J 

16 −8 5 2 . 25



.

Solution in parametric form:

 = U

30.

ñ óò

=


ñ 1 )

J

16 2 −8 5 , 25

where # =

 

2

( ï

+

–2

J :

−4 3

−2

$ =T

(’

2 2

31.

(ñó ò )–1 =


ñ

2

–2

+

’

U

32.

(ñó ò )2 =

J :

−4 3

( ï 1)


ñ

( ï

+

’

–1

= 'U 2

2

’

−1 2 ,

’ 1 )1 J 2 ,

= 'U

J :

−5 3

J ’ 5J 2, 1

−5 2

J :

−4 3

−2

−2 2

’

 J

−2 2

’ 32 ,

where

# =A

4 −1 1 2 , 3

2

−1 2 ,

where

# = −4 , $ = A



$ = −4  .

.

’ 32 ,

= 'U 4 J

3

:

’

 J 

4 1 2 −1 . 3

.

Solution in parametric form:

:  = U 4 J 3

3

.

−1 2 ,

−2

:

J # .

 

Solution in parametric form:

 = U

2

4 2 −2 5 3

Solution in parametric form:

:  = U 4 J 3

4 3

A

= 'U

J :

−4 3

−2

’

−2 2 2 ( 3

’

− 4’

3 2 ),

where

# =

16 −2 9



 , $ = 4 ˜# .

© 2003 by Chapman & Hall/CRC

Page 188

ê ( Ë , Ï , Ï ÒÍ

1.6. EQUATIONS OF THE FORM

33.

(ñó ò )–2 =
ñ –1 + ( ï . Solution in parametric form:

 = U

34.

J :

−4 3

’

J

−1 2 , 2

= 'U

J :

−2 3

−1

= 'U 4 J

3 2 ),

− 4’

J :

−2 3

−1

(ñ ó ò )–2 =
ñ 2 + ( ï –2 . Solution in parametric form:

 = U ]

−2 2 2 ( 3

’

(ñ ó ò )2 =
ñ –2 + ( ï 2 . Solution in parametric form:

:  = U 2 J 3 ’

35.

−2

’

’

−1 2 2 ( 3

J

= 'U 2 J

−1 2 3 1 2 , 2 ( 3 −4 2)

’

’

’

3

:

2

J

− 4’

3

:

−1 2 ,

’

’

3 1 2 , 2)

J

−1 2 , 2

# = 4˜$ , $ =

where

where # = 4 2  2 $ , $

² =



Æ =







1 2

an L 2 . One of = A 1), while the

(ñ ó ò )–1 =
ñ –2 + ( ï 2 . Solution in parametric form:

where # = −

−2

1 Te@ 2

2

−1

²

 , $ = − 21 

= 'U 2 ,

Æ ,

−1 −1



@ = £ |1 − 4 #2$ |,

.

(ñó ò )1 ) 2 =
ñ + ( ï –1 . Solution in parametric form:

 = U 2 ,

= 'U

−2

O²

−1

Æ −

(ñó ò )–1 ) 2 =
ñ –1 + ( ï . Solution in parametric form:

 = U 39.

.

for the upper sign, L 1 U Á + L 2 U −Á L 1 sin( @ ln U ) + L 2 cos( @ ln U ) for the lower sign, L 1 ln U + L 2 for @ = 0, Á Á (1 + @ ) L 1 U + (1 − @ ) L 2 U − for the upper sign, ( L 1 − @ L 2 ) sin( @ ln U )+( L 2 + @ L 1 ) cos( @ ln U ) for the lower sign, L 1 ln U + L 1 + L 2 for @ = 0.

 = U

38.

−2

$ = 4 2  2 # .

 

The expressions of ² and Æ contain two “arbitrary” constants L them can be fixed to set it equal to any nonzero number (for example, we can set L other constant remains arbitrary.

37.



16 −4 2 . 9

=

16 2 −4 , 9

# =

where

16 9

In the solutions of equations 36–46, the following notation is used:

x3y'z|{}~/€

36.

189

) = 0 CONTAINING ARBITRARY PARAMETERS

−2

O²

−1

Æ −

1 Te@ 2

2

P

,

1 Te@ 2

2

,

where

# =

−1

= 'U 2 ,

where

# =

1 Te@ 2

P

=
ñ –1 + ( ï –1 ) 2 . Solution in parametric form:

O−

J



1 2

2 rP

2

, $

=

˜$ , $ = 

2

1 Te@ 2

−1

O−



2 "P

˜# .

J

1 2

.

ñ óò

 = U 2 ² 2 ,

= 'UÆ ,

where

# = (−1 Aç@ 2 )



2

2

,

$ =

J .

−1 2

© 2003 by Chapman & Hall/CRC

Page 189

190 40.

FIRST-ORDER DIFFERENTIAL EQUATIONS (ñó ò )–1 =
ñ –1 ) 2 + ( ï –1 . Solution in parametric form: = 'U 2 ²

 = UÆ , 41.

= 'U 2 Æ

2

,

= 'U 2 [ Æ

$ = (−1 Aç@ 2 )

where

# = 2(−1 Ae@ 2 ) 

= 'U ² ,

where

# = 4

2

− (−1 Ae@ 2 ) ²

2

− (−1 Aç@ 2 ) ²

2

(ñ ó ò )2 =
ñ –2 + ( ï –1 . Solution in parametric form: = 'U [ Æ

2

],

 = U [ Æ

2

− (−1 Aç@ 2 ) ²

J

2 1 2

]

,

,



2

.

2

−2

= 'U ² ,

where

# = (−1 Ae@ 2 ) 

]

J

= 'U 2 ²

2

−2

.

−1

.

 , $ = (−1 Ae@ 2 ) 

# = 16

2 1 2

$ = 4

$ = 2(−1 Aç@ 2 )  1 J 2 

where

− (−1 Aç@ 2 ) ²

(ñó ò )–2 =
ñ –1 + ( ï –2 . Solution in parametric form:

2

−2

 J ,

−1 1 2

],

(ñó ò )–2 =
ñ 2 + ( ï . Solution in parametric form:

 = U 2 ² 2 ,

46.

J ,

−1 2

# = 

,

(ñó ò )2 =
ñ + ( ï 2 . Solution in parametric form:

 = U 2 [ Æ

45.

2

(ñ ó ò )–1 =
ñ + ( ï 1 ) 2 . Solution in parametric form:

 = U ² ,

44.

where

=
ñ 1 ) 2 + ( ï . Solution in parametric form:

 = U 2 Æ

43.

,

ñóò

 = U ² ,

42.

2

−2

−2

˜# .

$ , $ = 16 

−2

.

,

where

# = (−1 Ae@ 2 ) 

−1 2

−1 2

,

where

# =  2

= (−1 Aç@ 2 )  2 

−1

−1

, $

 $ , $ =

 .

# .

1.6.4. Other Equations 1.6.4-1. Equations containing algebraic and power functions with respect to
 . 1.

2.

= ï ñó ò + ô ï 2 + õ ñ ó ò + ö , ô ≠ 0. ü Differentiating the equation with respect to  and changing to new variables a =
 and ¨ (a ) = −2  , we arrive at an Abel equation of the form 1.3.1.32: ¨*¨pg
− ¨ = − 'a −1 J 2 .

ñ

– ï = ( ô ï + õÿñ ) (ñ ó ò )2 + 1. ü This is a special case of equation 1.8.1.5 with  (‡ ,  ) = ‡ + ' .

ï ñóò

3.

+ ï = ( ô ï + õÿñ ) (ñ ó ò )2 + 1. ü This is a special case of equation 1.8.1.6 with  (‡ ,  ) = ‡ + ' .

4.

ñ

ññóò

=

ï ñóò

+ ô (ñó ò )÷ .

= 3 L  + L  . In addition, there is a singular solution: !#  −1 B  = −(B − 1)  −1 , B ≠ 1.

Solution:

= #* 



−1

, where

© 2003 by Chapman & Hall/CRC

Page 190

1.6. EQUATIONS OF THE FORM

5. 6. 7.

8.

ê ( Ë , Ï , Ï ÒÍ

= ï ñ ó ò + ô ï ÷ (ñ ó ò )ú . This is a special case of equation 1.8.1.15 with  (¨ ) =  ¨p+ .

ñ

= ô ïƒ÷ (ñó ò )2÷ + 2 ï ñó ò . This is a special case of equation 1.8.1.51 with  (¨ ) =  ¨  .

ñ

= ï ñó ò + ô ï 2 + õ (ñó ò )2 + ö (ñó ò )ú +1 + ý , ô ≠ 0. Differentiating the equation with respect to  and passing to the new variables a =
 and ¨ (a ) = −2  , we arrive at the Abel equation ¨*¨ug
− ¨ = −4 'a − 2!( ( + 1)a + , whose solvable ¡ case are outlined in Subsection 1.3.1.

ñ

ô (ñ óò )÷

+ õ (ñó ò )ú = ï . 1 b . Solution in parametric form with B ≠ −1,

 = a  + 'a + ,

 =

ô (ñ óò )÷

 a

+ + 'a ,

B  a B −1

−1

 = L +  ln |a | +

11.

B +1

a

+1



+

¡

¡

ô ï (ñ óò )÷

 ¡

¡

¡

 ¡

¡

¡

+1

+1

a+

a+

+1

+1

.

.

≠ 1:

¡ ¡



+

≠ −1:

= L +  ln |a | +

−1

2 b . Solution in parametric form with B = 1,

10.

B

=L +

+ õ (ñó ò )ú = ñ . 1 b . Solution in parametric form with B ≠ 1,

 =L +

≠ −1:

¡

2 b . Solution in parametric form with B = −1,

9.

191

) = 0 CONTAINING ARBITRARY PARAMETERS

a+

−1

+ = a  + 'a .

,

≠ 1:

¡

−1

a+

−1

= a + 'a + .

,

+ õÿñ (ñó ò )ú + ö (ñó ò )  = 0. This is a special case of equation 1.8.1.12 with  (‡ ) = ‡  , (‡ ) = '‡ + , and % (‡ ) = (˜‡  . = ô ï ÷ ( ï ñ ó ò – ñ )ú . ¨ g
, The Legendre transformation  = u 2 ¨ + 2 ¨
a = ( g ) .

¨ g
− ¨ =a2

1 b . Solution in parametric form with

≠ −B , B ≠ −1:

ñ óò

¡

+

1 1 −  ¬ ¡ + B a O a  + L2® + +  ,  =O  P B +1  P

a

2 b . Solution in parametric form with 1

1

 =

1

¡

= 1, B = −1:

−

+ + +

.

= −B , B ≠ −1:

¡

1

¡

1

a  B a  = LȬ aO − 1® exp ¬ aO ® .  P · B + 1 ·P ≠ −B , B = −1:

+  V  (1 − ) ln |a | + L3W 1− + , ¡ a

4 b . Solution with

1

a  a  1− +B ¡ a O =¬ − L2®u¬ ¡ a O + L2® 1+B  P B +1  P

a  B a   = L­O exp ¬ aO ® ,  P · B + 1 ·P 3 b . Solution in parametric form with

(
 = a ) leads to a separable equation:

= L3 



= aÖ − V  (1 −

¡

) ln |a | + L3W

1 1−

+ .

−1 .

© 2003 by Chapman & Hall/CRC

Page 191

192 12.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñ óò

ô  ó (ñ óò

+ñ =

=   L 

Solution: 13.

14.

1 2

ô ï ÷ + (ñ óò )

+ õ (ï

ô 1 ïƒ÷ ú 1

1

(ï

+ù

ñóò =

+B

constraint  1 L

16.

−1

1

ñ ó ò (ñ ó ò + ô ï )÷

ô (ñ óò

L

−

2

ñ )ú

−



.

ö

+

= 0.

ô 2 ïƒ÷ ú

+

1

2

 − 1 −

1

2−B 1 + 1 +  2L 2

L

2

(ï

+

1

ö

ñóò

+ù

1

ñ )ú

and L

1

2

õ

+

22.

23.

1

1

≠

and L

ù

2.

2

are related by the

= 0.

õ  ó (ñ óò )

ô  ó (ñ óò

ô  _ ó (ñ óò

– ñ )  + õ [(ñ 1 2

1



2

óò

ö

+ −





= 0.

. Here, the constants L

)2 – ñ

2

1 2

−

= L 1 − L 2 L 2 )  + ( = 0.

+ Ė )  +

]÷ +

õ ÷ ` ó (ñ óò



ö

1

and L

2

are related by the constraint

= 0.

. Here, the constants L + ñ )÷ + ö = 0,

1



and L

2

L 1 −:  L 2 −  −   ; . Here, the constants L = −ý = −ý constraint L 1 + L 2 + ( = 0.

ô (ñ óò ô (ñ óò

=

cosh ï – ñ sinh ï ) ÷ + õ (ñó ò sinh ï – ñ cosh ï )  +

ö

1

cosh ï – ñ sinh ï ) ÷ = õ (ñó ò 2 – ñ

2

ñóò (ñ óò

sinh ï – ñ cosh ï ) ÷ = õ (ñó ò 2 – ñ

2

)ú

and L

2

are related by the

= 0. 1

and L

2

are related by the

1

and L

2

are related by the

cosh ï – ñ sinh ï ) ÷ + õ (ñó ò 2 – ñ 2 )  + ö = 0.

ñóò (ñ óò

are related by the constraint

≠ .

= L 1 sinh  + L 2 cosh  . Here, the constants L Solution: constraint L 1 +  ( L 12 − L 22 )  + ( = 0. 21.

ù

= 0,

= L 1 sinh  − L 2 cosh  . Here, the constants L Solution: constraint L 1 + L 2 + ( = 0. 20.

are related by the constraint

2

 −  2 . Here, the constants L

2

2−B +  = 0.

2

B

óò 2 + 2ôñ )ú

+ õ (ñ

+ ñ )÷ +

Solution:

19.

− 12 L 

The contact transformation m = 
 +  , n = (
 )2 + 2  , n o
= 2
 , where n = n (m ), leads to a separable equation: n o
= −2m −  ( ˜n + + ( ). The inverse contact transformation:  = 12  −1 (2m − n o
), = 18  −1 [4 n −( n o
)2 ], 
 = 12 n o
.

Solution: L 2 +  ( L 18.



.

+ ùñ )ú

ñóò

= L 1−L Solution: L 1 + L 2 + ( = 0. 17.

–1



= − L 1   + L 2 . Here, the constants L Solution:   ( L 1 B ) +  ( L 2 B ) + + ( = 0.

Solution:

15.

– ñ )

cosh ï .

The contact transformation m = (
 )2 − 2 , n = 
 cosh  − sinh  , n o
= leads to a separable equation: 2 'm + n o
= n  . )ú

cos ï + ñ sin ï ) ÷ + õ (ñ

Solution: = L 1 sin  − L L 1 + L 2 + ( = 0.

2

óò

cosh  (
 )−1

1 2

sinh  (
 )−1

sinh ï .

The contact transformation m = (
 )2 − 2 , n = 
 sinh  − cosh  , n o
= leads to a separable equation: 2 'm + n o
= n  .

ô (ñ óò

1 2

sin ï – ñ cos ï )  + ö = 0.

cos  . Here, the constants L

1

and L

2

are related by the constraint

© 2003 by Chapman & Hall/CRC

Page 192

1.6. EQUATIONS OF THE FORM

24.

25.

26.

ê ( Ë , Ï , Ï ÒÍ

) = 0 CONTAINING ARBITRARY PARAMETERS

cos ï + ñ sin ï ) ÷ + õ (ñ ó ò 2 + ñ 2 )  + ö = 0. Solution: = L 1 sin  + L 2 cos  . Here, the constants L L 1 +  ( L 12 + L 22 )  + ( = 0.

ô (ñ óò

1

and L

2

193

are related by the constraint

cos ï + ñ sin ï ) ÷ = õ (ñ ó ò 2 + ñ 2 )ú cos ï . The contact transformation m = (
 )2 + 2 , n = 
 cos  + sin  , n o
= to a separable equation: 2 'm + n o
= n  .

ñ ó ò (ñ ó ò

sin ï – ñ cos ï ) ÷ = õ (ñ ó ò 2 + ñ 2 )ú sin ï . The contact transformation m = (
 )2 + 2 , n = 
 sin  − cos  , n o
= to a separable equation: 2 'm + n o
= n  .

ñ ó ò (ñ ó ò

1 2

cos  (
 )−1 leads

1 2

sin  (
 )−1 leads

1.6.4-2. Equations containing exponential, logarithmic, and other functions with respect to 
.

ï

27.

= ô exp( ƒñ„ó ò ) + õ exp( “ñó ò ). This is a special case of equation 1.8.1.7 with  (¨ ) =  exp( Œ ¨ ) +  exp(“ ¨ ).

28.

ñ

29. 30.

= ô exp( ƒñ„ó ò ) + õ exp( “ñó ò ). This is a special case of equation 1.8.1.8 with  (¨ ) =  exp( Œ ¨ ) +  exp(“ ¨ ). = ï ñ ó ò + ô ï ÷ exp( ƒñ ó ò ). This is a special case of equation 1.8.1.15 with  (¨ ) =  exp( Œ ¨ ).

ñ

= ô ï exp( ƒñ„ó ò ) + õ exp( “ñó ò ). This is a special case of equation 1.8.1.11 with  (¨ ) =  exp( Œ ¨ ) and (¨ ) =  exp(“ ¨ ).

ñ

ï

31.

= ô sinh( ƒñ„ó ò ) + õ sinh( “ñ„ó ò ). This is a special case of equation 1.8.1.7 with  (¨ ) =  sinh( Œ ¨ ) +  sinh(“ ¨ ).

32.

= ô sinh( ƒñ ó ò ) + õ sinh( “ñ ó ò ). This is a special case of equation 1.8.1.8 with  (¨ ) =  sinh( Œ ¨ ) +  sinh(“ ¨ ).

33. 34.

ñ

= ï ñó ò + ô ïƒ÷ sinh ú ( ƒñó ò ). + This is a special case of equation 1.8.1.15 with  (¨ ) =  sinh ( Œ ¨ ).

ñ

= ô ï sinh( ƒñ„ó ò ) + õ sinh( “ñ„ó ò ). This is a special case of equation 1.8.1.11 with  (¨ ) =  sinh( Œ ¨ ) and (¨ ) =  sinh(“ ¨ ).

ñ

ï

35.

= ô cosh( ƒñ„ó ò ) + õ cosh( “ñ„ó ò ). This is a special case of equation 1.8.1.7 with  (¨ ) =  cosh( Œ ¨ ) +  cosh(“ ¨ ).

36.

= ô cosh( ƒñ„ó ò ) + õ cosh( “ñ„ó ò ). This is a special case of equation 1.8.1.8 with  (¨ ) =  cosh( Œ ¨ ) +  cosh(“ ¨ ).

37. 38.

ñ

= ï ñó ò + ô ïƒ÷ coshú ( ƒñó ò ). + This is a special case of equation 1.8.1.15 with  (¨ ) =  cosh ( Œ ¨ ).

ñ

= ô ï cosh( ƒñ„ó ò ) + õ cosh( “ñ„ó ò ). This is a special case of equation 1.8.1.11 with  (¨ ) =  cosh( Œ ¨ ) and (¨ ) =  cosh(“ ¨ ).

ñ

© 2003 by Chapman & Hall/CRC

Page 193

194 39.

FIRST-ORDER DIFFERENTIAL EQUATIONS ln ñ

+ï

óò

ñ óò

ôñ

+

+

õ

= 0.

1 b . Solution in parametric form with  ≠ 0,  ≠ −1: 1

 =

+ L3a − 

a

1 = − (a + ln a +  ).

1 +1 ,



2 b . Solution in parametric form with  = 0:

 =−

ln a + 

a

= L + (  − 1) ln a +

,

1 (ln a )2 . 2

3 b . Solutions with  = −1:

40. 41.

42. 43.

ï

= L3 + ln L +  + ln ñ„ó ò + ô (ñó ò + ñ )  +

õ

and

= ln(−1  ) +  − 1.

= 0.

This is a special case of equation 1.8.1.41 with  (‡ , ¨ ) = ln ‡ +  ¨  +  .

ñ

=

ï ñóò

+ô

ï

ñ

=

ï ñóò

+ô

ïƒ÷

=

ô

2

+ õ ln ñó ò + ö ,

ô

≠ 0.

Differentiating the equation with respect to  and changing to new variables a =
 and ¨ (a ) = −2  , we arrive at an Abel equation of the form 1.3.1.16: ¨*¨pg
− ¨ = −2 'a −1 . ln ú ( ƒñó ò ).

+ This is a special case of equation 1.8.1.15 with  (¨ ) =  ln ( Œ ¨ ).

ï

sin( ƒñ„ó ò ) + õ sin( “ñ„ó ò ).

This is a special case of equation 1.8.1.7 with  (¨ ) =  sin( Œ ¨ ) +  sin(“ ¨ ).

44.

ñ = ô sin(ƒñ„óò ) + õ sin( “ñ„óò ). This is a special case of equation 1.8.1.8 with  (¨ ) =  sin( Œ ¨ ) +  sin(“ ¨ ).

45.

ñ

=

ï ñóò

ñ

=

ô ï

=

ô

46. 47. 48. 49. 50. 51.

+ô

ïƒ÷

sin ú ( ƒñó ò ).

This is a special case of equation 1.8.1.15 with  (¨ ) =  sin + ( Œ ¨ ). sin( ƒñ„ó ò ) + õ sin( “ñ„ó ò ).

This is a special case of equation 1.8.1.11 with  (¨ ) =  sin( Œ ¨ ) and (¨ ) =  sin(“ ¨ ).

ï

cos( ƒñ„ó ò ) + õ cos( “ñó ò ).

This is a special case of equation 1.8.1.7 with  (¨ ) =  cos( Œ ¨ ) +  cos(“ ¨ ).

ñ

=

ô

cos( ƒñ„ó ò ) + õ cos( “ñó ò ).

ñ

=

ï ñ óò

ñ

=

ô ï

This is a special case of equation 1.8.1.8 with  (¨ ) =  cos( Œ ¨ ) +  cos(“ ¨ ). +ô

ï ÷

cosú ( ƒñ

óò

).

This is a special case of equation 1.8.1.15 with  (¨ ) =  cos + ( Œ ¨ ). cos( ƒñ

óò

) + õ cos( “ñ

óò

).

This is a special case of equation 1.8.1.11 with  (¨ ) =  cos( Œ ¨ ) and (¨ ) =  cos(“ ¨ ).

ï ñ óò

ï ÷

tanú ( ƒñ

óò ). This is a special case of equation 1.8.1.15 with  (¨ ) =  tan + ( Œ ¨ ). ñ

=

+ô

© 2003 by Chapman & Hall/CRC

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1.7. EQUATIONS OF THE FORM

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY FUNCTIONS

195

1.7. Equations of the Form ð ( ì , í ) í ï î = ë ( ì , í ) Containing Arbitrary Functions ]

Notation:  , , and % are arbitrary composite functions whose argument, indicated after the function name, can depend on both  and .

1.7.1. Equations Containing Power Functions 1.

= ! ( ô ï + õÿñ + ö ). In the case  = 0, we have an equation of the form 1.1.1. If  ≠ 0, the substitution ‡ ( ) =  +  + ( leads to a separable equation: ‡
 =  (‡ ) +  .

ñóò

2.

= ! (ñ + ô ïƒ÷ + õ ) – ôù ïƒ÷ –1 . The substitution ‡ = +   +  leads to a separable equation: ‡
 =  (‡ ).

3.

ñóò

ñóò

=

ñ ! (ƒï ÷ ñ ú ï

).

Generalized homogeneous equation. The substitution H =   + "H 
= B±H + H! ( H ).

leads to a separable equation:

¡

4.

5.

ñ óò = ! (ï )ñ 1+÷

+ " ( ï )ñ + % ( ï )ñ 1–÷ . ¨ 
= B… ( )¨ The substitution ¨ =  leads to a Riccati equation: 3

ñóò

=–

ù ñ þ ï

+ñ

 ! (ï )" (ƒï ÷ ñ ú

The substitution H =   + 6.

ñóò

=I ! O

ô ï  ï

2

+ B ( )¨ + B…% ( ).

).

leads to a separable equation: H 
=

+ õÿñ + ö . + ñ + P

¡



 − ´   ++ +  ( ) H +

−1

( H ).

= − (˜ý (ƒ − ´= , =  (‡ ) + leads to 1 b . For ¢ = ý − ˜ƒ ≠ 0, the transformation  = ‡ + ¢ ¢ an equation: ‡ + '  
=  O . ƒ…‡ + ý± P Dividing both the numerator and denominator of the fraction on the right-hand side by ‡ , we obtain a homogeneous equation of the form 1.1.6. 2 b . For ¢ = 0 and  ≠ 0, the substitution  ( ) =  +  + ( leads to a separable equation of the form 1.1.2: '  
=  +  O . ý± + = − (˜ýCP 3 b . For ¢ = 0 and ý ≠ 0, the substitution  ( ) = ƒ… + ý + = also leads to a separable equation: ' + (˜ý − =  
= ƒ + ýh*O .

ý±

7. 8.

P

= ï ÷ –1 ñ 1–ú ! ( ô ï ÷ + õÿñ ú ). ¨ 
=      +  + leads to a separable equation: 3 The substitution ¨ = 

ñ óò

ñ ÷ ñóò

+ ô ïƒ÷ + " ( ï ) ! (ñ ÷ +1 + ô The substitution ¨ =  +1 +  

ïƒ÷

+1

−1

[ B + 

¡

 (¨ )].

+1

) = 0. ¨ 
+ (B + 1) ( )  (¨ ) = 0. leads to a separable equation: 3

© 2003 by Chapman & Hall/CRC

Page 195

196 9.

FIRST-ORDER DIFFERENTIAL EQUATIONS

÷ ! (ñ

[ï

)+ï

" (ñ )]ñ óò

= % (ñ ).

This is a Bernoulli equation with respect to  =  ( ) (see Subsection 1.1.5). 10.

[ï

2

+

ï ! (ñ

óò

) + " (ñ )]ñ

= % (ñ ).

This is a Riccati equation with respect to  =  ( ) (see Section 1.2). 11.

= [ ! ( ï )ñ + " (ï )] (ñ – ô )(ñ – õ ).

ñóò

The substitution ‡  ( ) − ( ). 12.

Q

ñ

ü = ( −  ) ( −  ) leads to a Riccati equation: A 2‡
 = [  ( ) + ( )]‡

ñ

+ï

!IO ï P

2

D-%IO ï Sñóò P

=3 " O

ï

ñ P

+ñ

ñ ï D –1 I % O ï

P

2

−

.

The substitution = a leads to a Bernoulli equation with respect to  =  (a ): [ (a )− aª (a )] g
=  (a ) + % (a )  +1 . 13.

ï h ú (ï , ñ )]ñ óò . Darboux equation. Here,

[g

÷ (ï , ñ

)+

i ÷ (ï , ñ

=

)+ñ

h ú (ï , ñ

).

 and Æ  are homogeneous polynomials of order B , and ² + is a homogeneous polynomial of order . Dividing the Darboux equation by   leads to an ¡ equation of the form 1.7.1.12. 14.

ï

[ ! (ô

+

õÿñ

õ ï " (ô ï

)+

+

õÿñ )]„ñ óò

ï

= % (ô

+

õÿñ

ï " (ô ï

)–ô

+

õÿñ

).

  +  leads to a linear equation with respect to  The substitution a =  [  (a ) + % (a )] g
=  (a ) +  (a ). 15.

[ ! (ô

ï

+

õÿñ

õÿñ" (ô ï

)+

õÿñ

ï " (ƒï ÷ ñ ú

)]ñó ò = ñ [ % ( ïƒ÷

ñ ú

)–ù

 " (ƒï ÷ ñ ú

)]ñó ò = ñ [ % ( ïƒ÷

ñ ú

)–

The substitution a =  +  − ´ (a ) + % (a ). 16.

17.

ï [ ! (ƒï ÷ ñ ú

)+þ

ï [ ! (ƒï ÷ ñ ú

) + þxñ

õÿñ )]„ñ óò

= % (ô

ï

+

+

) – ôñ" (ô

ï

+

õÿñ

=  (a ):

).

leads to a linear equation with respect to = (a ): [  (a )+ % (a )]
g =

ï " (ƒï ÷ ñ ú

)].

) – ùñ

 " (ƒï ÷ ñ ú

)].

ü!

ñ

The transformation a =   + , H =  −  leads to a linear equation with respect to H = H (a ): a [B… (a ) + % (a )]H g
= − ,  (a ) H − , (a ). ¡ ¡

18.

The transformation a =   + , H = −  leads to a linear equation with respect to H = H (a ): a [B… (a ) + % (a )]H g
= − , % (a ) H + ,!B (a ). ¡ À ï s ï ïw÷ ú ï s ïƒ÷ ú

19.

[ ! (ñ ) + ôþ

[s ! (

ñ

) – þj" (

)]ñó ò = ñ [ù#" (

The transformation a =   + , ¨ =  

ï ÷ ñ ú

–1

Solution: X¯ ( ) 20.

ñ

ï ñóò

– ñ = ! (ï

2

M

]ñ

+ " (ï ) +

M

(

)].

¨ g
= ¨ (¨ ). leads to a separable equation: aª (a )u

ôù ï ÷ –1 ñ ú

+ Xw ( )  +  

= 0.

+ =L .

+ ñ 2 )(ññó ò + ï ).

Setting  = & (a ) cos a , a = XÈ& −1  (& 2 ) & + L .

^_

óò

ñ

= & (a ) sin a and integrating, we obtain a solution in implicit form:

M

Reference: G. W. Bluman, J. D. Cole (1974, p. 100).

© 2003 by Chapman & Hall/CRC

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â ( Ë , Ï ) Ï ÒÍ

1.7. EQUATIONS OF THE FORM

21.

= ã ( Ë , Ï ) CONTAINING ARBITRARY FUNCTIONS

197

ï ñóò

– ñ = ! ( ï 2 – ñ 2 )(ññó ò – ï ). Setting  = & (a ) cosh a , = & (a ) sinh a and integrating, we obtain a solution in implicit form: a = − X & −1  (& 2 ) & + L .

M

22.

23.

[ ï ! ( ï 2 + ñ 2 ) + ñ" ( ï 2 + ñ 2 ) + % ( ï 2 + ñ 2 )](ññ ó ò + ï ) = ï ñ ó ò – ñ . The transformation  = & cos Y , = & sin Y leads to an equation of the form 1.7.4.11 with respect to Y = Y (& ): Y
Ÿ =  (& 2 ) cos Y + (& 2 ) sin Y + & −1 % (& 2 ).

[ ï ! ( ï 2 – ñ 2 ) + ñ" ( ï 2 – ñ 2 ) + % ( ï 2 – ñ 2 )](ññ ó ò – ï ) = ï ñ ó ò – ñ . 1 b . For  > , the transformation  = & cosh Y , = & sinh Y leads to an equation of the form 1.7.2.18 with respect to Y = Y (& ): Y
Ÿ = −  (& 2 ) cosh Y − (& 2 ) sinh Y − & −1 % (& 2 ). 2 b . For  < , the transformation  = H sinh j , = H cosh j leads to an equation of the form 1.7.2.18 with respect to j = j ( H ): j ‘
= −  (− H 2) sinh j − (− H 2) cosh j − H −1 % (− H 2 ).

24.

ï ñóò

–ñ =

! ï

2

+ñ

2

Setting  = & (a ) cos a ,

ï

ï ñóò

–ñ =

! ï

2

–ñ

2

 "I”

Setting  = & (a ) cosh a ,

! ( ï , ñ )ñ ó ò

+ " ( ï , ñ ) = 0,

,

2

ñ

ï

2

a

(cos a , sin a )

ï

ï

,

 (& 2 ) & +L . & M

=ù

ñ

ï

(ññó ò – ï ).

–ñ –ñ ü ü = & (a ) sinh a and integrating, we obtain the solution:

ù 26.

2

M

ù 25.

ï

(ññó ò + ï ). +ñ +ñ 2• ü ü = & (a ) sin a and integrating, we obtain the solution:

 "I”

2

2

M

a

2

(cosh a , sinh a ) where

k ï k

!

•

2

 (& 2 ) & =L . & M

+ù

=

k

"

ñ k

.

Exact differential equation.   Solution: X  ( 0 , a ) a + X  (a , ) a = L , where  M M 0 0

0

and

0

are arbitrary numbers.

1.7.2. Equations Containing Exponential and Hyperbolic Functions 1. 2. 3.

4.

ó

ñóò

ó

=  –  ! (  ñ ).  The substitution ‡ = ž 

leads to a separable equation: ‡
 =  (‡ ) + Œ ‡ .

=  ] ! (  ] ï ). The substitution ‡ = ž   leads to a separable equation: ‡
 = Œ ‡

ñ óò

ó

2

 (‡ ) + ‡ .

= ñ! (  ^ ñ ú ). This equation is invariant under “translation–dilatation” transformation. The substitution  H =  9 + leads to a separable equation: H 
= ƒhH + H! ( H ).

ñóò ñóò

=

ƒï ÷ ï ! ( ^] 1

¡

).

This equation is invariant under “dilatation–translation” transformation. The substitution H =    9  leads to a separable equation: "H 
= B±H + ƒhH! ( H ).

© 2003 by Chapman & Hall/CRC

Page 197

198 5.

6.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñ óò

= ! (ï )

ñ óò

=–ï

ñóò

=–

+ " ( ï ).

]

The substitution ‡ =  −   leads to a linear equation: ‡
 = − Œ ( )‡ − Œ" ( ).

ù

÷ ]

+ ! ( ï )" ( ï

).

The substitution H =    leads to a separable equation: H 
=  ( ) Hž ( H ). 7.

8. 9. 10.

11.

þ



ñ

+ñ

 ! ( ï )" (  ^ ó ñ ú

).

 The substitution H =  9 + leads to a separable equation:  ++ ƒ H 
= exp (1 − , ) S  ( ) H + ¡ Q ¡

ñóò

= ! ( ï ) ] + " ( ï ) + % ( ï )

ñóò

=

ñóò

= ! (ñ +

ñóò

=–

–

]

−1

( H ).

.

The substitution ‡ = ž leads to a Riccati equation: ‡
 = Œ" ( )‡

ó ó  ^ –_ l] ! (ô  ^

õ _ ]

+

2

+ Œ ( )‡ + Œ"% ( ).

).

  The substitution ¨ =   9 +   :  leads to a separable equation: ¨ 
=  9 [ !ƒ + 'ýh (¨ )].

ô ó

+ õ ) – ô

ó

.

 ¨ 
=  (¨ ). The substitution ¨ = +  ž +  leads to a separable equation: u

ù

 ï

+

! (ï ÷  ^ ]

)

ï ñ

.

The substitution a =    9  leads to a linear equation with respect to −B + ƒs (a ). 12.

ñ óò

ù

 ï

=–

+

! (ï ÷  ^ ]

ï ñ

)

2

[ ! (ô

ï

[ ! (ô

ï

õ ^ ]m" (ô ï

+

õ ^ ó " (ô ï

+

õÿñ

)+

õÿñ

)+

)+

ô ]ñ óò

+

2

aª (a )
g =

.

The substitution a =    9  leads to a Riccati equation: ƒ 13.

= (a ): ƒ

õÿñ )]ñ óò

= % (ô

õÿñ )]ñ óò

= % (ô

ï

+

2

aª (a )
g = −B

õÿñ

) – ô

^ ]n" (ô ï

+

õÿñ

) – ô

^ ó " (ô ï

+

õÿñ

).

õÿñ

).

2

+ ƒs (a ).

The transformation a =  +  , H =  − 9  leads to a linear equation with respect to H = H (a ): [  (a ) + % (a )] H g
= − ƒs% (a ) H + ƒh´ (a ).

14.

+

The transformation a =  +  , H =  9 [  (a ) + % (a )] H g
= − ƒs (a ) H − ƒs (a ).



15.

[

^ ó ! (ñ

Solution: X 16.

17.

ï [ ! (ƒï ÷  ^ ]

_ ]" (ï

+

 −:   ( ) M

)+

ô 

−

ï

+

leads to a linear equation with respect to H = H (a ):

= 0.

+ X  − 9 ( )  −   − 9

) + ñ" ( ïƒ÷  ^ ] )] ñ óò

M

= % ( ïƒ÷  ^ ]

 −:  =L .

) – ùñ" ( ïƒ÷

 ^ ]

).

The substitution a =    9  leads to a linear equation with respect to a [B… (a ) + ƒs% (a )] g
= −B (a ) + % (a ). [ ! (

^ ó ñ ú

)+þ

ï " ( ^ ó ñ ú

)]ñó ò = ñ [ % ( 

^ ó ñ ú

)–

ï " ( ^ ó ñ ú

)].  The substitution a =  9 + leads to a linear equation with respect to  a [ ƒs (a ) + % (a )] g
= (a ) +  (a ).

¡

=

(a ):

=  (a ):

¡

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1.7. EQUATIONS OF THE FORM

18. 19. 20.

21.

22. 23. 24. 25.

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY FUNCTIONS

ñ óò

= ! ( ï ) sinh( ƒñ ) + " ( ï ) cosh( ƒñ ) + % ( ï ).

ñóò

= ! ( ï ) sinh2 ( ƒñ ) + " ( ï ) cosh2 ( ƒñ ) + % ( ï ) sinh(2 ƒñ ) + s( ï ).

ñóò

= ! ( ï ) sinh 

Ė

+ " (ï )  .

ñóò

= ! ( ï ) cosh 

Ė

+ " (ï )  .

ñ óò

=

! (ñ ú

sinh ï ).

The substitution ‡ =   leads to a Riccati equation: 2‡
 = Œ (  + )‡ The substitution ¨ = tanh( Œ The substitution ¨ = Œ 
( ). The substitution ¨ = Œ 
( ). coth ï

ñ

=

ñóò

=

ñóò

=

ï

–1

! (ñ ú

+ 2 Œ"% ¨ + Œ ( − ° ).

¨ 
= Œ" ( ) cosh ¨ + + ( ) leads to an equation of the form 1.7.2.18: I

tanh ñB! ( ïƒ÷ sinh ñ ).

tanh ï

2

¨ 
= Œ" ( ) sinh ¨ + + ( ) leads to an equation of the form 1.7.2.18: I

The transformation a =   , H = sinh

ñ

+ 2 Œ"% ‡ + Œ ( −  ).

¨ 
= Œ (  + ° )¨ ) leads to a Riccati equation: p

The transformation a = sinh  , H = +

ñóò

2

199

cosh ï ).

leads to an equation of the form 1.7.1.3: a›H g
=

¡

H! (a›H ).

leads to an equation of the form 1.7.1.3: B/a›H g
= H! (a›H ).

The substitution a = cosh  leads to an equation of the form 1.7.1.3: a
g =  (a + ).

ï

–1

coth ñB! ( ïƒ÷ cosh ñ ).

leads to an equation of the form 1.7.1.3: "H 
= H! (  H ).

The substitution H = cosh

1.7.3. Equations Containing Logarithmic Functions 1.

ñóò

= ! ( ï )ñ ln2 ñ + " ( ï )ñ ln ñ + % ( ï )ñ .

ñóò

=

The substitution ‡ = ln 2.

ï –1 ñ ú

+1

! (ñ ú

leads to a Riccati equation: ‡
 =  ( )‡

ln ï ).

The substitution a = ln  leads to an equation of the form 1.7.1.3: 3.

ñóò

=

ï –÷ –1 ñ ! (ïƒ÷

ln ñ ).

The substitution H = ln 4.

ñóò

=

ï –1  ]l! ( ]

ln ï ).

6.

ñ –ó ! ( ó

ñóò

=

ñóò

= –ù

+ ( )‡ + % ( ).


g = a

[a +  (a + )].

H  (  H ) leads to an equation of the form 1.7.1.3: H 
= .    H

The substitution a = ln  leads to an equation of the form 1.7.2.4: 5.

2


1 g = [a    (a   )]. a

ln ñ ).

  ( H )
. The substitution H = ln leads to an equation of the form 1.7.2.3: H  = H   H

ï –1 ñ

ln ñ + ñ! ( ï )" (ïw÷ ln ñ ).

The substitution ¨ ( ) =   ln

¨ 
=    ( ) (¨ ). leads to a separable equation: u

© 2003 by Chapman & Hall/CRC

Page 199

200

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñóò

7.

=–

ù ñ þ ï

+

ñ! (ï ÷ ñ ú ï ln ñ

)

.

The transformation a =   + , H = ln 2 aª (a ) H g
= −B±H +  (a ).

¡

8.

ñóò

=–

ù ñ þ ï

¡

+

ñ! (ï ÷ ñ ú ) . ï (ln ñ )2

The transformation a =   + , H = ln

9.

leads to a linear equation with respect to H = H (a ):

leads to a Riccati equation:

ï [ ! (ƒï ÷ ñ ú

¡

2

aª (a ) H g
= −B±H 2 +  (a ). ¡

) + þ ln ño" ( ïƒ÷ ñ ú )]ñó ò = ñ [ % ( ïƒ÷ ñ ú ) – ù ln ño" ( ïƒ÷ ñ ú )]. The transformation a =   + , H = ln leads to a linear equation with respect to H = H (a ): a [B… (a ) + % (a )]H g
= −B (a ) H + % (a ).

¡

10.

ï [ ! (ƒï ÷ ñ ú

) + þ ln ï " ( ïƒ÷ ñ ú )]ñó ò = ñ [ % ( ïƒ÷ ñ ú ) – ù ln ï " ( ïƒ÷ ñ ú )]. The transformation a =   + , H = ln  leads to a linear equation with respect to H = H (a ): a [B… (a ) + % (a )]H g
= (a ) H +  (a ).

¡

¡

1.7.4. Equations Containing Trigonometric Functions 1. 2. 3. 4. 5. 6. 7. 8.

9.

10.

= ñ ú +1 sin ï H (ñ ú cos ï ). This is an equation of the type 1.7.4.3 with  (ˆ ) = ˆž (ˆ ).

ñ óò

= ñ ú +1 cos ï H (ñ ú sin ï ). This is an equation of the type 1.7.4.4 with  (ˆ ) = ˆž (ˆ ).

ñóò

= ñ tan ï ! (ñ ú cos ï ). The substitution a = cos  leads to an equation of the form 1.7.1.3: a
g = −  (a + ).

ñóò

= ñ cot ï ! (ñ ú sin ï ). The substitution a = sin  leads to an equation of the form 1.7.1.3: a
g =  (a + ).

ñóò

= ï –1 tan ñB! ( ïƒ÷ sin ñ ). The transformation a =   , H = sin

ñóò

= ï –1 cot ñB! ( ïƒ÷ cos ñ ). The transformation a =   , H = cos

ñóò

= ï –1 sin 2ñB! ( ïƒ÷ tan ñ ). The transformation a =   , H = tan

ñóò

= ï –1 sin 2ñB! ( ïƒ÷ cot ñ ). The transformation a =   , H = cot

ñóò

ñ

leads to an equation of the form 1.7.1.3: B/a›H g
= H! (a›H ). leads to an equation of the form 1.7.1.3: B/a›H g
= − H! (a›H ). leads to an equation of the form 1.7.1.3: B/a›H g
= 2 H! (a›H ). leads to an equation of the form 1.7.1.3: B/a›H g
= −2 H! (a›H ).

! (ñ ú tan ï ). sin 2 ï The substitution a = tan  leads to an equation of the form 1.7.1.3: 2a
g =  (a + ).

ñ óò

=

ñ

! (ñ ú cot ï ). sin 2 ï The substitution a = cot  leads to an equation of the form 1.7.1.3: 2a
g = −  (a + ).

ñóò

=

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1.7. EQUATIONS OF THE FORM

11. 12.

â ( Ë , Ï ) Ï ÒÍ

= ã ( Ë , Ï ) CONTAINING ARBITRARY FUNCTIONS

= ! ( ï ) cos(ôñ ) + " ( ï ) sin( ôñ ) + % ( ï ). The substitution ‡ = tan(  2) leads to a Riccati equation: 2‡ 
=  ( % −  )‡ 2 +2 ´ !‡ +  (  + % ).

ñ óò

= ! ( ï ) cos2 ( ôñ ) + " ( ï ) sin2 ( ôñ ) + % ( ï ) sin(2 ôñ ) + s( ï ). The substitution ‡ = tan( ) leads to a Riccati equation: ‡
 =  ( + ° )‡

ñ óò

2

+ 2 % ‡ +  (  + ° ).

13.

= ! (ñ + ô tan ï ) – ô tan2 ï . The substitution ‡ = +  tan  leads to a separable equation: ‡
 =  +  (‡ ).

14.

sin 2ñ ! (tan ï tan ñ ). sin 2 ï The transformation a = tan  , H = tan

15.

16.

17.

18. 19. 20.

21.

22.

23.

201

ñóò ñóò

=

= cot ï tan ñB! (sin ï sin ñ ). The transformation a = sin  , H = sin

ñ óò

! (ï

leads to an equation of the form 1.7.1.3: a›H g
= H! (a›H ).

" (sin ï sin ñ ). cos ñ The substitution ¨ ( ) = sin  sin leads to a separable equation: ¨ 
= sin 3 ( ) (¨ ).

ñóò

= – cot ï tan ñ +

leads to an equation of the form 1.7.1.3: a›H g
= H! (a›H ).

)

sin 2ñ + cos2 ñB! ( ï )" (tan ï tan ñ ). sin 2 ï ¨ 
= tan 3 ( ) (¨ ). The substitution ¨ ( ) = tan  tan leads to a separable equation: p

ñóò

=–

= –ù ï –1 sin 2ñ + cos2 ñB! ( ï )" ( ï 2÷ tan ñ ). The substitution ¨ ( ) =  2  tan leads to a separable equation: ¨ 
=  2   ( ) (¨ ).

ñóò

(1 + tan2 ñ )ñó ò = ! ( ï ) tanú +1 ñ + " ( ï ) tan ñ + % ( ï ) tan1–ú ñ . leads to a Riccati equation: ‡
 =  ( )‡ The substitution ‡ = tan +

¡

2

+

¡

( )‡ +

¡

% ( ).

= ! ( ï ) sin  ƒñ + " ( ï )  . ¨ 
= Œ" ( ) sin ¨ + The substitution ¨ = Œ + ( ) leads to an equation of the form 1.7.4.11: I
 ( ).

ñóò

= ! ( ï ) cos  ƒñ + " ( ï )  . ¨ 
= Œ" ( ) cos ¨ + The substitution ¨ = Œ + ( ) leads to an equation of the form 1.7.4.11: I
 ( ).

ñóò

= ! ( ï ) sin2  ƒñ + " ( ï )  . ¨ 
= Œ" ( ) sin2 ¨ + The substitution ¨ = Œ + ( ) leads to an equation of the form 1.7.4.12: I
 ( ).

ñóò

= ! ( ï ) cos2 pƒñ + " ( ï )  . ¨ 
= Œ" ( ) cos2 ¨ + The substitution ¨ = Œ + ( ) leads to an equation of the form 1.7.4.12: I
 ( ).

ñ óò

1.7.5. Equations Containing Combinations of Exponential, Logarithmic, and Trigonometric Functions 1.

ó

= – sin 2ñ + cos2 ñB! ( ï )" (  2 tan ñ ).  ¨ 
=  2   ( ) (¨ ). The substitution ¨ ( ) =  2 tan leads to a separable equation: u

ñ óò

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Page 201

202 2.

FIRST-ORDER DIFFERENTIAL EQUATIONS

ñóò

=

ó

cos ñ ) . sin ñ

H (  ó

This is an equation of the type 1.7.5.5 with  (ˆ ) =  (ˆ ) ˆ . 3.

ñ óò

=

 ]

cos ï

H ( ]

sin ï ).

This is an equation of the type 1.7.5.7 with  (ˆ ) = ˆž (ˆ ). 4.

ñóò

ó

= tan ñB! ( 

sin ñ ).

 leads to an equation of the form 1.7.2.3: H 
= H! (  H ).

The substitution H = sin 5.

ñ óò

ó

= cot ñB! ( 

cos ñ ).

 leads to an equation of the form 1.7.2.3: H 
= − H! (  H ).

The substitution H = cos 6.

7.

8.

9.

10.

ñóò

= tan ï

! (  ]

cos ï ).

ñ óò

= cot ï

! ( ]

sin ï ).

ñóò

= sin 2ñB! (

ñ óò

= sin 2ñB! (

The substitution a = cos  leads to an equation of the form 1.7.2.4: a
g = −  (a  ). The substitution a = sin  leads to an equation of the form 1.7.2.4: a
g =  (a  ).

ó

tan ñ ).



The substitution H = tan  leads to an equation of the form 1.7.2.3: H 
= 2 H! (  H ).

ó

cot ñ ).

 The substitution H = cot  leads to an equation of the form 1.7.2.3: H 
= −2 H! (  H ).

ñ óò

=

ó

H (  ó

sin ñ ) . cos ñ

This is an equation of the type 1.7.5.4 with  (ˆ ) =  (ˆ ) ˆ . 11.

ñóò

=

]

sin ï

H (  ]

cos ï ).

This is an equation of the type 1.7.5.6 with  (ˆ ) = ˆž (ˆ ). 12.

tan ï ) . sin 2 ï The substitution a = tan  leads to an equation of the form 1.7.2.4: 2a
g =  (a  ).

ñóò

=

! ( ]

13.

ñ óò

=

cot ï ) . sin 2 ï The substitution a = cot  leads to an equation of the form 1.7.2.4: 2a
g = −  (a  ).

14.

ñóò

=

! ( ]

ó

 – !

(

ï

+ ln ñ ).

The substitution ‡ = Œ  + ln 15.

ñ óò

=

 ] ! ( Ė

+ ln ï ).

The substitution ‡ = Œ

leads to a separable equation: ‡
 = 

−



+ ln  leads to a separable equation: ‡
 = Π

 (‡ ) + Œ .



 (‡ ) + 1.

© 2003 by Chapman & Hall/CRC

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1.8. EQUATIONS OF THE FORM

ê ( Ë , Ï , Ï ÒÍ

) = 0 CONTAINING ARBITRARY FUNCTIONS

203

1.8. Equations of the Form c ( ì , í , í ï î ) = 0 Containing Arbitrary Functions 1.8.1. Some Equations 1.8.1-1. Arguments of arbitrary functions depend on  and . 1.

(ñó ò )2 + [ ! ( ï ) + " (ï )]ñ„ó ò + ! ( ï )" ( ï ) = 0.

The equation can be factorized: [
 +  ( )][
 + ( )] = 0, i.e., it falls into two simpler equations 
+  ( ) = 0 and 
+ ( ) = 0. Therefore, the solutions are:

 ( )  = L M

+X 2.

(ñó ò )2 + 2 !„ññó ò + "Iñ

= (" –

2

!

2

ó

) exp O –2 X



=

3.

ï ñ óò

– ñ = ! (ï

2









P

.



− 2  +L  ü M P  M P L exp O − X   M P      cosh O X  2−  +L exp O − X   ü P M P M exp O − X






M

!ý ï

D

Here,  =  ( ), = ( ). Solution:






+ X ( )  = L .

and





óò

+ ñ 2 ) (ñ

ü



sin O X

if >  2 , if ≡  2 , if <  2 .

)2 + 1.

Raising the equation to the second power and applying the transformation  = & (a ) cos a , = & (a ) sin a , one arrives at the relation & 4 =  2 (& 2 )[(& g
)2 + & 2 ]. Solving it for & g
yields a separable equation:  (& 2 )& g
= A"& & 2 −  2 (& 2 ). 4.

ñ ñóò + ï

= ! (ï

+ ñ ) (ñ

2

2

ü

óò

ü

)2 + 1.

Raising the equation to the second power and applying the transformation  = & (a ) cos a , = & (a ) sin a , one arrives at the relation & 2 (& g
)2 =  2 (& 2 )[(& g
)2 + & 2 ]. Solving it for & g
yields a &ž (& 2 ) . separable equation: & g
= A & 2 −  2 (& 2 )

^_ 5.

ü

Reference: G. W. Bluman, J. D. Cole (1974, p. 100).

ï ñ óò

–ñ =

ü

ï

2

+ñ

2

!e”

ï

ü

ï

2

+ñ

2

,

ñ

ü

ï

2

+ñ

2

+ñ

2

• ü

(ñ

óò

)2 + 1.

Raising the equation to the second power and applying the transformation  = & (a ) cos a , = & (a ) sin a , one arrives at the relation & 2 =  2 (cos a , sin a )[(& g
)2 + & 2 ]. Solving it for & g
yields a separable equation:  (cos a , sin a )& g
= A"& 1 −  2 (cos a , sin a ). 6.

ññ óò

+ï =

ü

ï

2

+ñ

2

!e” ü

ï

ü

ï

2

+ñ

2

,

ü

ï

ñ 2

• ü

(ñ

óò

)2 + 1.

Raising the equation to the second power and applying the transformation  = & (a ) cos a , = & (a ) sin a , one arrives at the relation & 2 (& g
)2 =  2 (cos a , sin a )[(& g
)2 + & 2 ]. Solving it for & g
yields a separable equation: 1 −  2 (cos a , sin a ) & g
= A"&ž (cos a , sin a ).

ü

© 2003 by Chapman & Hall/CRC

Page 203

204

FIRST-ORDER DIFFERENTIAL EQUATIONS

1.8.1-2. Argument of arbitrary functions is "
 . 7.

ï

= ! (ñó ò ). Solution in parametric form:

 =  (a ), 8.

9.

ñ

= ! (ñó ò ). Solution in parametric form:

! (ñ óò ) + ô ï

+ õÿñ + s = 0. Solution in parametric form:

= X ªa  g
(a ) a + L .

M

a  = Xv g
(a ) M + L , a

=  (a ).

 g
(a ) a M ,  = −  − ° −  (a ).  + 'a In addition, there is a particular solution = ƒ… + ý , where ƒ and ý are determined by solving  =L −X

the system of two algebraic equations:  + ˜ƒ = 0,

10.

11.

12.

13.

14.

15.

16.

ñ = ï ñóò

 ( ƒ ) + 'ý + ° = 0.

+ ! (ñó ò ). = L3 +  ( L ). The Clairaut equation. Solution: In addition, there is a singular solution, which may be written in the parametric form as:
 = −  g
(a ), = −aª g (a ) +  (a ).

= ï ! (ñó ò ) + " (ñó ò ). The Lagrange–d’Alembert equation. For the case  (a ) = a , see equation 1.8.1.10. Having differentiated with respect to  , we arrive at a linear equation with respect to  =  (a ), where a = 
 : [a −  (a )] g
=  g
(a ) + g
(a ). See also 1.8.1.12.

ñ

ï ! (ñ óò

) + ñ" (ñ ó ò ) + % (ñ ó ò ) = 0. The Legendre transformation m [  (m ) + m (m )] n o
− (m ) n + % Inverse transformation:  = n

= 
 , n =  
 − , n o
=  (m ) = 0.


o , = mn o − n ,  = m .

leads to a linear equation:

= ï 2 ! (ñó ò ) + ï " (ñó ò ) + % (ñó ò ). Having differentiated with respect to  , we arrive at an Abel equation with respect to  =  (a ), where a = 
 : V 2  (a ) + (a ) − aÖW g
= −  g
(a ) 2 − g
(a ) − % g
(a ) (see Subsection 1.3.4).

ñ

ï

= ñ 2 ! (ñó ò ) + ñ" (ñó ò ) + % (ñó ò ). Having differentiated with respect to  , we arrive at an Abel equation with respect to = (a ), where a = 
 : V 2aª (a ) + aÝ (a ) − 1W g
= −aª g
(a ) 2 − aÝ g
(a ) − aª% g
(a ) (see Subsection 1.3.4).

= ï ÷ ! (ñ ó ò ) + ï ñ ó ò . Differentiating with respect to  and denoting a = 
, we obtain a Bernoulli equation for  =  (a ): B… (a ) g
−  g
(a ) −  2−  = 0.

ñ

( ï ñ ó ò – ñ )÷ ! (ñ ó ò ) + ñ" (ñ ó ò ) + ï % (ñ ó ò ) = 0. The Legendre transformation  = ‡ g
, = aև g
− ‡ [aÝ (a ) + % (a )]‡ g
= (a )‡ +  (a )‡  .

(
 = a ) leads to a Bernoulli equation:

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1.8. EQUATIONS OF THE FORM

ê ( Ë , Ï , Ï ÒÍ

) = 0 CONTAINING ARBITRARY FUNCTIONS

205

1.8.1-3. Arguments of arbitrary functions are linear with respect to
 . 17.

ñ

= ô (ñ

óò

)2 + ! ( ï – 2 ôñ

óò

).

1 =  (L ) + ( − L )2 . In addition, there is a singular solution, which can be Solution: 4 represented in parametric form as:

 = a + 2  g
(a ), 18.

ñóò

= ! (ñó ò + ô

ï

The contact transformation leads to a linear equation: n Inverse transformation: 19.

ñóò

= ! (ñó ò + ô

ï )(ñ óò 2 + 2ôñ

ïƒ÷  ^ ]

= ! (ï

ñóò

).

The substitution

22.

23.

24.

25.

o

m




).

= 
 +  , n = 12 (
 )2 +  , n o
= 
 , where n = n (m ), = 2 (m ) n +  (m ).

=  −1 (m − n o ), = 12  −1 [2 n − ( n o )2 ], 
 = n o
.

) + " (ñó ò +

ô ï )(ñ óò 2 + 2ôñ )

.

The contact transformation m = 
+  , n = 12 ( 
)2 +  , n o
= 
, where n = n (m ), leads to a Bernoulli equation: n o
= 2  (m ) n + 2  (m ) n  . Inverse transformation:  =  −1 (m − n o
), = 12  −1 [2 n − ( n o
)2 ], 
 = n o
.

20.

21.

ô ï )(ñ óò 2 + 2ôñ

) + " (ñó ò +

=  (a ) +  [  g
(a )]2 .

ï

= ! (ñó ò )" ( ï

ñóò

¨ 
¨ ). leads to an equation of the form 1.8.1.32:   ¨ 9 =  ( 2

= ln ¨

– ñ ).

The Legendre transformation m = 
 , n =  
 − , n o
=  leads to a separable equation: n o
=  (m ) ( n ). Inverse transformation:  = n o
, = mn o
− n , 
 = m .

ï ! (ï ñ óò

– ñ )ñó ò + ï

" (ï ñ óò

ï

= ! (ï

ñóò

+ ñ )" ( ï

2

= ! (ï

ñ óò

+ñ )+ï

2

– ñ )(ñó ò )  = ô .

The modified Legendre transformation m =  
 − , n = 
 , n o
= 1  leads to a Bernoulli equation: !n o
=  (m ) n + (m )n  . Inverse transformation:  = ( n o
)−1 , = n ( n o
)−1 − m , 
 = n .

ñóò

).

The contact transformation m =  
 + , n =  2 
 , n o
=  , where n = n (m ), leads to a separable equation: n o
=  (m ) ( n ). Inverse transformation:  = n o
, = m − n ( n o
)−1 , 
= n ( n o
)−2 .

ï

ñ óò " (ï ñ óò

+ ñ ).

The contact transformation m =  
 + , n =  2 
 , n o
=  , where n = n (m ), leads to a linear equation: n o
= (m ) n +  (m ). Inverse transformation:  = n o
, = m − n ( n o
)−1 , 
= n ( n o
)−2 .

ï ñóò ! (ï ñ óò

+ñ )+

ï 3 (ñ óò )2" (ï ñ óò

+ñ )=ô .

The contact transformation m =  
 + , n =  2 
 , n o
=  , where n = n (m ), leads to a Bernoulli equation: !n o
=  (m ) n + (m )n 2 . Inverse transformation:  = n o
, = m − n ( n o
)−1 , 
= n ( n o
)−2 .

© 2003 by Chapman & Hall/CRC

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206 26.

FIRST-ORDER DIFFERENTIAL EQUATIONS + ï ) = ñ 2 (ñó ò 2 + 1). Setting ‡ ( ) = ! 
+  and differentiating with respect to  , we obtain

! (ñ ñóò

‡
 [  
(‡ ) − 2‡ + 2 ] = 0.

(1)

= −( − L ) + $ . Substituting Equating the first factor to zero, after integrating we find the latter into the original equation yields $ =  ( L ). As a result we obtain the solution: 2 =  ( L ) − (  − L )2 . There is also a singular solution that corresponds to equating the second factor of (1) to zero. This solution in parametric form is written as: 2

27.

28.

 = ‡ − 12  
(‡ ),

=  (‡ ) − 14 [  
(‡ )]2.

2

= ! (ññ ó ò – ï ) + ñ 2 (ñ ó ò 2 – 1)" (ññ ó ò – ï ). The contact transformation  = nIn o
− m , = n [( n o
)2 − 1]1 J 2 , 
 = n o
[( n o
)2 − 1]−1 J 2 , ú = n 2. where n = n (m ), leads to the equation nCn o
=  (m ) + n 2 (m ), which is linear in




! 2 1 J 2 2 −1 Inverse transformation: m =  −  , n = − [(  ) − 1] , n o = −  [(  ) − 1] J 2 .

ññ óò

ñóò

=

1 ! Oiñóò ï 2I

ñ ï

+

P

+ Oiñó ò –

ñ " Oiñóò ï P3

+




ï

ñ

P

.

The contact transformation m =  +   , n =  equation: mn o
= 2 (m ) n + 2mR (m ). Inverse transformation: m n o
 1  =A mn o
− n , =A m ü 2 mn 29.

ï D –1 I ! Oiñóò

–ô

ï

ñ

P

+ Oiñó ò –

ñ "3Oiñóò ï P

–ô

For  ≠ 1, the contact transformation m = to a linear equation: ˜n o
= (m ) n +  (m Inverse transformation: 1 1  = ( n o
) 1−  , = (mn 1− 30.

2

ï D +1 I ! Oiñóò

+ô

ï

ñ

P

= " (ï

D +1 ñ óò

–ï

D¦ñ

ï

ñü

P

2

(
 )2 −

− 2n ,
o −n

2

, n o
= 2 2 
 leads to a linear


=

=õ .


 −    , n =  ).



 o − n )( n o ) 1−  ,

1−

m 2n o
. 2(mn o
− n )

 
 −  −  , n o
=  
=

1−

mn o
− !n . (1 −  ) n o


).

For  ≠ −1, the contact transformation m = 
 +    , n =   +1 
 −   , n o
=   to a separable equation:  (m ) n o
= ( n ). Inverse transformation:  1 1 mn o
+ !n 
= (mn o
− n )( n o
)−  +1 ,  = ( n o
)  +1 , = .  +1 (  + 1) n o
31. 32.

 leads

ó  ^ ñ ÷

= ! (ñó ò ¦ñ ). The substitution  = ln a leads to an equation of the form 1.8.1.32: a9

ï ÷ ñ ú

+1

leads

 =  (a 
g  ).

= ! ( ï ñ ó ò ¦ñ ). We pass to a new variable ¨ ( ) =  
 , divide both sides of the equation by   + , and differentiate with respect to  . As a result we arrive at a separable equation:   ?
(¨ )¨2
= ( ¨ + B )  (¨ ). ¡ Solution in parametric form:  ?
(¨ ) ¨ + L ,   + =  (¨ ). ln | | = X M ( ¨ + B )  (¨ )

¡

In addition, there are singular solutions equation # + −  (−B ) = 0.

 J + , where #  are roots of the algebraic = #   −´

¡

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Page 206

ê ( Ë , Ï , Ï ÒÍ

1.8. EQUATIONS OF THE FORM

33.

34.

35.

36.

ó

ñ óò

ó

=  ! ( ñ óò ) – ñ .  The contact transformation m =  
 , n = 
 + , n separable equation: nCn o
=  (m ). m n Inverse transformation:  = − ln n o
, = n − 

ó

ó

 ! ( ñóò

) – " (ñó ò + ñ ) = 0.  The contact transformation m =  
 separable equation: ( n ) n o
= −  (m Inverse transformation:  = − ln n

ó

ó

! ( ñ óò

) + " (  ñ ó ò )(ñ ó ò + ñ The contact transformation linear equation: !n o
= (m Inverse transformation:

ó

ó


o =  − , where n = n (m ), leads to a


o ,  = mn o .

 = 
 + , n o
=  − , where n


o ,

) – " (ñó ò + ñ ) = ï . This equation can be rewritten in the form 1.8.1.34:


o =  − , where n = n (m ), leads to a


o ,  = mn o .

! (  ñóò

where

 (‡ ) = − ln  (‡ ),

sin ï – ñ cos ï ) = ñ„ó ò ! (ñóò cos ï + ñ sin ï ). The contact transformation


cos  + sin  £ (
 )2 + 2 , 2 leads to the homogeneous equation: n o
=  ( n m ).

H ( ïƒ÷

+1

1 = £ 
(  )2 +

,

H (ï 2 ñ óò

n o
=
( 
sin  − cos  ) 

n =

ñóò , ï ñ óò

Solution:  (− L 1 B , L

39.

( ) = − ln 4 ( ).

ñ (ñ óò

m

38.

= n (m ), leads to a

= n − mn o
, 
 = mn o
.

) = ô – .  m =  
 , n = 
 + , n ) n +  (m ).  = − ln n o
, = n −  m n

    (  
) − 4 ( 
+ ) = 0,

37.

, n ).

207

) = 0 CONTAINING ARBITRARY FUNCTIONS

+ ùñ ) = 0. = L 1  −  + L 2 . Here, the constants L 2 B ) = 0.

1

and L

2

are related by the constraint

+ 2 ï ñ , ï 3 ñ ó ò + ï 2 ñ ) = 0. = L 1  −1 + L 2  −2 . Here, the constants L 1 and L 2 are related by the constraint Solution:  ( L 1 , − L 2 ) = 0. The singular solution can be represented in parametric form as:

  (‡ ,  ) + " (‡ ,  ) = 0,

 (‡ ,  ) = 0,



‡ =  2 a + 2 ,

where

 =  3a + 

2

.

The subscripts ‡ and  denote the respective partial derivatives, and a is the parameter. 40.

+ þ ïƒ÷ ñ , ïƒú +1 ñó ò + ù ïƒú ñ ) = 0. = L 1  −  + L 2  − + . Here, the constants L Solution:   L 1 ( − B ), L 2 (B − )  = 0.

H ( ïƒ÷

+1

ñóò

¡

41.

42.

and L

2

are related by the constraint

¡

ó H (  ñ óò , ñ óò

+ ñ ) = 0.  = L 1 − + L Solution:  (− L 1 , L 2 ) = 0.

ó H (  ^ ñ óò

1

ó

ó

2.

Here, the constants L

1

and L

2

are related by the constraint

ó

+ J ^ ñ ,  _ ñó ò + q _ ñ ) = 0.   Solution: = L 1  − 9 + L 2  −: . Here, the constants L   L 1 (ý − ƒ ), L 2 ( ƒ − ý )  = 0.

1

and L

2

are related by the constraint

© 2003 by Chapman & Hall/CRC

Page 207

208 43.

FIRST-ORDER DIFFERENTIAL EQUATIONS

H (ñ óò

cosh ï – ñ sinh ï ,

ñ óò

= L 1 sinh  + L Solution: constraint  ( L 1 , − L 2 ) = 0. 44.

H (ï ñ óò , ñ

–ï

45.

HwO

ñ óò ñ

, ln ñ –

Solution:  ( L 2 , ln L 46.

HwO

ï ñ óò ñ

1)

47.

H (ñ óò

ln  + L

1

ï

ñóò ñ

P

ï ñ óò ñ

ñ óò

,

ñ

–

r

r òó ñ óò P

= L Solution:  ( L 1 , L 2 ) = 0. 49.

óò ñ

2

are related by the

1

and L

2

are related by the constraint

2

 ). Here, the constants L

1

and L

2

are related by the constraint

ln ï

= 0.

P

and L

1

2

cos  . Here, the constants L

r

= 0,

r (ï

=

 ( L 1 , − L 2 ) = 0.

and L

1

2

are related by the constraint

).

Y ( ) + L 2 . Here, the constants L

–

are related by the constraint

2

sin ï – ñ cos ï ) = 0.

ñ óò

ñ óò r òóñ – r ñ óò s r , r óò – óò óò – r óò P s = L Y ( s ) + L s j ( ). s Solution: 1 2 HwOr s

and L

1

k 2 . Here, the constants L

= L 1 1 ) = 0.

cos ï + ñ sin ï ,

HwO r òó

1

Here, the constants L

2.

Solution: = L 1 sin  + L  ( L 1 , − L 2 ) = 0. 48.

cosh  . Here, the constants L

= 0.

= L 1 exp( L = 0.

, ln ñ –

Solution:  ( L 2 , ln L

2

ln ï ) = 0.

ñ óò

= L Solution:  ( L 1 , L 2 ) = 0.

sinh ï – ñ cosh ï ) = 0.

r

= 0,

=

and L

1

r (ï

),

Here, the constants L

1

are related by the constraint

2

=

s

and L

s 2

( ï ). are related by the constraint

The singular solution can be represented in parametric form as:

 (‡ ,  ) = 0,

j·  (‡ ,  ) + Ys (‡ ,  ) = 0,

where ‡ =



j Y j h




− aÖj , 
− j·Y


 =

Y Y j h




− a›Y . 
− j·Y


The subscripts ‡ and  denote the respective partial derivatives, and a is the parameter. 50.

H r ó

+

r ñ óò , r ]

– ï (r

r ñ óò )  ] ºLt , Y  = ºLt

ó

Here, Y = Y ( , ), Y  =

º

+

º

= 0.



(Y  + Y

. Differentiating with respect to  , we obtain




)
 ( 



− "



) = 0,

where   = º Ž  and  = º Ž are partial derivatives of function  (‡ ,  ). Equating the first º  solution: factor to zero,º we find the

Y ( , ) = L3 + # ,

where

It remains to be checked whether the equation  which of them satisfy the original equation.



 ( L , # ) = 0.

− "



= 0 possesses any solutions and

© 2003 by Chapman & Hall/CRC

Page 208

1.8. EQUATIONS OF THE FORM

ê ( Ë , Ï , Ï ÒÍ

209

) = 0 CONTAINING ARBITRARY FUNCTIONS

1.8.1-4. Arguments of arbitrary functions are nonlinear with respect to
 . 51. 52.

= ! (ï ñ óò 2) + 2ï ñ óò . Solution: [ −  ( L )]2 = 4 L3 .

ñ

= 2 ô (ñ„ó ò )3 + ! ( ï – 3 ôñó ò 2 ). This is a special case of equation 1.8.1.72 with B = 3.  − L 3J 2 . In addition, there is the following singular solution Solution: =  ( L ) + 2 rO 3 P written in parametric form:

ñ

 = a + 3  [  g
(a )]2 ,

53.

54.

55.

56.




=  (a ) + 2 [  g (a )]3 .

= ! ( ôñó ò 2 – õ ï ) + (2 ôñó ò 3 – 3 õÿñ )" ( ôñó ò 2 – õ ï ), õ ≠ 0. 

2 The contact transformation m =  (  ) − ' , n = 2  (  )3 − 3  , n o
= 3
 leads to a linear equation: n o
= 3 (m ) n + 3  (m ). 1 −1  [2  ( n o
)3 − 27 n ], 
 = 13 n o
. Inverse transformation:  = 19  −1 [  ( n o
)2 − 9m ], = 81

ñóò

ñóò

= ! (ñó ò + ô

ï )"

( 21 ñó ò 2 + ôñ ).

The contact transformation m = 
 +  , leads to a separable equation: n o
=  (m ) Inverse transformation:  =  −1 (m − n

ñóò (ñ óò

– ñ ) = ! (ñó ò 2 – ñ 2 ).

The contact transformation m = (
 )2 − (separable) equation: 2  (m ) n o
= n .

ó

ñ óò

= ! (ñ ó ò 2 – ñ 2 )( ô + õ The contact transformation

–

ó

2

n = 12 (
 )2 +  , n o
= 
 , where n = n (m ), ( n ).



o ), = 12  −1 [2 n − ( n o )2 ],  = n o .

  , n =  (
 − ), n o
= 12  ( 
)−1 leads to a linear

).

    n = 
(   +   − ) − (   −   − ), leads to a separable equation: 2  (m ) n o
= 1. m

57.

58.

59.

60.

ñóò

= ( 
)2 −

2

ó

,

ó

ó

=  ! (ñó ò 2 – ñ 2 )" (  ñó ò –  ñ ). The contact transformation m = (
 )2 − separable equation: 2  (m ) ( n ) n o
= 1.

2

, n

  n o
= 12 (   +   − )( 
)−1



=  (
 − ), n o
=

1 2

  (
 )−1 leads to a

= ! (ñó ò 2 – ñ 2 ) cosh ï . The contact transformation m = ( 
)2 − 2 , n = 
cosh  − sinh  , n o
= leads to a separable equation: 2  (m ) n o
= 1.

ñóò ñóò

= ! (ñó ò 2 – ñ

2

1 2

cosh  ( 
)−1

1 2

sinh  ( 
)−1

) sinh ï .

The contact transformation m = (
 )2 − 2 , n = 
 sinh  − cosh  , n o
= leads to a separable equation: 2  (m ) n o
= 1.

= ! (ñó ò 2 – ñ 2 )( ô cosh ï + õ sinh ï ). The contact transformation

ñóò

m = ( 
)2 − 2 ,

n = 
(  cosh  +  sinh  )− (  sinh  +  cosh  ),

leads to a separable equation: 2  (m ) n o
= 1.

 cosh  +  sinh  n o
= 2 


© 2003 by Chapman & Hall/CRC

Page 209

210 61.

62.

63.

64. 65.

FIRST-ORDER DIFFERENTIAL EQUATIONS = ! (ñó ò 2 – ñ 2 )" (ñó ò cosh ï – ñ sinh ï ) cosh ï . The contact transformation m = (
 )2 − 2 , n = 
 cosh  − sinh  , n o
= leads to a separable equation: 2  (m ) ( n ) n o
= 1.

ñóò

= ! (ñó ò 2 – ñ 2 )" (ñó ò sinh ï – ñ cosh ï ) sinh ï . The contact transformation m = (
 )2 − 2 , n = 
 sinh  − cosh  , n o
= leads to a separable equation: 2  (m ) ( n ) n o
= 1.

ñóò

) + ôñó ò 2 cosh ï – ôññó ò sinh ï = cosh ï . The contact transformation m = (
 )2 − 2 , n = 
 cosh  − sinh  , n o
= leads to a linear equation: 2 n o
= !  n +  (m ).

ñóò ! (ñ óò 2 – ñ

68.

1 2

cosh  (
 )−1

= ! (ñó ò 2 + ñ 2 )( ô cos ï + õ sin ï ). The contact transformation

ñóò

n o
= 12 (  cos  +  sin  )( 
)−1

= ! (ñó ò 2 + ñ 2 )" (ñó ò cos ï + ñ sin ï ) cos ï . The contact transformation m = (
 )2 + 2 , n = 
 cos  + sin  , n o
= to a separable equation: 2  (m ) (n ) n o
= 1.

ñóò

) + ôñó ò 2 cos ï + ôññó ò sin ï = cos ï . The contact transformation m = (
 )2 + 2 , n = 
 cos  + sin  , n o
= to a linear equation: 2 n o
= !n +  (m ).

ñóò ! (ñ óò 2 + ñ

= ! (ñó ò 2 + ñ 2 )" (ñó ò sin ï – ñ cos ï ) sin ï . The contact transformation m = (
 )2 + 2 , n = 
 sin  − cos  , n o
= to a separable equation: 2  (m ) (n ) n o
= 1.

ñóò

ï 2 ñ óò

1 2

cos  (
 )−1 leads

1 2

cos  (
 )−1 leads

1 2

sin  (
 )−1 leads

2

ñ ï ï P " ( 2 ñ óò 2 – ñ

=I ! Oiñóò +

2

).

The contact transformation m = 
 +   , n =  2 (
 )2 − equation: n o
= 2  (m ) ( n ). Inverse transformation:

 =A

71.

sinh  (
 )−1

= ! (ñó ò 2 + ñ 2 ) cos ï . This is a special case of equation 1.8.1.65 with  = 1 and  = 0.

69.

70.

1 2

ñóò

n = 
(  cos  +  sin  )+ (  sin  −  cos  ), leads to a separable equation: 2  (m ) n o
= 1.

67.

cosh  (
 )−1

2

m = ( 
)2 + 2 ,

66.

1 2

m

1

ü

mn o
− n ,

=A

mn o
− 2 n , 2 mn o
− n ü

2

, n o
= 2 2 
 leads to a separable


=

m 2n o
. 2(mn o
− n )

= ! ( ôñó ò 2 – õ ï )" (2 ôñó ò 3 – 3 õÿñ ), õ ≠ 0. 
2 The contact transformation m =  (  ) − ' , n = 2  (
 )3 −3  , n o
= 3
 leads to a separable equation: n o
= 3  (m ) ( n ). 1 −1  [2  ( n o
)3 − 27 n ], 
 = 13 n o
. Inverse transformation:  = 19  −1 [  ( n o
)2 − 9m ], = 81

ñóò

ñ

= ! (ï

ñóò ÷

Solution:

)+

ù

ù

–1

ï ñóò

.

 −1 B…L =  (L  ) +   . B −1

© 2003 by Chapman & Hall/CRC

Page 210

1.8. EQUATIONS OF THE FORM

72.

ê ( Ë , Ï , Ï ÒÍ

) = 0 CONTAINING ARBITRARY FUNCTIONS

211

= ô (ù – 1)(ñ„ó ò )÷ + !  ï – ôù (ñó ò )÷ –1  . Differentiating with respect to  , we obtain a factorized equation:

ñ

V 1 − B (B − 1)( 
) 

 B ( 
) where a =  − 

−2



 W V 
−  g
(a )W = 0,

(1)

−1

. Equate the first factor to zero and integrate the obtained equation. Substituting the expression obtained into the original equation, we find the solution:

 −L  B P

=  ( L ) +  (B − 1) O



−1

.

Equating the second factor in (1) to zero, we have another solution that can be written in parametric form as:

 = a + B [  g
(a )] 

73.

−1

=  (a ) +  (B − 1)[  g
(a )]  .

,

= !  ô (ñó ò )  – õ ï  "  ôƒü (ñó ò )  +1 – õ ( ü + 1)ñ  . The contact transformation (  ≠ 0, , ≠ −1)

ñóò

=  ( 
)  − ' ,

m

n = , ( 
) 

+1

n o
= ( , + 1) 


−  ( , + 1) ,

leads to a separable equation: n o
= ( , + 1)  (m ) ( n ). Inverse transformation:

 (n o
)  m − ,   ( , + 1) 

 = 74.

=

, ( n o
)   ( , + 1) 

+1 +2

−

n

 ( , + 1)

= !u ô (ñ ó ò )  – õ ï  +  ôƒü (ñ ó ò )  +1 – õ ( ü + 1)ñ  * "  ô (ñ The contact transformation (  ≠ 0, , ≠ −1)

óò )

ñ óò


=  (  )  − ' ,

m

n = , ( 
) 

+1

–

−  ( , + 1) ,

leads to a linear equation: n o
= ( , + 1) (m ) n + ( , + 1)  (m ). Inverse transformation:

 (n o
)  m − ,  ( , + 1)  

 = 75.

76.

H  ññóò

+ Solution:

ó

H (  ñóò

ô ï ,ñ

–

2

78.

, ( n o
)   ( , + 1) 

+1 +2

−

n

 ( , + 1)

ñ óò 2 + ô

 = 0. = −  + 2 L 1  + L 2 . Here, the constants L ü

2

ó ñ , ñ óò 2 – ñ 

= L 1 + L Solution:  (−2 L 2 , −4 L 1 L 2 ) = 0.

77.

=

1

and L

  L 1 , ü L 12 + L 2  = 0 if   L 1 , − ü L 12 + L 2  = 0 if

n o
.


=

,

õï

, +1

.

n o
= ( , + 1) 


n o
.


=

,

, +1

are related by the constraint

2

> 0, < 0.

2

) = 0. − . Here, the constants L 2

1

and L

cosh ï – ñ sinh ï , ñ ó ò 2 – ñ 2 ) = 0. = L 1 sinh  + L 2 cosh  . Here, the constants L Solution: constraint  ( L 1 , L 12 − L 22 ) = 0.

2

are related by the constraint

H (ñ óò

cos ï + ñ sin ï , ñ ó ò 2 + ñ 2 ) = 0. Solution: = L 1 sin  + L 2 cos  . Here, the constants L  ( L 1 , L 12 + L 22 ) = 0.

H (ñ óò

1

and L

1

and L

2

are related by the constraint

2

are related by the

© 2003 by Chapman & Hall/CRC

Page 211

212

FIRST-ORDER DIFFERENTIAL EQUATIONS

1.8.2. Some Transformations 1.

2.

ï

= ! (ñ , ñó ò ). Substituting a = 
and differentiating both sides of the equation with respect to  , we obtain an equation with respect to = (a ): – – where g = º g ,  = º . [1 − aª ( , a )] g
= aªg ( , a ),   º º If = (a ) is the solution of the latter equation, the solution of the original equation can be represented in parametric form as:  =  ( (a ), a ), = (a ).

= ! ( ï , ñó ò ). Differentiating with respect to  and setting a = "
 , we obtain an equation with respect to  =  (a ): – – where g = º g ,   = º  . [a −   ( , a )] g
= g ( , a ), º º If  =  (a ) is the solution of the latter equation, the solution of the original equation can be represented in parametric form as:  =  (a ), =  ( (a ), a ).

ñ

ïƒ÷ ñ ú

3.

=I ! O À ï

Set H =  

ñ

s

,

and ¨

ï ñ óò ñ =

P  


. . Divide both sides of the equation by   +

and differentiate

with respect to  . As a result we arrive at the following equation with respect to ¨ = ¨ ( H ): ‘ H ( ° ¨ + , )(  +  ? ¨ ‘
) = ( ¨ + B )  , where  =  ( H , ¨ ), ¡ which is usually simpler than the original equation, since it is readily solved for the derivative. If ¨ = ¨ ( H ) is the solution of the Àequation obtained, the solution of the original equation is written in parametric form as:   = H ,   + =  ( H , ¨ ( H )). 4. 5. 6.

7.

8.

]

ïƒ÷  ^ ]

= ! ( ïƒú The substitution

ó  ^ ñ ÷

 _ ] , ï ñ óò

ó

). = ln ‡ leads to an equation of the form 1.8.2.3:   ‡ 9 =    + ‡ : , ‡
 ‡  .

= ! (  _ ñ ú , ñ ó ò ¦ñ ). The substitution  = ln a leads to an equation of the form 1.8.2.3: a 9

! (ï , ï ñ óò

 =   a : + , a g
  .

– ñ , ñó ò ) = 0. The Legendre transformation  = ‡ g
, = aև g
− ‡ (
 = a ), where ‡ = ‡ (a ), leads to the equation  (‡ g
, ‡ , a ) = 0. Inverse transformation: a = 
, ‡ =  
− , ‡ g
=  .

(ñ ó ò )2 = ƒñ + ! ( ï ). For Œ ≠ 0, the transformation Œ ¨ = 2 £ Œ +  ( ) leads to an Abel equation of the second kind, ¨*¨ 
= ¨ + Y ( ), where Y = 2 Œ −2  
( ), which is outlined in Subsection 1.3.1 for specific functions Y .

= ï ñó ò + ô ï 2 + ! (ñó ò ), ô ≠ 0. Differentiating the equation with respect to  and changing to new variables a = 
and ¨ = −2  , we arrive at an Abel equation of the second kind, ¨*¨ g
= ¨ + Y (a ), where Y = −2  g
(a ), which is outlined in Subsection 1.3.1 for specific functions Y .

ñ

For information about contact transformations, see Subsection 0.1.8.

© 2003 by Chapman & Hall/CRC

Page 212

Chapter 2

Second-Order Differential Equations 2.1. Linear Equations 2.1.1. Representation of the General Solution Through a Particular Solution 1 b . A homogeneous linear equation of the second order has the general form

 2 ( ) 

 +  1 ( ) 
+  0 ( ) = 0. Let 0 = 0 ( ) be a nontrivial particular solution ( solution of equation (1) can be found from the formula: =

0

OL

1

+L

2

X

 −Ž

2 0

 , M P

0

(1)

1 0) of this equation. Then the general

where

 =X 



1 2

M

 .

(2)

For specific equations described below in 2.1.2–2.1.9, often only particular solutions are given, while the general solutions can be obtained with formula (2) (see also Paragraph 0.2.1-1, Item 3 b ). x3y'z|{}~/€ Only homogeneous equations are considered in Subsections 2.1.2 through 2.1.8; the solutions of the corresponding nonhomogeneous equations can be obtained using relations (7) and (8) of Subsection 0.2.1. 2 b . Suppose a particular solution of a homogeneous linear equation is obtained in the closed form = [  ( )]  , with this formula valid for  ( ) ≥ 0. If the equation makes sense in a range of  where  ( ) < 0, then the function = |  ( )|  will be a particular solution of the equation in that range.

3 b . Suppose = L 1  ( )[ ( )]  + L 2  ( )[ ( )]  is the general solution of the homogeneous linear equation with  ≠  , where  and  are free parameters. Then the function = L 1  ( )[ ( )]  + L 2  ( )[ ( )]  ln ( ) will be the general solution of this equation with  =  .

2.1.2. Equations Containing Power Functions 2.1.2-1. Equations of the form 

 +  ( ) = 0. 1.

2.

ñóò ò ó

+ ôñ = 0. Equation of free oscillations.   L 1 sinh( £ | | ) + L 2 cosh( £ | | ) if  < 0, = L 1 + L 2 Solution: if  = 0, L 1 sin( £  ) + L 2 cos( £  ) if  > 0.

– ( ô ï + õ )ñ = 0, ô ≠ 0. −2 J 3 (  +  ) leads to the Airy equation: The substitution ˆ = 

ñóò ò ó

ªŠ

Š − ˆ

= 0,

(1)

213 © 2003 by Chapman & Hall/CRC

Page 213

214

SECOND-ORDER DIFFERENTIAL EQUATIONS which often arises in various applications. The solution of equation (1) can be written as: =L

1

Ai(ˆ ) + L

2

Bi(ˆ ),

where Ai( ˆ ) and Bi(ˆ ) are the Airy functions of the first and second kind, respectively. The Airy functions admit the following integral representation: 1

Ai(ˆ ) =

X ÷ cos  13 a 3 + ˆa  a , 0 M



Bi(ˆ ) =



1

X ÷ exp  − 31 a 3 + ˆa  + sin  13 a 3 + ˆa  S a . 0 Q M

The Airy functions can be expressed in terms of the Bessel functions and the modified Bessel functions of order 1 3 by the relations:

ˆ V® Bi(ˆ ) = Å 13 ˆ V ®

Ai(ˆ ) =

1 3

H ) − ® 1 J 3 ( H )W , Ai(−ˆ ) = 13 ˆ V ø ü 1 V Å ø −1 J 3 ( H ) + ® 1 J 3 ( H )W , Bi(− ˆ ) = 3ˆ −1 3 (

J

ü

J H ) + ø 1 J 3 ( H )W ,

−1 3 (

J H ) − ø 1 J 3 ( H )W ,

−1 3 (

where H = 23 ˆ 3 J 2 . For large values of ˆ , the leading terms of the asymptotic expansions of the Airy functions are: 1 −1 J 4 1 −1 J 4 ˆ ˆ exp(− H ), Ai(−ˆ ) = £ sin OH +  , Ai(ˆ ) = £ 4P 2   1 −1 J 4 1 −1 J 4 ˆ ˆ exp( H ), Bi(−ˆ ) = £ cos OH +  . Bi(ˆ ) = £ 4P





The Airy equation (1) is a special case of equation 2.1.2.7 with  = B = 1. 3.

ñ óò ò ó

– (ô

2

ï

2

+ ô )ñ = 0.

Particular solution: 4.

ñóò ò ó

ï

– (ô

2

0

= exp 

1 2

 2  .

+ õ )ñ = 0.

The Weber equation (two canonical forms of the equation correspond to  = A

‘

1 4 ).

1 b . The transformation H =  2 £  , ‡ =  J 2 leads to the degenerate hypergeometric equation 1  1 + 1 ‡ = 0. 2.1.2.70: H´‡ ‘'

‘ + O − H ‡ ‘
− O £ 2 4 P P  2 b . For  = ,

2

> 0,  = −(2B + 1) , , where B = 1, 2,  , there is a solution of the form: = exp  − 12 ,! 2  5   £ ,·  ,

, > 0,



where 5  ( H ) = (−1)  exp  H 2  M  exp  − H 2  is the Hermite polynomial of order B . H See also Subsection S.2.10.M

^_ 5.

ñ óò ò ó

References: H. Bateman and A. Erd´elyi (1953, Vol. 2), M. Abramowitz and I. A. Stegun (1964).

+

ô 3ï

(2 – ô

ï )ñ

Particular solution: 6.

ñóò ò ó

– (ô

ï

2

+

õï

= 0. 0

= exp  − 12  2 

2

+   .

+ ö )ñ = 0.

The substitution ˆ =  +



2

leads to an equation of the form 2.1.2.4:

ªŠ

Š − Oˆ 2 + ( − 

2

4 ·P

= 0.

© 2003 by Chapman & Hall/CRC

Page 214

215

2.1. LINEAR EQUATIONS

7.

– ô ï ÷ ñ = 0. 1 b . For B = −2, this is the Euler equation 2.1.2.123, while for B = −4, this is the equation 2.1.2.211 (in both cases the solution is expressed in terms of elementary function).

ñ óò ò ó

2 b . Assume 2 (B + 2) = 2

=







1−2





 (

1−2

(

+ 1, where

¡

& c )+

+1

)− + ¬L

&c

¬L

1

1

¡

exp ”

is an integer. Then the solution is:

£ 



£ 

exp ”

 &

 &



•

+L

• +L

2

exp ” −

exp ” −

2

£ 

£ 





 &

 & •

•

®

≥ 0,

if

®

if

¡ ¡

< 0,

B +2

1 = . 2 2 +1 ¡ M 3 b . For any B , the solution is expressed in terms of the Bessel functions and modified Bessel functions of the first or second kind (see 2.1.2.126 and 2.1.2.127):

where c



=



where 8.

ñ óò ò ó



= M ,



=





L 1£ 

10.

1 2

”

£ −



 &

&

•

+L

2

£ 2n

1 2

”

£ −



 &

• & £ £    & + L 2£  ; 1 ”  & L 1 £ ® 1 ”  •  • 2 2 & &






if  < 0, if  > 0,

= 12 (B + 2).

– ô (ô

ï 2÷

+ù

ï ÷

Particular solution: 9.

ø

–1 0

)ñ = 0.

= exp O



B +1

 

+1

– ô ï ÷ –2 ( ô ï ÷ + ù + 1)ñ = 0. Particular solution: 0 =  exp    B  .

.

P

ñ óò ò ó

+ ( ô ï 2÷ + õ ï ÷ –1 )ñ = 0. The substitution ˆ =   +1 leads to a linear equation of the form 2.1.2.108: (B + 1) 2 ˆ 
Šª
Š + B (B + 1)
Š + ( ˆ +  ) = 0.

ñ óò ò ó

2.1.2-2. Equations of the form 

 +  ( )
 + ( ) = 0. 11.

ñóò ò ó

+ ôñó ò + õÿñ = 0. Second-order constant coefficient linear equation. In physics this equation is called an equation of damped
vibrations. V 1 1 1 2 2  exp  − 2   L 1 exp  2 Œ   + L 2 exp  − 2 Œ   W if Œ =  − 4  > 0, Solution:

12.

=




exp  − 12   V L

Œ   +L 1     L 1 + L 2  exp  − 2  1

sin 

1 2

2

cos 

1 2

Π W

+ ôñó ò + ( õ ï + ö )ñ = 0. 1 b . Solution with  ≠ 0:

if Œ

2

= 4 − 

if 

2

= 4 .

2

> 0,

ñóò ò ó

= exp  − 12   where

ø

ü

ˆ VL

ø

£ ˆ 3 J 2  + L 2 n 2£  ˆ 3J 2  W , J  1J 3  3 "

2 1 1 3 3

ˆ = +

4( −  4

2

,

J H ) and n 1 J 3 ( H ) are the Bessel functions.

1 3(

2 b . For  = 0, see equation 2.1.2.11.

© 2003 by Chapman & Hall/CRC

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216 13.

14.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï

ñóò ò ó

+

ôñóò

– (õ

ñóò ò ó

+

ôñóò

+ õ (– õ

+ ö )ñ = 0.

2

= ¨ exp  12  2 £   leads to a linear equation of the form 2.1.2.108: The substitution ¨2

 + (2 £ " +  )¨2
+ (  £ " − ( + £  )¨ = 0.

ï

2

Particular solution: 15.

ñóò ò ó

+

ôñóò

+

õï

(– õ

ï

Particular solution: 16.

ñóò ò ó

+

ôñóò

17.

ñóò ò ó

+

ôñóò

ñ óò ò ó

+

ï ñ óò

ñóò ò ó

– 2ï

+ô

ï

+ô

ïƒ÷

+ 2)ñ = 0.

= exp  − 13 ' 3  .

= exp O −

0

–ô 0

ïƒ÷

+ù

ïƒ÷



ïƒ÷

+ù

= exp O −

B +1

ù



= M  ¦ exp  − 21  2  L  Q

M

ñóò

+ 2ùñ = 0,



ù

 

–1



)ñ = 0.

–1

B +1

+ (ù + 1)ñ = 0,

Solution: 19.

3

ï 2÷

+ õ (– õ

Particular solution: 18.

= exp  − 12 ' 2  .

0

Particular solution:

+ 1)ñ = 0.

0

ï 2÷

+ õ (– õ

ô ï

+

+1

.

P

)ñ = 0.

 

+1

− 

= 1, 2, 3, 1

+L

2

= 1, 2, 3,

P

.

uuu

X exp  12  2   S § . M

uuu

Solution: = exp( 2 ) M  ¦ exp(− 2 ) L 1 + L 2 X exp( 2 )  S"§ .  Q M For L 1 = (−1)  andM L 2 = 0, this solution defines the Hermite polynomials. 20.

ñóò ò ó

+

ñ óò ò ó

+

ô ï ñóò

+

ô ï ñ óò

+

õÿñ

= 0.

= L 1 `  12  −1  , 12 , − 21  2  + L 2 i  12  −1  , 12 , − 21  2  , where ` (  ,  ;  ) and Solution: i (  ,  ;  ) are the degenerate hypergeometric functions (see equation 2.1.2.70 and Subsection S.2.7). 21.

õï ñ

= 0.

 =   J  V L 1 `  12  −3  2 , 12 , − 12 ˆ 2  + L 2 i  12  −3  2 , 12 , − 21 ˆ 2  W , ˆ =  − 2  −2  , Solution: where ` (  ,  ;  ) and i (  ,  ;  ) are the degenerate hypergeometric functions (see equation −

2.1.2.70 and Subsection S.2.7).

22. 23. 24.

+ ô ï ñó ò + ( õ ï + ö )ñ = 0. This is a special case of equation 2.1.2.108 with 

ñóò ò ó

26.

=

1

= 0 and 

+ 2 ô ï ñó ò + ( õ ï 4 + ô 2 ï 2 + ö ï + ô )ñ = 0. This is a special case of equation 2.1.2.49 with B = 1 and

ñóò ò ó ñ óò ò ó

+ (ô

ï

+ õ )ñ

óò

Particular solution:

25.

2

+ ôñ = 0. 0

= exp  − 12 

+ ( ô ï + õ )ñó ò – ôñ = 0. Particular solution: 0 =  +  .

2

¡

2

= 1.

= 2.

− '  .

ñóò ò ó

+ ( ô ï + õ )ñó ò + ö ( ô ï + õ – ö )ñ = 0.  Particular solution: 0 =  − v .

ñóò ò ó

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217

2.1. LINEAR EQUATIONS

27.

ñóò ò ó

ï

Particular solution: 28. 29.

0

31.

32.

34. 35.

ñóò ò ó

ï

37.



−

= 

õ 2 )ñ

= 0.

.

+ õ )ñó ò + ö [( ô – ö ) ï

+ (ô

0

= exp 

2

õï

+

(˜  . 2

− 12

2

= 1.

+ 2( ô ï + õ )ñó ò + ( ô 2 ï 2 + 2 ôƒõ ï + ö )ñ = 0. The substitution ‡ = exp  12  2 + '  leads to a constant coefficient linear equation of the form 2.1.2.1: ‡

 + ( ( −  −  2 )‡ = 0.

+ ( ô ï + õ )ñó ò + (  ï 2 +  ï + )ñ = 0. The substitution = ‡ exp(°˜ 2 ), where ° is a root of the quadratic equation 4 ° 2 + 2 !° + ƒ = 0, leads to an equation of the form 2.1.2.108: ‡

 + [(  + 4 ° ) +  ]‡
 + [(ý + 2 ˜° ) + = + 2 ° ]‡ = 0.

ñóò ò ó ñóò ò ó

ï

+ (ô

+ õ )ñó ò + ö (– ö 0

ï 2÷

ïƒ÷

+ô

= exp O −

+1

(

B +1

õ ïƒ÷

+

 

+1

+ ô ( ï 2 – õ 2 )ñó ò – ô ( ï + õ )ñ = 0. Particular solution: 0 =  −  .

+ù

ïƒ÷

–1

)ñ = 0.

.

P

ñóò ò ó

+ ( ô ï 2 + õ )ñó ò + ö ( ô ï Particular solution: 0 = 

ñóò ò ó ñ óò ò ó

ï

+ (ô

2

óò

+ 2 õ )ñ

2 −

+ õ – ö )ñ = 0. .

ï

2

= 

−

+ ( ôƒõ 0

v

ï

–ô





õ 2 )ñ

+

= 0.

.

+ (2 ï 2 + ô )ñó ò + ( ï 4 + ô ï 2 + 2 ï + õ )ñ = 0. The substitution ‡ = exp  13  3  leads to a constant coefficient linear equation of the form 2.1.2.11: ‡ 

 + ‡
 + '‡ = 0.

ñóò ò ó

+ ( ô ï 2 + õ ï )ñ ó ò + (  ï 2 +  ï + )ñ = 0. 1 b . This is a special case of equation 2.1.2.146 with B = 1.

ñ óò ò ó

38.

+ ( ôƒõ ï 2 + õ ï + 2 ô )ñó ò + ô 2 ( õ ï 2 + 1)ñ = 0.  Particular solution: 0 = (  + 1)  −  .

39.

ñóò ò ó

0

=  exp  − 31 

3

− 12 '  2  .

ñóò ò ó

ï

+ (ô

2

+

õï

+ ö )ñó ò + ï ( ôƒõ

Particular solution:

ñóò ò ó

ï

+ (ô

2

+

õï

ñóò ò ó

+ (ô

ï

3

0

− 13

= exp 

+ ö )ñó ò + ( ôƒõ

Particular solution: 41.

2

+ 1]ñ = 0.

2 b . Let ƒ = 0, ý = 3  , = = 2  . Particular solution:

40.

= 0 and 

ñóò ò ó

Particular solution: 36.



+ ( ô ï + õ )ñó ò + ( ö ï + ý )ñ = 0. This is a special case of equation 2.1.2.108 with 

Particular solution: 33.

–ô +

ñóò ò ó

Particular solution: 30.

ï

+ 2 õ )ñó ò + ( ôƒõ

+ (ô

0

= exp 

ï

3

= 

−

+ 2 õ )ñó ò + ( ôƒõ

Particular solution:

0





ï

− 12 –ô

ï

2

+ 3

 3

+

'

2

ï

2

õ^ö

+ 2 ô )ñ = 0.

− (˜  .

ô„ö ï

2

+ õ )ñ = 0.

− ˜(   .

+

õ 2 )ñ

= 0.

.

© 2003 by Chapman & Hall/CRC

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218 42.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

+ (ô

ï

3

õ ï )ñ óò

+

Particular solution: 43.

ñ óò ò ó

+ ( ôƒõ

ï

3

õï

+

ñóò ò ó

+

ô ƒï ÷ ñ óò

óò

+ 2 ô )ñ

2

+ õ )ñ = 0.

2

4

− 12 '  2  .

3

+ 1)ñ = 0.

=  exp  − 14 

0

ï

+ ô 2(õ

−  . 0 = (  + 1) 

Particular solution: 44.

ï

+ 2(2 ô

= 0.

This equation is encountered in the theory of diffusion boundary layer. =L

Solution: 45.

46.

ô ƒï ÷ ñ óò

ñóò ò ó

+

ñóò ò ó

+ 2ô

+

+L

1

õ ïƒ÷ –1 ñ

48.

ïƒ÷ ñóò

ï 2÷

+ ô (ô

ñóò ò ó

+

ô ƒï ÷ ñ óò

+ (õ

ñóò ò ó

+

ô ƒï ÷ ñ óò

– õ (ô

ï 2÷

ñóò ò ó

ïƒ÷ ñóò

+ 2ô

+ö

ïƒ÷ +ú 2

ñóò ò ó

+ (ô

ïƒ÷

ñ óò ò ó

+ (ô

ï ÷

−1

óò

+ 2 õ )ñ

+ ( ôƒõ

ïƒ÷

+

ñóò ò ó

+ ( ôƒõ

ïƒ÷

+ 2õ

ñóò ò ó

+

ïƒ÷ [ô ï

2

)ñ = 0.

   B +1

+1

.

P

)ñ = 0.

ñ óò ò ó

+ (ô

ï ÷

+

ïƒ÷

–1

õ

+

−

 

 +

+1

−

0

2

0



–ô



ï ÷

–1

ï )ñ óò

0

–1

ïƒú

+ö

–1

)ñ = 0.

leads to a linear equation of the form 2.1.2.10:

P

ï ÷

+

ïƒ÷

+ ô ( ôƒõ

õ^ö ]„ñ óò

= +( . = .

.

P

õ 2 )ñ

= 0.

.

 = (  + 2)  −  .

– (ô

)ñ = 0.

.

ï ÷

–ô

–1

– ö )ñ = 0.

+ 2 ô )ñó ò + ô 2 ( õ

–1

+1

ôù ïƒ÷

−  . 0 = (  + 1) 

õ ï ú )ñ ó ò

Particular solution:

+1

B +1

ïƒ÷

= 



ïƒú

+þ

+



)¨ = 0.

+ ( ôƒõ

¡

õ ï 2ú

+ ( ô„ö + õ ) ï +

Particular solution: 55.

õ ï 2ú

exp O

0

õ ïƒ÷

Particular solution: 54.

+

v 0 = 

Particular solution: 53.

–1

+

+ õ )ñó ò + ö ( ô

Particular solution:

ñóò ò ó

ï 2÷

=

Particular solution:

52.

ïƒ÷

= exp O

0

+ (ô

¨ 

 + ( ' 2 + + (˜ +

51.

–1

+1

The substitution ˆ =   +1 leads to a linear equation of the form 2.1.2.108: (B + 1) 2 ˆ Šª

Š + (B + 1)( ˆ + B ) Š
+ ( 'ˆ + ( ) = 0.

The substitution ¨

50.

ïƒ÷

+ù

=  exp O −

0

Particular solution: 49.

= 0.

For B = −1, we obtain the Euler equation 2.1.2.123. For B ≠ −1, the substitution H =   ‘'‘ ‘ leads to an equation of the form 2.1.2.108: (B + 1) 2 H 

+ (B + 1)(!H + B )
+  = 0.

Particular solution: 47.

  +1  . B + 1 P3M

X exp O −

2

–1

+

+ 1)ñ = 0.

ïƒ÷

– ïƒ÷ ( ô

õï ú

–1

ï

+

õ ïƒ÷

–1

–ô

2

ï )ñ

= 0.

+ õ )ñ = 0.

)ñ = 0.

© 2003 by Chapman & Hall/CRC

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219

2.1. LINEAR EQUATIONS

56. 57.

ñóò ò ó

+ (ô

ïƒ÷

+

õ ïƒú )ñ óò

+ ( ôù

ñ óò ò ó

+ (ô

ï ÷

+

õ ï ú )ñ ó ò

+ [ ô (ù + 1)ï

ñóò ò ó

ïƒ÷

+ (ô

ñóò ò ó

õ ïƒú )ñ óò

+

ïƒ÷

+ (ô

ñ óò ò ó

−

õ ïƒú )ñ óò

+

ï ÷

+ (ô

+ ö (ô

v 0 = 

Particular solution: 60.

=  exp O −

0

Particular solution: 59.

–1

õÿþ ïƒú

+

)ñ = 0.

–1

Integrating yields first-order linear equation: "
 + (   + ' + ) = L .

Particular solution: 58.

ïƒ÷

óò

+ ö )ñ

Particular solution:



 

õ ïƒú

+1

ú

–1

 +

+1

+ 1) ï

+ õ (þ



−

+1

¡

]ñ = 0. .

P

– ö )ñ = 0.

.

ïƒú +÷

+ ( ôƒõ

= exp O −

0

–1

B +1 +

=  exp O −

0

õï ú

+

+ [ ôƒõ

ïƒ÷

÷

 ¡

+ õ (þ +1

ï ú +÷ 

B +1

 

 + + +1

+ 1) ïƒú +1

–1

ïƒ÷

–ô

–1

]ñ = 0.

.

P

õ^ö ï ú − (˜

+ ôù

P

ï ÷

–1

)ñ = 0.

.

2.1.2-3. Equations of the form (  +  )

 +  ( )
 + ( ) = 0. 61.

62.

ï ñóò ò ó

+

1 2

ñóò

=

ï ñóò ò ó

+

ôñóò

= 0.

cos £ 4  + L 2 sin £ 4  if  > 0, L 1 cosh £ 4|  | + L 2 sinh £ 4|  | if   < 0.

L

™

Solution: +

ôñ

+

1

õÿñ

= 0.

1 b . The solution is expressed in terms of Bessel functions: 1− 2

=

 V L

1

Á± 2 £ '  + L 2 n Á± 2 £ '  W ,

ø

@ = |1 −  |.

where

2 b . For  = 12 (2B + 1), where B = 0, 1,  , the solution is:



=

63.

ï ñóò ò ó

+

ôñóò

+




  if  > 0, M  cos £ 4  + L 2 M  sin £ 4     M  M L 1 M  cosh 4|  | + L 2 M  sinh 4|  | if   < 0.   ü ü M M L




õï ñ

1

= 0.

1 b . The solution is expressed in terms of Bessel functions: 1− 2

=

 V L

1

Á £ "    + L 2 n Á  £    W ,

ø

where

@ = 12 |1 −  |.

2 b . For  = 2B , where B = 1, 2,  , the solution is: =

64.

ï ñóò ò ó

+

ôñóò







L

1

O

1



 3P

cos   £   + L

2

O



1

M

 3P

sin   £  

M1   L 1O cosh   £ −   + L 2 O sinh   £ −   M M  P  3P  3 M M

+ (õ

ï

1 M

M

+ ö )ñ = 0.

This is a special case of equation 2.1.2.108 with 

2

= 1 and 

1

=

2

if  > 0, if  < 0.

= 0.

© 2003 by Chapman & Hall/CRC

Page 219

220 65.

SECOND-ORDER DIFFERENTIAL EQUATIONS + ùñó ò + õ ï 1–2÷ ñ = 0. For B = 1, this is the Euler equation 2.1.2.123. For B ≠ 1, the solution is:

ï ñóò ò ó

=

66. 67.






70.

L

£ 

1

sin ”

1

exp ”

B −1 £ −

1−

 

B −1



1−

+L

•  •

+ (1 – 3ù )ñ„ó ò – ô 2 ù 2 ï 2÷ –1 ñ = 0. = L 1 (   + 1) exp(−   ) + L Solution:

2

cos ” exp ”

£ 

 B −1 −£ −

1−

if  > 0,

•

1−



+L

2

2 (−

  + 1) exp(  ).

B −1





if  < 0.

•

+ ôñó ò + õ ïƒ÷ ñ = 0. If B = −1 and  = 0, we have the Euler equation 2.1.2.123. If B ≠ −1 and  ≠ 0, the solution is expressed in terms of Bessel functions:

ï ñóò ò ó

ï ñ óò ò ó

1− 2



¬L

1

ø

ÁhO

2£    B +1

+ ôñ ó ò + õ ï ÷ (– õ ï ÷

Particular solution: 69.

L

ï ñóò ò ó

= 68.





0

+1

+

ô

P

+L

2

n Á…O

2£    B +1

+1 2

P

® ,

where

@ =

|1 −  | . B +1

+ ù )ñ = 0.

= exp O −

+ ô ï ñó ò + ôñ = 0.  Particular solution: 0 =   −  .

ï ñóò ò ó

+1 2

   B +1

+1

.

P

ï ñóò ò ó

+ ( õ – ï )ñó ò – ôñ = 0. The degenerate hypergeometric equation. 1 b . If  ≠ 0, −1, −2, −3,  , Kummer’s series is a particular solution:

½ ( )    , ` ( ,  ;  ) = 1 + ÷  =1 (  )  , ! where (  )  =  (  + 1)  ( + , − 1), (  )0 = 1. If  >  > 0, this solution can be written in terms of a definite integral:

1  g ( ) ` ( ,  ;  ) = w X  a  −1 (1 − a )  −  ( ) (  −  ) 0 w w ‘ −g −1 a is the gamma function. where ( H ) = X ÷  a w M 0 If  is not an integer, then the general solution has the form:

=L

1

` ( ,  ;  ) + L 2 

1−

 ` (

−1

M

a,

−  + 1, 2 −  ;  ).

Table 14 gives some special cases where ` is expressed in terms of simpler functions. The function ` possesses the properties:

 ` (  ,  ;  ) =  ` (  −  ,  ; − );

 ( ) M  ` (  ,  ;  ) =  ` (  + B ,  + B ;  ).  ( ) M

The following asymptotic relations hold:

` ( ,  ;  ) = w ` ( ,  ;  ) =

w

w

( )   − 1 S    1 + [­O ( ) | | P Q ( ) 1 (− )−  1 + [­O S w ( −  ) | | P Q

if ED

+F ,

if ED

−F .

© 2003 by Chapman & Hall/CRC

Page 220

221

2.1. LINEAR EQUATIONS

TABLE 14 Special cases of Kummer’s function ` (  ,  ; H )



H 







1

2

2



 +1

−

1 2

3 2

−B

1 2



−B

3 2



`

Conventional notation

 

1 

 sinh 



Incomplete gamma function

 −  = (  ,  ) £

2

−

 2

= ( ,  ) = X

2 2

2

0

−



ga

−1

M

a

Error function  2 X exp(−a 2 ) a erf  = £

erf 

1 − B ! O− 5 2  ( ) (2B )! 2 P 1 − B ! O− 5 2  +1 ( ) (2B + 1)! 2 P

2





M

0

Hermite polynomials

2  2 5  ( ) = (−1)   M   −  ,  M  B = 0, 1, 2, 3, Laguerre polynomials

−B

@ +

1 2

B +1





2@ + 1

2

2B + 2

     −9 M    −   + 9  , B !  ƒ =  M − 1, (  )  =  (  + 1)  (  + B − 1)

0 ( 9 ) (

B ! ( −1) 0  ( ) ( ) 

w

(1 + @ ) 

2

OB +

w



 O

−

Á

2P

3    O 2P 2P



®Á (

− − 12

®

) + 12 (

)=

Modified Bessel functions ®Á ( )

 )

2 b . The following function is a solution of the degenerate hypergeometric equation:

i ( ,  ;  ) =

w

(1 −  ) w ` ( ,  ;  ) + (  −  + 1)

w

(  − 1)  ( )

w

1−

 ` (

−  + 1, 2 −  ;  ).

Calculate the limit as ‹D„B (B is an integer) to obtain

i ( , B ;  ) = ½

(−1)  −1 ™ ` ( , B +1;  ) ln  B ! ( − B )

w

+ ÷

Ÿ =0

+

Ÿ

( )Ÿ V j (  + & ) − j (1 + & ) − j (1 + B + & )W  (B + 1) Ÿ &!

 (B − 1)! ½ ( ) Ÿ

w

−1 =0

¢

Ÿ ( − B )Ÿ  − , (1 − B ) Ÿ & !

where B = 0, 1, 2,  (the last sum is omitted for B = 0), j ( H ) = [ln ( H )]
‘ is the logarithmic w derivative of the gamma function:

j (1) = −= ,

j ( B ) = −= +

½ 

−1

,

−1

,

= = 0.5772  is the Euler constant.

=1

© 2003 by Chapman & Hall/CRC

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222

SECOND-ORDER DIFFERENTIAL EQUATIONS If  is a negative number, then the function i can be expressed in terms of the one with positive second argument using the relation

 i (

1−

i ( ,  ;  ) = 

−  + 1, 2 −  ;  ),

which holds for any value of  . 3 b . For  ≠ 0, −1, −2, −3,  , the general solution of the degenerate hypergeometric equation can be written in the form: =L

1

` (  ,  ;  ) + L 2 i (  ,  ;  ),

while for  = 0, −1, −2, −3,  , it can be represented as: 1−

=

 V L 1 ` (

−  + 1, 2 −  ;  ) + L

2

i (  −  + 1, 2 −  ;  )W .

The functions ` and i are described in Subsection S.2.7 in more detail; see also the books by Abramowitz & Stegun (1964) and Bateman & Erd´elyi (1953, Vol. 1). 71.

ï ñóò ò ó

ï

+ õ )ñó ò + ö [( ô – ö ) ï + õ ]ñ = 0.

+ (ô

Particular solution: 72.

ï ñ óò ò ó

ï

+ (2 ô

ï ñóò ò ó

óò

+ õ )ñ =

Solution: 73.

0

™ 

−

=

+ ô (ô

  −  (L −

.

ï

+ õ )ñ = 0.

1−  ) if  ≠ 1, ln |  |) if  = 1. 2

+L 1+L 1

(L

+ [( ô + õ ) ï + ù + þ

Here, B and

v



2

]ñ„ó ò + ( ôƒõ

ï

ôù

+

are positive integers;  ≠  or B ≠

¡

 + = L 1  M +  M

−1

74.

+ ( ô ï + õ )ñó ò + ( ö ï + ý )ñ = 0. This is a special case of equation 2.1.2.108.

75.

ï ñóò ò ó

ï ñóò ò ó

ï

Particular solution: 76.

ï ñóò ò ó

ï

– (2 ô

ï ñóò ò ó

+ (ô

0

 =  L Q

ï

78.

ï ñóò ò ó

– (2 ô

ï

Solution:

2

79.

ï ñóò ò ó

+ ( ôƒõ

ï

0

öï

3

ô 2ï

+

(– ö

ï

2

= exp 

+ 1)ñó ò +

+ ô )ñ = 0.

õ ï 3ñ

+

− 12

ô ï

2

+

(˜  . 2

cos 

õ

2

+

õ

= 0.

– 5)ñó ò + 2 ô 2 ( õ – 2) ï

+ ( ô ï 2 + õ ï )ñ Particular solution:

ï ñ óò ò ó

0

óò

= ( 

1 2

2

 2£   S .

+ 1) = 0.

= L 1 exp V 12   + £  2 −    2 W + L

Particular solution: 80.

ï

£   +L 1 1 sin  2 

Particular solution:

.

+ ô )ñ = 0.

2

+ õ )ñó ò +

= 0.

= exp  − 12 ' 2  .

+ 1)ñ„ó ò + ( õ

Solution: 77.

õ ï 2(õ ï

+ 1)ñó ò –

õÿþ )ñ

¡  −1 V  −  (  −  ) W + L 2  −   M V  −+  (  −  ) W .  −1  M

−1

−

Solution:

– (ô

+

3

ñ

2

exp V

1 2

  − £  2 −    2W .

= 0. 2

+ 1) exp(−  ).

– [ ô„ö ï 2 + ( ô + v . 0 =  

õ^ö

+ ö 2 )ï +

õ

+ 2 ö ]ñ = 0.

© 2003 by Chapman & Hall/CRC

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223

2.1. LINEAR EQUATIONS

81. 82. 83.

+ ( ô ï 2 + õ ï + 2)ñó ò + õÿñ = 0. Particular solution: 0 =  +   .

ï ñóò ò ó

+ ( ô ï 2 + õ ï + ö )ñó ò + (2 ô ï + õ )ñ = 0. Integrating, we obtain a first-order linear equation:  "
 + ( 

ï ñóò ò ó ï ñóò ò ó

ï

+ (ô

2

+

õï

v. 0 = 

+ ' + ( − 1) = L .

+ õ )ñ = 0.

1−

Particular solution: 84.

ï

+ ö )ñó ò + ( ö – 1)(ô

2

+ ( ô ï 2 + õ ï + ö )ñó ò + (
ï 2 + ( ï + = )ñ = 0. 1 b . Let # = , , $ = , (  − , ), L  = (, , where , is an arbitrary number. Particular solution: 0 =  −  .

ï ñóò ò ó

2 b . Let # =  (  + , ), $ =  ( ( + 1) − , (  + , ), L = − (, . ,  . Particular solution: 0 = exp  − 12  2 + ! 3 b . Let # =  (  + , ), $ = 2  − , − , 2 , L =  ( ( − 1) + , ( ( − 2). Particular solution: 0 =  1− v exp  − 12  2 + ,!  . 4 b . Let # = − , , $ =  ( ( − 1) − , (   + , ), L Particular solution: 0 =  1− v   .

85.

86.

+ ( ô ï 2 + õ ï + 2)ñó ò + ( ö ï 2 + ý ï + õ )ñ = 0. ¨ 

 + (  +  )2 ¨ 
+ The substitution ¨ =  leads to a linear equation of the form 2.1.2.108: I ¨ ( (˜ + −  ) = 0.

ï ñóò ò ó ï ñóò ò ó

M

ï

+ (ô

3

+ õ )ñó ò + ô ( õ – 1) ï

ï ñóò ò ó

ï

+ ï (ô

2

. 0 = 

89. 90.

0

+ ( ôƒõ ï 3 + õ ï Particular solution:

ï ñóò ò ó ï ñóò ò ó

+ (ô

ï

3

+

õï

2

ï ñ óò ò ó

+

ô ï ÷ ñ óò

ï ñ óò ò ó

+ (ô

ï ÷

+ö

2

Particular solution: 93.

= 0.

+ õ )ñ = 0.

=  exp  − 13 

3

− '  .

ô ï

– 1)ñó ò + ô 2 õ  = (  + 1)  −  .

ï

0

óò

1−

ï ÷

0

–ô

= 

+ô =

ï 3ñ

+ ý )ñó ò + ( ý – 1)(ô

x 0 = 

+ ( ôƒõ

+ 2)ñ

+ 0

Particular solution: 92.

2

ñ

+ ( ô ï 3 + õ ï 2 + 2)ñó ò + õ ï ñ = 0. Particular solution: 0 =  +   .

ï ñóò ò ó

Particular solution: 91.

ï

+ õ )ñó ò + (3 ô

Particular solution: 88.

2

1−

Particular solution: 87.

=  (( − 1) + , ( ( − 2).

ï

2

+

õï

+ ö )ñ = 0.

.

ï ÷

−





ï ÷ –1 ñ −1

= 0.

–1

–

õ 2ï

+ 2 õ )ñ = 0.

. = 0.

.

+ ( ïƒ÷ + 1 – ù )ñó ò + õ ï 2÷ –1 ñ = 0. 1 b . For  ≠ 14 , the solution has the form: = L 1 exp  ý 1    + L are roots of the quadratic equation: B 2 ý 2 + B/ý +  = 0.

ï ñóò ò ó

2 b . For  = 14 , the solution has the form:

= (L

1

+L

2

2 exp

  ) exp  − 12 B

 ý 2    . Here, ý −1

1

and ý

2

  .

© 2003 by Chapman & Hall/CRC

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224 94.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï ñóò ò ó

+ (ô

ïƒ÷

Particular solution: 95.

ï ñóò ò ó

+ (ô

ïƒ÷

0

ï ñóò ò ó

+ (ô

ïƒ÷

0

ï ñ óò ò ó

+ (ô

ï ÷

ï ñóò ò ó

+ ( ôƒõ

óò

+ õ )ñ

ï ñóò ò ó

+ (ô

ïƒ÷

ïƒ÷

1−

=

–1

.

õ

+

ï ÷

+ ö (ô

ï

–ö

−v  . 0 = 

ñ

= 0.

–1

ñ

= 0.

+ õ )ñ = 0.

– 3ù + 1)ñ„ó ò + ô

2

– ù )ï

ù (õ

÷ ñ

2 –1

   + 1) exp(−  ). 0 = (

Particular solution: 99.

exp(−   B ).



   B ). 0 = exp(− 

Particular solution: 98.

=

= 0.

+ õ )ñó ò + ô ( õ + ù – 1)ïw÷

Particular solution: 97.

1−

+ õ )ñó ò + ô ( õ – 1) ïƒ÷

Particular solution: 96.

ôù ïƒ÷ –1 ñ

+ õ )ñó ò +

ï 2÷

+ õ )ñó ò + ( ö

–1

+

ý ïƒ÷

–1

= 0.

)ñ = 0.

This is a special case of equation 2.1.2.146 with = = 0. 100.

ï ñóò ò ó

+ (ô

ïƒ÷

õ ïƒ÷

+

–1

ï ñ óò ò ó

+ (ô

ï ÷

õ ï )ñ ó ò

+

ï ñóò ò ó

+ ( ôƒõ

ïƒ÷

õ ïƒ÷

+

ï ñóò ò ó

+ (ô

ïƒ÷

ï ñóò ò ó

+ ( ôƒõ

ïƒ÷ +ú

0

+ ôù

ï

1

ï

ï

+ô

+ ô 0 )ñó ò ò ó + ( õ

– 1)ñó ò + ô −

=

1−

ïƒ÷

v.

+

+ ö )ñó ò +

Particular solution: 106. ( ô

–1

õ ïƒú

2

– õ )ñ = 0.

õ ƒï ÷ ñ

ïw÷

–1

+

= 0.

õ ïƒú

+ 1 – 2ù )ñó ò + ô

   + 1) exp(−  ). 0 = (

Particular solution: 105. ( ï + ô )ñ„ó ò ò ó + ( õ

ôù ï ÷

+ ö )ñó ò + ( ö – 1)(ô

Particular solution: 104.

+

. 0 = (  + 1) 

õ ïƒú

+

ï ÷

+ ( ôƒõ

–1

Particular solution: 103.

= 0.

   B ). 0 =  exp(− 

Particular solution: 102.

õ ïƒ÷ –2 ñ

0 =  +  .

Particular solution: 101.

+ 2)ñó ò +

0

1

ï

õÿñ

õ 0 )ñ óò

2

)ñ = 0.

õÿù ï 2÷ +ú

–1

ñ

= 0.

= 0.

' + ( − 1  .  + ?M P

= exp O − X +

–1

–þ

õ 1ñ

= 0.

= 1, 2, 3,  , a polynomial of order in  is a particular solution of the equation, ¡ ½+ 1  ,ž + ®´ −+ −1 [( 1  +  0)c 2 +  0 c ] .   + , where O− which can be represented as: 0 = If

¡

 Á



+1

=0

with @ ≠ −1. c = M , ´®  Á =  @ +1 M À ï ï 2

107. ( ô

+ õ )ñ„ó ò ò ó + s( ö

Particular solution:

 1P

+ ý )ñó ò – s [( ô + ö ) ï + 0

=



õ

+ ý ]ñ = 0.

.

© 2003 by Chapman & Hall/CRC

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225

2.1. LINEAR EQUATIONS

TABLE 15 Solutions of equation 2.1.2.108 for different values of the determining parameters

  −“ =   ¨ ( H ), where H =

Solution:

,

Constraints

 2 ≠ 0, 2  1 ≠ 4 0 

2

 2 = 0,  1≠0  2 ≠ 0,  = 4 0  2 1

Œ

£ c − 2 2  0 −  1 −

2

 2 =  1 = 0,  0≠0

−



2



2

−

1



2

2 2 , + 



2 1

1

−



2

y

2

2 2 , + 



−

2

Notation: c

1



1



2 2

2 1 − 4 0 2,

=

Œ

Parameters

 = $ ( , ) (2  2 , +  1 ),  = (  2  1 −  1  2 )  −2 2

 = $ ( , ) (2  1), ý = −  (2  )

( ,  ; H )

  , 12 ; ý…H 2 

y

1

: H ÁJ2 / Á ý £ H 

2

 21 − 4  0  4 0  2

1

2



−

1

1

¨ “

 

@ = 1 − (2  2 , +  1 )  −1 2 , ý = 2 $ (, ) ü 1J 2 2  ý = O 0 3 2P

: H 1 J 2 1 J 3  ý…H 3 J 2  see also 2.1.2.12

$ (, ) =  2 , 2 +  1 , + 

2

0

108. ( ô 2 ï + õ 2 )ñ ó ò ò ó + ( ô 1 ï + õ 1 )ñ ó ò + ( ô 0 ï + õ 0 )ñ = 0. Let the function (  ,  ;  ) be an arbitrary solution of the degenerate hypergeometric equation :  

 + (  −  )
 y −  = 0 (see 2.1.2.70), and the function Á ( ) be an arbitrary solution of 


the Bessel equation  2  +   + ( 2 − @ 2 ) = 0 (see 2.1.2.126). The results of solving the original equation are presented in Table 15. 109. ( ï + )ñ

óò ò ó

+ (ô

ï ÷

õï ú

+

Particular solution:

0

+ ö )ñ

óò

= exp O − X

+ ( ôù ï ÷ –1 + õÿþ ï ú –1 )ñ = 0.   + ' + + ( − 1  .  +Œ M P

2.1.2-4. Equations of the form  2 

 +  ( )
 + ( ) = 0. 110.

ï 2 ñ óò ò ó

+ ôñ = 0. g This is a special case of equation 2.1.2.123. The substitution  =  leads to a constant


coefficient linear equation: g g − g +  = 0.

111.

+ ( ô ï + õ )ñ = 0. This is a special case of equation 2.1.2.132.

112.

ï 2 ñ óò ò ó

ï 2 ñ óò ò ó

+ [ô

Solution: 113.

ï 2 ñ óò ò ó ï 2 ñ óò ò ó

– ù (ù + 1)]ñ = 0, ù = 0, 1, 2, uuu L cos    + L 2 sin  , where c   +1 = ( 3 c )  O 1 2  −1

– (ô

P



– [ô 2 ï

Solution: 114.

ï

2 2

2

+ ù (ù + 1)]ñ = 0,

ù

= 0, 1, 2,

uuu

= M .

M



  L 1   + L 2  −  +1 3  = ( c ) O , where c = M .    2  −1 P M

ï

2 2

+ 2 ôƒõ

Particular solution:

ï

0

+

õ

2

– õ )ñ = 0.

 =    .

© 2003 by Chapman & Hall/CRC

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226 115.

116.

SECOND-ORDER DIFFERENTIAL EQUATIONS + ( ô ï 2 + õ ï + ö )ñ = 0. The substitution =   ‡ , where Œ is a root of the quadratic equation Œ 2 − Œ + ( = 0, leads to an equation of the form 2.1.2.108: ‡

 + 2 Œ ‡
 + (  +  )‡ = 0. For  = − 14 ,  = , , and ( = 14 − 2 , the original equation is referred to as Whittaker’s ¡ equation.

ï 2 ñ óò ò ó

ï 2 ñ óò ò ó

– 

ô ï

3



5 16

+

Particular solution: 117.

ï 2 ñ óò ò ó

– [ô

ï

2 4

ñ

= 0. 0

J exp  2 £ r 3J 2  . 3  

−1 4

=

+ ô (2 õ – 1)ï

0 =  

exp  − 21  2  .

−

Particular solution:

+ õ ( õ + 1)]ñ = 0.

2

118.

+ ( ô ï ÷ + õ )ñ = 0. This is a special case of equation 2.1.2.132.

119.

ï 2 ñ óò ò ó

ï 2 ñ óò ò ó

– [ô

ï ÷

2 2

+ ô (2 õ + ù – 1) ïw÷ + õ ( õ – 1)]ñ = 0. exp(   B ).



0

=

õ ï 2÷

+

Particular solution: 120.

+ ( ô ï 2÷ + õ ï ÷ + ö )ñ = 0. This is a special case of equation 2.1.2.146.

121.

ï 2 ñ óò ò ó

ï 2 ñ óò ò ó

+ 

ô ï 3÷

+

1 4

1 4

–

ù

2



ñ

The transformation ˆ =   +  , ¨ ¨2Šª

Š + ( B )−2 ˆ ¨ = 0.

= 0.

122.

ï 2 ñ óò ò ó

+ Vô

ï 2÷ ( õ ï ÷

+ ö )ú

1 4

+

–

1 4

The transformation ˆ = '  + ( , ¨ ¨2Šª

Š +  ( 'B )−2 ˆ +2¨ = 0. 123.



= 

ù

2

W

ñ

−1 2

leads to an equation of the form 2.1.2.7:

= 0.



= 

−1 2

leads to an equation of the form 2.1.2.7:

+ ô ï ñó ò + õÿñ = 0. The Euler equation. Solution:

ï 2 ñ óò ò ó

=











| |

1− 2

| |

1− 2

| |

1− 2





ÝL 1 | |  + L 2 | |−  

if (1 −  )2 > 4  ,

(L

if (1 −  )2 = 4  ,

 V L

1

+L

1

sin(“ ln | |) + L

2

ln | |) 2

cos(“ ln | |)W

if (1 −  )2 < 4  ,

where “ = 12 |(1 −  )2 − 4  |1 J 2 . 124.

125.

+ ï ñó ò + V ï 2 –  ù + 12  Wñ = 0, ù = 0, 1, 2, uuu This is a special case of equation 2.1.2.126.  1 sin  cos  OL 1 S . =   +1 J 2 O +L 2 Solution: M

ï 2 ñ óò ò ó

2

3P

Q 

M





P

+ ï ñó ò – V ï 2 +  ù + 12  Wñ = 0, ù = 0, 1, 2, This is a special case of equation 2.1.2.127.   1  −  +1 J 2 OL 1 + L 2 O S . = Solution: M

ï 2 ñ óò ò ó

Q 

M

2

3P



uuu

åP

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227

2.1. LINEAR EQUATIONS

126.

ï 2 ñ óò ò ó

+ ï ñó ò + ( ï 2 – b The Bessel equation.

2

)ñ = 0.

1 b . Let @ be an arbitrary noninteger. Then the general solution is given by: =L

ø

1

Á ( ) + L 2 n Á ( ),

(1)

where ø Á ( ) and n Á ( ) are the Bessel functions of the first and second kind:

Á ½ (−1)  ( 2) +2  ø Á ( ) cos  @ − ø − Á ( ) . , n Á ( ) = Áø ( ) = ÷ , ! ( @ + , + 1) sin @  =0 w  : Solution (1) is denoted by = Á ( ) which is referred to as the cylindrical function.

(2)

The cylindrical functions possess the following properties:

:

2@

:

Á ( ) =  [ Á

:

 ) + Á +1 ( )], : : M [ − Á Á ( )] = − − Á Á 

−1 (

Á : Á ( )] =  Á : Á ( ), M [ −1 +1 (  ).  M ø Á ( ) and n Á ( ) can be expressed M in terms of definite integrals (with  > 0): The functions

 ø 

Á ( ) = X

n Á ( ) = X

z

0

9

9

cos( sin − @

z

)

9

9

sin( sin − @ )

0

M M

9

− sin @ X ÷ exp(− sinh a − @!a ) a ,  0 M

9

− X ÷ ( 0

Á˜g + − ˜Á g cos @ ) −   

sinh



g a. M

2 b . In the case @ = B + 12 , where B = 0, 1, 2,  , the Bessel functions are expressed in terms of elementary functions:

ø 

+ 12 (

 )=



2  

+ 12

 sin  2   O− M , ø −  − 1 ( ) = 2  3P   M n  + 12 ( ) = (−1)  +1 ø −  − 12 ( ). 1

+ 12

1

O 

M

M

 cos  , 3P 

3 b . Let @ = B be an arbitrary integer. The following relations hold:

n −  ( ) = (−1)  n  ( ). The solution is given by formula (1) in which the function ø ( ) is obtained by substituting @ = B into formula (2), while n  ( ) is found by taking the limit as @*D B and for nonnegative B

ø

−

 ( ) = (−1)  ø  ( ),

becomes

2

n  ( ) = −



ø

1 ½



where j (1) = −z , j (B ) = −z +



( ) ln



2

÷ (−1)  O 7 

=0 −1 =1

,

−1

−



2P



 1 ½ 

−1

 +2

=0

(B − , − 1)! 2  O , ! 2P

−2



 j ( , + 1) + j (B + , + 1) , , ! (B + , )!

, z = 0.5772  is the Euler constant, j ( ) = [ln ( )]


is the logarithmic derivative of the gamma function.

w

For nonnegative integer B and large  , we can write

 ø 2  ( ) = (−1)  (cos  + sin  ) + [ ( −2 ),  £  ø 2  +1 ( ) = (−1)  +1 (cos  − sin  ) + [ ( −2 ).  The function ! ø  ( ) can be expressed in terms of a definite integral: 1 z B = 0, 1, 2,  ø  ( ) = X 0 cos( sin a − B/a ) M a ;  The Bessel functions are described in Subsection S.2.5 in more detail; see also the books £

by Abramowitz & Stegun (1964) and Bateman & Erd´elyi (1953, Vol. 2).

© 2003 by Chapman & Hall/CRC

Page 227

228 127.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï 2 ñ óò ò ó

+ ï ñó ò – ( ï 2 + b 2 )ñ = 0. The modified Bessel equation. It can be reduced to equation 2.1.2.126 by means of the substitution  = ™  ¯ (™ 2 = −1). Solution: = L 1 ®Á ( ) + L 2 ; Á ( ), where ®Á ( ) and ; Á ( ) are the modified Bessel functions of the first and second kind:

½

®Á

( ) = ÷



=0

Á ( 2)2  + , , ! ( @ + , + 1)

; Á (

w

)=

 ) − ®Á ( 2 ® − Á (sin  @

)

.

The modified Bessel function ®Á ( ) can be expressed in terms of the Bessel function:

Á (  z © J 2 ), ™ 2 = −1. The case @ = B + 12 , where B = 0, 1, 2,  , is given in 2.1.2.125. If @ = B is a nonnegative integer, we have  −1 1½  2 + −  (B − − 1)!   +1 + (  ) = (−1) ® (  ) ln (−1) + O ¡ ;|  2 2+ 2P ! =0 ¡ ½   +2 + j (B + + 1) + j ( + 1) 1 , ¡ ¡ + (−1)  ÷ O 2 ! (B + )! + =0 2 P ¡ ¡ where j ( H ) is the logarithmic derivative of the gamma function (see 2.1.2.126, Item 3 b ); for B = 0, the first sum is omitted. As þD + F , the leading terms of the asymptotic expansion are:  £   ®Á ( ) ¬ £ , ; Á ( ) ¬ £   − . 2  2

®Á

( ) = 

−

z

Á© J

2

ø



The modified Bessel functions are described in Subsection S.2.6 in more detail; see also the books by Abramowitz & Stegun (1964) and Bateman & Erd´elyi (1953, Vol. 2). 128. 129.

ï 2 ñ óò ò ó

+ 2ï Solution: 

ï 2 ñ óò ò ó

– 2ô

Solution: 130.

ï 2 ñ óò ò ó

– 2ô

Solution: 131.

132.

– ( ô 2 ï 2 + 2)ñ = 0.   = L 1 (  − 1)   + L 2 (  + 1)  −  .

ñóò

2

ï ñóò =

™

+ [ õ 2 ï 2 + ô ( ô + 1)]ñ = 0. | |  ( L 1 sin ' + L 2 cos ' ) if  ≠ 0, L 1 | |  + L 2 | |  +1 if  = 0.

ï ñ óò =

™

+ [– õ

ï

2 2

+ ô ( ô + 1)]ñ = 0.

  | |  ( L 1   + L 2  −  ) if  ≠ 0, L 1 | |  + L 2 | |  +1 if  = 0.

+ ï ñó ò + ( ô ï 2 + õ ï + ö )ñ = 0. The substitution =   ‡ , where , is a root of the quadratic equation , 2 + ( Œ − 1) , + ( = 0, leads to an equation of the form 2.1.2.108: ‡

 + ( Œ + 2 , )‡
 + (  +  )‡ = 0.

ï 2 ñ óò ò ó

+ ô ï ñ ó ò + ( õ ï ÷ + ö )ñ = 0, ù ≠ 0. The case  = 0 corresponds to the Euler equation 2.1.2.123. For  ≠ 0, the solution is:   1−  2 2 =  2 L 1 ø Á O £ " 2 + L 2 n Á O £  2 S ,

ï 2 ñ óò ò ó

Q

B

P

B

P

where @ =  1 (1 −  )2 − 4 ( ; ø Á ( H ) and n Á ( H ) are the Bessel functions of the first and second ü kind.

© 2003 by Chapman & Hall/CRC

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229

2.1. LINEAR EQUATIONS

133.

134.

135.

136. 137.

+ ô ï ñó ò + ïƒ÷ ( õ ïƒ÷ + ö )ñ = 0. The substitution ˆ =   leads to an equation of the form 2.1.2.108: B ( 'ˆ + ( ) = 0.

ï 2 ñ óò ò ó

+ ( ô ï + õ )ñó ò + ö^ñ = 0. ‘ = H   ¨ , where , is a root of the quadratic equation The transformation  = H −1 , , 2 + (1 −  ) , + ( = 0, leads to an equation of the form 2.1.2.108:

H ¨ '‘

‘ + [(2 −  )H + 2 , + 2 −  ]¨ ‘
+ [(1 −  )H + 2 , + 2 −  − , ]¨ = 0.

+ ô ï 2 ñó ò + ( õ ï 2 + ö ï + ý )ñ = 0. = ¨ exp  − 12   leads to a linear equation of the form 2.1.2.115: The substitution

1 2 ¨2 2 2  + [( 4  +  ) + (˜ + ]¨ = 0. 

ï 2 ñ óò ò ó

M

ï 2 ñ óò ò ó

+ (ô ï

2

ï 2 ñ óò ò ó

+ (ô

ï

2

+ õ )ñó ò + ö [( ô – ö ) ï  Particular solution: 0 =  − v . +

õ ï )ñ óò

ï 2 ñ óò ò ó

+ (ô

ï

2

+

140. 141.

2

=

õ ï )ñ óò

−

   −

.

+ [ü (ô – ü )ï

− −   .  0 = 

ô 2 ï 2 ñ óò ò ó

+ õ ]ñ = 0.

– õÿñ = 0.

0

Particular solution:

139.

ˆ 
Šª
Š + B (B − 1 +  ) Š
+

ï 2 ñ óò ò ó

Particular solution: 138.

2

2

+ ( ôù +

– 2 üIù )ï + ù ( õ – ù – 1)]ñ = 0.

õü

+ ( ô 1 ï 2 + õ 1 ï )ñ ó ò + ( ô 0 ï 2 + õ 0 ï + ö 0 )ñ = 0. The substitution =   ¨ , where , is a root of the quadratic equation  2 , 2 +(  1 −  2 ) , + ( 0 = 0, leads to an equation of the form 2.1.2.108:  2  ¨2

 +(  1  +2  2 , +  1 )¨2
+(  0  +  1 , +  0 )¨ = 0. + [ ô ï 2 + ( ôƒõ – 1) ï + õ ]ñ„ó ò + ô 2 õ  Particular solution: 0 = (  + 1)  −  .

ï 2 ñ óò ò ó

ï ñ

= 0.

– 2 ï ( ï 2 – ô )ñó ò + {2ù ï 2 + [(–1)÷ – 1] ô }ñ = 0. For B = 0, 1, 2,  , particular solutions are polynomials:

ï 2 ñ óò ò ó 0

.

=  ( ), where

.

0(

 ) = 1,

.

1(

 )= ,

.

2(

 ) = 2 2 − 1 − 2  ,

.

3(

 ) = 2 3 − (3 + 2  ) , 

The polynomials contain only even powers of  for even B and only odd powers of  for odd B . 142.

+ ï ( ô ï 2 + õ ï + ö )ñó ò + (
ï 3 + ( ï 2 + = ï + F )ñ = 0. 1 b . The substitution =   ¨ , where , is a root of the quadratic equation , leads to an equation of the form 2.1.2.84 (see also 2.1.2.80–2.1.2.83):

ï 2 ñ óò ò ó

 ¨ 

 + ( 

2

+ ' + ( + 2 , )¨ 
+ [ #*

2

2

+( ( −1) , + c = 0,

+ ( $ + , ) + L + , ] = 0.

2 b . LetÀ ° and & be arbitrary parameters. For # = & , $ = !° + '& − & 2 , L = ˜° + (˜& − 2&° , c = ° ( ( − ° − 1), a particular solution is: − −Ÿ  À .  0 =  For # =  (  − & ), $ =  ( ( − ° + 1) + & (  − & ), L = ˜° + (˜& − 2&° , c = ° ( ( − ° − 1), a particular solution is: 0 =  − exp(− 21  2 − & ). 143.

ï 2 ñ óò ò ó

+

ô ƒï ÷ ñ óò

– ( ôƒõ

Particular solution:

ïƒ÷

+

ô„ö ïƒ÷

v  0 =   

–1

+

õ 2ï

2

+ 2 õ^ö

ï

+

ö

2

– ö )ñ = 0.

.

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230 144.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï 2 ñ óò ò ó

+

ô ƒï ÷ ñ óò

Particular solution: 145.

ï 2 ñ óò ò ó

+ ï (ô

ï ÷

147.

148.

óò

+ õ )ñ

–

õ 2 ï 4ú

=  − + exp O −

0

Particular solution:

146.

ïƒ÷ +2ú

+ ( ôƒõ

ï ÷

+ õ (ô

2



+1

¡

ïƒ÷

+ ôþ

 2+

+1

–1

–þ

2

–þ

)ñ = 0.

.

P

– 1)ñ = 0.

.

−

=

0

+2

+ ï ( ô ïƒ÷ + õ )ñó ò + (  ï 2÷ +  ïƒ÷ + )ñ = 0. The transformation H =   , ¨ = H −  , where , is a root of the quadratic equation B 2 , 2 + B (  − 1) , + = = 0, leads to a linear equation of the form 2.1.2.108: B 2 H ¨2‘'

‘ + [B±!H + 2 ,!B 2 + B (B − 1 +  )]¨3‘
+ ( ƒhH + ,!B± + ý )¨ = 0.

ï 2 ñ óò ò ó

+ ï (2 ô ïƒ÷ + õ )ñó ò + [ ô 2 ï 2÷ + ô ( õ + ù – 1)ïw÷ +  ï 2ú +  ïƒú + ]ñ = 0. The substitution ¨ = exp(  B ) leads to a linear equation of the form 2.1.2.146:  2 ¨2

 + ' ¨2
+ ( ƒ… 2 + + ý± + + = )¨ = 0.

ï 2 ñ óò ò ó ï 2 ñ óò ò ó

+ (ô

ïƒ÷

+2

õï

+

+ ö )ñó ò + ( ôù

2

Particular solution:

0

= exp O −

2.1.2-5. Equations of the form ( 

2



B +1

ïƒ÷

+1

 

+

ô„ö ïƒ÷

+1

− '

P

+

õ^ö )ñ

= 0.

.

+ ' + ( )

 +  ( )
 + ( ) = 0.

149. (1 – ï 2 )ñó ò ò ó + ù (ù – 1)ñ = 0, ù = 0, 1, 2, uuu This equation is encountered in hydrodynamics when describing axially symmetric Stokes flows. 1 b . For B ≥ 2, the solution is given by: =L

1

 ( ) + L 2 ‚ {  ( ),

y

where  ( ) and {‚ ( ) are the Gegenbauer functions which can be expressed in terms of the y functions of the first and second kind (see 2.1.2.153) as follows: Legendre

 ( ) =

y

.



.

 ) −  ( ) , 2B − 1

−2 (

2 b . For B = 0 and B = 1, the solution is:

150. ( ï

2

– ô 2 )ñ

óò ò ó

+

õÿñ óò

– 6ñ = 0.

Particular solution:

0

=L



{‚ ( 1

)=

+L

2

 .

Æ 

 ) − Æ  ( ) . 2B − 1

−2 (



= (4 −  )| +  | 2   | −  | 2   . 2 +

2 −

151. ( ï 2 – 1)ñó ò ò ó + ï ñó ò + ôñ = 0. 1 b . For  = , 2 > 0, the solution is: =

L

cos( , arccosh | |) + L 2 sin( , arccosh | |) if | | > 1, L 1 exp( , arccos  ) + L 2 exp(− , arccos  ) if | | < 1,

™

1

where arccosh  = ln   + £  2 b . For  = − ,

2

2

−1 .

< 0, the solution is: =

™

L

exp( , arccosh | |) + L 2 exp(− , arccosh | |) if | | > 1, L 1 cos( , arccos  ) + L 2 sin( , arccos  ) if | | < 1. 1

3 b . For  = −B 2 , where B is a nonnegative integer, particular solutions are the Chebyshev { polynomials:  ( ) = cos(B arccos  ).

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231

2.1. LINEAR EQUATIONS

ù = 0, 1, 2, uuu 152. (1 – ï 2 )ñó ò ò ó – ï ñó ò + ù 2 ñ = 0, This is a special case of equation 2.1.2.151 with  = −B [ ´J B ½

=

2 +

{ where

2]

(−1) +

¡

=0

. Particular solution:

(−1)  £ 1− 2   12  

{

=  ( ) = cos(B arccos  ) =

0

2

(B − − 1)! (2 )  ¡ ! (B − 2 )!

−2

2

M

 M  V (1 −  2 )  

− 21

W

+ ,

¡

 ( ) is the Chebyshev polynomial of the first kind, (  )  =  (  + 1)  ( + B − 1), and [  ] stands for the integer part of a number  .

153. (1 – ï 2 )ñó ò ò ó – 2 ï ñó ò + ù (ù + 1)ñ = 0, The Legendre equation. The solution is given by: =L

.

ù 1

.

= 0, 1, 2,

uuu

 ( ) + L 2 Æ  ( ),

where the Legendre polynomials  ( ) and the Legendre functions of the second kind Æ  ( ) are given by the formulas:

.

 ( ) =

1 B ! 2

. The functions .

0(

 ) = 1,

.

M

 M  ( 

2

½  1. 1. 1+ Æ  ( ) = (  ) ln − +  2 1− + =1 ¡

− 1)  ,

.

−1 (

.

 )  − + ( ).

 =  ( ) can be conveniently calculated by the recurrence relations: 1(

 )= ,

.

2(

.

1 2

 ) = (3 2 −1),  ,



+1(

 )=

.

2B +1  B +1

B .  ( )− B +1 

−1(

 ).

Three leading functions Æ  = Æ  ( ) are:

Æ

0(

 )=

1 1+ ln , 2 1−

Æ

1(

 )=

. zeros of the polynomial



2

ln

1+ − 1, 1−

Æ

2(

 )=

3

2

3 −1 1+ ln −  . 4 1− 2

All B  ( ) are real and lie on the interval −1 <  < 1; the . functions  ( ) form an orthogonal system on the closed interval −1 ≤  ≤ 1, with the following relations taking place:

X

.

1 −1

 (

. )

a

+ ( )  = M

0

2 2B + 1

if B ≠ if B =

,

¡

¡

.

154. (1 – ï 2 )ñó ò ò ó – 2 ï ñó ò + b ( b + 1)ñ = 0. The Legendre equation; @ is an arbitrary number. The case @ = B where B is a nonnegative integer is considered in 2.1.2.153. The substitution H =  2 leads to the hypergeometric equation. Therefore, with | | < 1 the solution can be written as: =L

1

@ 1+@ 1 , ;  O − , 2

2

2

P

+L

2

"O

1−@ @ 3 , 1+ , ;  , 2 2 2 P

where  ( ƒ , ý , = ;  ) is the hypergeometric series (see 2.1.2.171). The Legendre equation is discussed in the books by Abramowitz & Stegun (1964), Bateman & Erd´elyi (1953, Vol. 1), and Kamke (1977) in more detail (see also Subsection S.2.9).

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232

SECOND-ORDER DIFFERENTIAL EQUATIONS

155. (1 – ï 2 )ñó ò ò ó – 3 ï ñó ò + ù (ù + 2)ñ = 0, Particular solution: 0

= ’  ( ) =

½ ´J

[

=

+

2]

sin[(B + 1) arccos  ]

£ 1−

(B − )! (2 )  ¡ ! (B − 2 )!

(−1) +

¡

=0

2

ù = −2

= 1, 2, 3,

uuu

(−1)  (B + 1) 1 £ 1− 2 +1  1  2  +1

2

M

 M  V (1 −  2 )  

+ 21

W

+ ,

¡

where ’  ( ) is the Chebyshev polynomial of the second kind, (  )  =  (  + 1)  ( + B − 1), and [  ] stands for the integer part of a number  .

156. ( ï

2

– 1)ñ

óò ò ó

+ 2(ù + 1)ï



= M  

Solution: 2.1.2.154. 157. ( ï

2

M

ñ óò

– ( b + ù + 1)( b – ù )ñ = 0,

ù

uuu

= 1, 2, 3,

Á ( ), where Á ( ) is the general solution of the Legendre equation

– 1)ñó ò ò ó – 2(ù – 1)ï

ñ„óò 

– ( b – ù + 1)( b + ù )ñ = 0,

Solution: = | 2 − 1|  M   M equation 2.1.2.154. 158. ( ï 2 – 1)ñó ò ò ó + (2 ô + 1)ï 1 b . Particular solution: 0

uuu

= 1, 2, 3,

Á ( ), where Á ( ) is the general solution of the Legendre

ñ„óò

– õ (2 ô + õ )ñ = 0.

=

(2  +  )   2  +  , −  ,  + 12 ; (  + 1) (2 )

w

ù

w

w

1 2

− 12   ,

(1)

where  ( ƒ , ý , = ; H ) is the hypergeometric function (see equation 2.1.2.171 and Subsection S.2.8). 2 b . For  = B , where B = 0, 1,  , the right-hand side of (1) defines the Gegenbauer polynomials,

½ (2  + B ) (  + , ) (2  + B + , )( − 1)  w w . L  (  ) ( ) = w   2  + B , −B ,  + 12 ; 12 − 12   = (B + 1) (2 ) , ! (B − , )! 2  (  ) (2  + 2 , ) =0  w w w w

159. (1 – ï 2 )ñ ó ò ò ó + (2 ô – 3)ï Particular solution: 2  0 (  ) = (1 −  )

ñ óò

+ (ù + 1)(ù + 2 ô – 1)ñ = 0,

J L (  ) ( ) = (1 −  2 )  

−1 2

ù

= 0, 1, 2,

uuu

 (  + , ) (2  + B + , )( − 1)  J ½ , w w   =0 , ! (B − , )! 2 w (  ) w (2  + 2 , )

−1 2

where L  (  ) ( ) are the Gegenbauer polynomials.

160. (1 – ï 2 )ñó ò ò ó + V  –  – (  +  + 2) ï WNñó ò + ù (ù + Particular solution:



+  + 1)ñ = 0,

ù

= 0, 1, 2,

uuu

 . (−1)  (1 −  )− 9 (1 +  )−: M  (1 −  ) 9 +  (1 +  ): +  S  ) =  9 ,: (  ) =   Q 2 B ! M  ½ + + + + = 2−  L  + L   +−: ( − 1)  − ( + 1) , 9 + =0 . where  9 ,: ( ) are the Jacobi polynomials and L  are binomial coefficients.  0(

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233

2.1. LINEAR EQUATIONS

161. (1 – ï

2

)ñó ò ò ó + V  –  + (  +  – 2) ï WUñó ò + (ù + 1)(ù +  +  )ñ = 0,

Particular solution: 0 ( ) = (1 −  ) 9 (1 +  ):

.

mials (see 2.1.2.160).

ï

162. ( ô

2

+ õ )ñ

óò ò ó

ô ï ñ óò

+

The substitution H = X 163. ( ï

2

2

– ô 2 )ñó ò ò ó + 2 õ =L

Solution: 165. ( ï

2

+

ô 2)ñ óò ò ó

ï

166. ( ô

2

+ õ )ñ

óò ò ó

1



leads to a constant coefficient linear equation: 
‘'
‘ + (

+

2

= |

0

= 0.

+  |1−  .

2

ï ñóò

+ õ ( õ – 1)ñ = 0.

ï ñóò

+ õ ( õ – 1)ñ = 0.

 −  |1−  + L 2 | +  |1−  .

+ 2õ =L

Solution:

1|

uuu

+ 2( õ – 1)ñ = 0.

Particular solution: 164. ( ï

= 0, 1, 2,

= 0.

£ M

ï ñóò

+ ô )ñó ò ò ó + 2 õ

ö^ñ

+

ù

. ,: ,:  9 ( ), where  9 ( ) are the Jacobi polyno-



2

+  2

1− 2



sin Y + L

ï ñ óò

+ (2ù + 1)ô

+

ö^ñ

2



2

1− 2

+  2

ù

= 0,



cos Y , where Y = (1 −  ) arctan(   ).

uuu

= 1, 2, 3,

This equation can be obtained by B -fold differentiation of an equation of the form 2.1.2.162: (  2 +  )‡

 + ‡
 + ( ( − B 2 )‡ = 0. = ‡ (  ) . Solution: 167. (1 – ï

2

–ï

óò ò ó

)ñ

ñ óò

ï

+ (2 ô

2

+ õ )ñ = 0.

This is an algebraic form of the Mathieu equation. The substitution  = cos H leads to the ‘'‘ Mathieu equation 2.1.6.29: 

+ (  +  +  cos 2 H ) = 0. 168. (1 – ï

2

óò ò ó

)ñ

+ (ô

ï

óò

+ õ )ñ

ö^ñ

+

= 0.

1 b . The substitution 2 H = 1 +  leads to the hypergeometric equation 2.1.2.171: H (1 − H )/
‘'
‘ + ‘ [ !H + 12 (  −  )]
+ ( = 0. 2 b . For  = −2

¡

− 3,  = 0, and ( = Œ , the Gegenbauer functions are solutions of the equation.

3 b . In the special case  = − ƒ − ý − 2,  = ý − ƒ , and ( = B (B + ƒ + ý + 1), solutions of the equation are the Jacobi polynomials:

½ 9 : ( ) = 2−  L  + + L   +−: + ( − 1)  − + ( + 1) + , 9 + =0

.

( , )

where L  are binomial coefficients (see Paragraph S.2.1-2). 169. ( ô

ï



2

+ õ )ñó ò ò ó + ( ö

ï

Particular solution: 170. ( ô

ï

2

+ õ )ñ

óò ò ó

2

+ ý )ñó ò + [( ö – ô ) ï 0

0

+

ý

– õ ]ñ = 0.

=  .

+ [ (ö + ô )ï

Particular solution:



−

2

2

+ ( ö – ô ) ï + 2 õ ]ñ −



óò

+

2

(ö

ï

2

+ õ )ñ = 0.

= ( Π + 1)   .

© 2003 by Chapman & Hall/CRC

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234 171.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï (ï

– 1)ñ ó ò ò ó + [(  +  + 1) ï – ]ñ ó ò + ñ = 0. The Gaussian hypergeometric equation. For = ≠ 0, −1, −2, −3,  , a solution can be expressed in terms of the hypergeometric series:

½ ( ƒ )  (ý )    , ( ƒ )  = ƒ ( ƒ + 1)  (ƒ + , − 1),  (ƒ , ý , = ;  ) = 1 + ÷ , !  =1 (= )  which, a fortiori, is convergent for | | < 1. For = > ý > 0, this solution can be expressed in terms of a definite integral:  (ƒ , ý , = ;  ) =

(= ) X ( ý ) (= − ý )

w

w

where (ý ) is the gamma function.

w

w

1 0

a:

−1

(1 − a );

:

− −1

(1 − aÖ )− 9

a, M

If = is not an integer, the general solution of the hypergeometric equation has the form: =L

1

 (ƒ , ý , = ;  ) + L 2 

1−

;  ( ƒ − = + 1, ý − = + 1, 2 − = ;  ).

In the degenerate cases = = 0, −1, −2, −3,  , a particular solution of the hypergeometric equation corresponds to L 1 = 0 and L 2 = 1. If = is a positive integer, another particular solution corresponds to L 1 = 1 and L 2 = 0. In both these cases, the general solution can be constructed by means of the formula given in 2.1.1. Table 16 presents some special cases where  is expressed in terms of elementary functions. Table 17 gives the general solutions of the hypergeometric equation for some values of the determining parameters. The function  possesses the following properties:

 ( ƒ , ý , = ;  ) =  (ý , ƒ , = ;  ),  ( ƒ , ý , = ;  ) = (1 −  ) ; − 9 −:  (= − ƒ , = − ý , = ;  ),  ),  ( ƒ , ý , = ;  ) = (1 −  )− 9  ( ƒ , = − ý , = ;  −1  ( ƒ )  (ý )   ( ƒ + B , ý + B , = + B ;  ). M   (ƒ , ý , = ;  ) =  (= )  M

The hypergeometric functions are discussed in the books by Abramowitz & Stegun (1964) and Bateman & Erd´elyi (1953, Vol. 1) in more detail; see also Subsection S.2.8.

172.

+ ô )ñ„ó ò ò ó + ( õ ï + ö )ñó ò + ýIñ = 0. The substitution  = − !H leads to the hypergeometric equation 2.1.2.171: H (1 − H )
‘'
‘ + = 0. [( (  ) − ˜H ]
‘ −

ï (ï

M

173. 2 ï (ï – 1)ñ„ó ò ò ó + (2 ï – 1)ñ„ó ò + ( ô ï + õ )ñ = 0. The substitution  = cos2 ˆ leads to the Mathieu equation 2.1.6.29: 
Šª
Š −(  +2  +  cos 2ˆ ) = 0. 174. ( ï

2

ï

+ 2ô

Solution:

+ õ )ñó ò ò ó + ( ï + ô )ñó ò – þ = L 1  + + £ 

2

ñ

= 0.

+ 2  +  

2

+

+L

2

 + +£ 

2

+ 2  +  

−

+

.

175. ( ô ï 2 + õ ï + ö )ñó ò ò ó + ( ý ï + ü )ñó ò + ( ý – 2 ô )ñ = 0. Integrating yields a first-order linear equation: (  2 + ' + ( )
 + [( − 2  ) + , −  ] = L . 176.

(ô ï

2

+õ ï

+ ö )ñ ó ò ò ó + ( ü ï

Particular solution:

0

+ ý )ñ óò – = ,! + .

üIñ

M

= 0.

M

© 2003 by Chapman & Hall/CRC

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235

2.1. LINEAR EQUATIONS

TABLE 16 Some special cases in which the hypergeometric function  ( ƒ , ý , = ; H ) is expressed in terms of elementary functions

=

ƒ

H

ý =

−B

ý

−B



−B −

ý

 ½ 

 ¡

ý ƒ

ƒ + 12

1 2



2

ƒ

ƒ + 12

3 2



2

ƒ

−ƒ

1 2

−

2

ƒ

1− ƒ

1 2

−

2

ƒ

ƒ − 12

2ƒ − 1



ƒ

−ƒ

1 2

sin 

ƒ

1− ƒ

3 2

sin2 

ƒ

2− ƒ

3 2

sin2 

ƒ

1− ƒ

1 2

sin2 

ƒ

ƒ + 12

1 2

− tan2 

ƒ

ƒ +1 ƒ

ƒ + 12 ƒ

ƒ + 12 1 2

=0



=0

(−B )  (ý )    , (−B − )  , !

½ 

ƒ

1 2



(−B )  (ý )    , (= )  , !

¡

 ý



2ƒ + 1



where B = 1, 2, 3, 

(1 −  )− 9

V (1 +  )−2 9 + (1 −  )−2 9 W

(1 +  )1−2 9 − (1 −  )1−2 9 2 (1 − 2 ƒ )

V  £ 1 +  2 +   29 +  £ 1 +  2 −   29 W 2 −1 2 −1  £ 1+ 2 +  9 +  £ 1+ 2 −  9 2£ 1 +  2

1 2

22 9

ƒ

1 2

1 2

where B = 1, 2, 3, 

 1+ £ 1− 

−2

2−2

cos(2 ƒ… ) sin[(2ƒ − 1) ( ƒ − 1) sin(2 sin[(2ƒ − 2) ( ƒ − 1) sin(2 cos[(2 ƒ − 1) cos 

] ) ] ) ]

cos2 9  cos(2 ƒ… ) (1 +  )(1 −  )− 9

−1

−2  12 + 12 £ 1 −   9 −1 J 2 1  1−   2 + 12 £ 1 −  



2ƒ

1 2

3 2



1

3 2

−

2

1

1

2

−

1 2

1

3 2



1 2

1 2

3 2

−

B +1

B + +1 ¡

B + + , +2 ¡





2

 

−1

arcsin 

−1

arctan 

−1

ln( + 1)

1−2

9

1+ 1 ln 2 1−

2 2

9



−1

ln   + £ 1 + 

2



  (−1) + (B + + , + 1)!  + + ¡ M  + + ™ (1 −  ) + +  M   B ! , ! (B + )! ( + , )!  ¡ −1 ¡ M M  = − ln(1 −  ), B , , , = 0, 1, 2, 3, 

¢

,

¡

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236

SECOND-ORDER DIFFERENTIAL EQUATIONS

TABLE 17 General solutions of the hypergeometric equation for some values of the determining parameters

ý

= ý

=

ƒ

ƒ + 12

2ƒ + 1

ƒ

ƒ − 12

1 2

L

ƒ

ƒ + 12

3 2

£  QL

ƒ 0

L

ƒ ý

ƒ +1

ï

+ 2õ

The substitution ˆ = X form 2.1.2.1: Šª

Š +

ï

178. ( ô

2

+ 2õ

ï

+ ö )ñó ò ò ó + ( ô

ï

£ 

2

= 0.

M

ï

+ ö )ñó ò ò ó + 3( ô

M



ýIñ

−2

 1+ £  

1

1

9 + L 2

1−2

 1+ £   −1

| |− 9 OL

+ õ )ñó ò +

X | |−; | − 1|; −:

2

 1+ £ 1− 

| − 1|−: OL

ƒ

2

L

| |1−; | − 1|; −:

ý

ï

1

1+

1

ƒ

177. ( ô

L

= ý

1

= ( )

Solution:

OL

1+ 1+

L

9 +L

1−2

1+

L

2 2

L

−2

2

2

1−2

 1− £  

X | |;

−1



9  1 + £ 1 −   29

−2

9

1−2

| − 1|: −;

X | |− 9 | − 1|:

X | | 9

M

 1− £  

9 +L 2

−1

| − 1|−:

−1

9 S  M P

 M P

 M P

= 0. leads to a constant coefficient linear equation of the

+ 2 ' + (

+ õ )ñó ò + ýIñ = 0.

|  2 + 2 ' + ( | leads to a linear equation of the form 2.1.2.177: The substitution ¨ = ü (  2 + 2 ' + ( )¨2

 + (  +  )¨2
+ ( −  )¨ = 0. M

179. ( ô

2

ï

Let Œ

õ 2ï

+ ö 2 )ñó ò ò ó + ( õ

2

+

1

and Œ

1 b . For Œ

1

2

1

ï

+

ö 1 )ñ óò

+ö

0

ñ

= 0.

are roots of the quadratic equation  2 Œ

≠ Œ 2 , the substitution H =

H (1 − H )
‘'
‘ − ( #* + $ )
‘ − L

Œ

 −Œ 2

−Œ

1 1

2

+  2Π+ (

2

= 0.

leads to the hypergeometric equation 2.1.2.171:

= 0, where # =





1 2

, $

=

 1Œ 1 + ( 1 (0 , L = .  2(Œ 2 − Œ 1)  2 = ˆ  ‡ , where , is a root of the

2 b . For Œ 1 = Œ 2 = Œ , the transformation  = Œ + ˆ −1 , quadratic equation  2 , 2 + (  2 −  1 ) , + ( 0 = 0, leads to a linear equation of the form 2.1.2.108:  2 ˆ‡ Šª

Š − [( ( 1 + Œ" 1 )ˆ +  1 − 2  2 ( , + 1)]‡ Š
− , ( ( 1 + Œ" 1 )‡ = 0. 3 b . Let ( 0 = −  2 B (B − 1) −  1 B , where B is a positive integer. Then, among solutions there exists a polynomial of degree ≤ B .

ï

180. ( ô

2

+

õï

+ ö )ñó ò ò ó – ( ï

Particular solution: 181. ( ô

ï

2

+

õï

0

+ ö )ñó ò ò ó + ( ï

Particular solution:

0

2

–

ü 2)ñ óò

+ ( ï + ü )ñ = 0.

= −, . 3

+

ü 3 )ñ óò

– (ï

2

–

ü ï

+

ü 2 )ñ

= 0.

= +, .

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237

2.1. LINEAR EQUATIONS

2.1.2-6. Equations of the form (  3  182. 183.

184. 185. 186.

188.

+ ( ô ï 2 + õ )ñó ò + ö ï ñ = 0. ‘'‘ ‘ The substitution  = 1 H leads to an equation of the form 2.1.2.139: H 2 

+ H (2 −  − ˜H )
+ ( = 0.

ï 3 ñ óò ò ó

+ ( ô ï 2 + õ ï )ñ Particular solution:

ï 3 ñ óò ò ó

óò

+ õÿñ = 0. =  −2+  .

0

+ ( ô ï 2 + õ ï )ñó ò + ö^ñ = 0. The substitution  = 1 H leads to an equation of the form 2.1.2.108: H
‘'
‘ +(2−  − ˜H ) ‘
+ (

ï 3 ñ óò ò ó

= 0.

+ ( ô ï 2 + õ ï )ñó ò + ( ö ï + ý )ñ = 0. 1 b . The substitution =   ‡ , where , = −  , leads to a linear equation of the form M 2.1.2.134:  2 ‡

 + [(  + 2 , ) +  ]‡
 + [ , (  + , − 1) + ( ]‡ = 0.

ï 3 ñ óò ò ó

=  (  − 2), a particular solution is:

M

+ ( ô ï 3 – ï 2 + ôƒõ ï + õ )ñ ó ò + ô 2 õ  Particular solution: 0 = (  + 1)  −  .

ï 3 ñ óò ò ó

+ ï ( ô ïƒ÷ + õ )ñó ò – ( ô ïƒ÷ – ôƒõ Particular solution: 0 =  exp(   ).

ï 3 ñ óò ò ó ï (ô ï

+ õ )ñó ò ò ó + 2( ô Particular solution:

190.

ï (ï

ï

2

+ õ )ñó ò – 2 ô =  +   .

2 0

2 + ô )ñó ò ò ó + ( õ ï

ï ñ

ïƒ÷

–1

ï ñ

0

 = J .

= 0. + õ )ñ = 0.

= 0.

+ ö )ñó ò + s ï ñ = 0. ‘'‘ The substitution !H = − 2 leads to the hypergeometric equation 2.1.2.171: H (1 − H )"

+ ‘ 
1 V 1 −1 − (1 +  ) HžW − 4 ° = 0. 2 1 + (

ï 2(ô ï

191.

+ õ )ñó ò ò ó + [ ö Particular solution:

192.

ï 2(ô ï ï 2(ô ï

2

ï

+ õ )ñó ò ò ó – 2 ï ( ô

Solution:

L 1

=

+ (2 õ + ô ) ï +

 ). 0 = exp( Œ

2

ï

+ 2 õ )ñ„ó ò + 2( ô

+ L 2 .  +  2

=







ï 2(ï

õ ]„ñ óò ï

)ï

2

– õ (ù + þ

L 1 | |  + L 2 | | + +1  +  | |  ( L 1 + L 2 ln | |)  + 

) ï ]ñó ò + [ ôþ

(ù – 1)ï +

õÿù (þ

+ 1)]ñ = 0.

≠ B − 1,

if if

+ ( ö – 2 ô )ñ = 0.

+ 3 õ )ñ = 0.

3

+ õ )ñó ò ò ó + [ ô (2 – ù – þ

Solution:

194.

+  1  +  0 )

 +  ( )
 + ( ) = 0.

2

+ ( ô ï + õ )ñ = 0. This is a special case of equation 2.1.2.132 with B = −1.

189.

193.

+  2

ï 3 ñ óò ò ó

2 b . If ( = 0 and 187.

3

¡

¡

= B − 1.

+ ô 2 )ñó ò ò ó + ï ( õ 1 ï + ô 1 )ñó ò + ( õ 0 ï + ô 0 )ñ = 0. The substitution =   ‡ , where , is a root of the quadratic equation  2 , 2 + , (  1 −  2 )+  0 = 0, leads to a linear equation of the form 2.1.2.172:  ( +  2 )‡

 + [(2 , +  1 ) + 2 , 2 +  1 ]‡
 + [ , 2 + , (  1 − 1) +  0 ]‡ = 0.

© 2003 by Chapman & Hall/CRC

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SECOND-ORDER DIFFERENTIAL EQUATIONS

ï

195. ( ô

3

+

õï

+ö

2

ï )ñ óò ò ó

Particular solution:

ï

196. ( ô

3

+

õï

+ö

2

ï )ñ óò ò ó

Particular solution:

ï

197. ( ô

3

+

õï

+ö

2

ï )ñ ó ò ò ó

Particular solution:

ï

198. ( ô

3

+

õï

+ö

2

ï )ñ óò ò ó

Particular solution: 199. ( ô

ï

3

+

õï

+ö

2

ï )ñ óò ò ó

Particular solution:

ï

+ ( 0

ï

+

ï

2

=

ï

2

1−

 .

= ( +

200. ( ô ï 3 + õ ï 2 + ö ï )ñó ò ò ó + (ù With the constraint

ï

+þ

ï

2

2

¡

)

2

ï

+þ

a particular solution has the form:

2

2

+

õï

The substitution ˆ = X O 2
Šª
Š + Œ

203.

204.

ï (ô ï

ï

+ ö )ñó ò ò ó + ( ô

= 0.



+ õ ï + 1)ñ ó ò ò ó + (  The substitution =  1−; ¨ 2

2

2

ï



óò

+ 2( ô

ï

+ 1)ñ = 0.

+ ( ( + , )(2 − , ).

+ ü )ñó ò + ( ü – 1)[(ù – ôƒü ) ï + þ

+ (2 ö^þ

+ (2  + 4 (

– 1)ï – ö ]ñ„ó ò + (–2þ

–õ

ï

= 0.

+ 1)ñ = 0.

− 1)( + ( ).

¡

ü ]ñ

+ ü )ñó ò + [–2( ô + ù ) ï + 1]ñ = 0.

0

+ 1)[

¡

= (2  + B )

201. ( ô ï 3 + ï 2 + õ )ñó ò ò ó + ô 2 ï ( ï 2 – õ )ñó ò – ô  Particular solution: 0 = (  + 2)  −  .

ï

ï

+ 2 ö )ñó ò – ( 

2(2  + B )(( + , ) + (2  + 2

202. 2 ï (ô

.

– ( õ + 1) ï + ü ]ñ

– ô )ï

+ [(þ 0

ï

= ( , +  − 1)

+ (ù 0

−1

+ 2 õ –  )ñ = 0. Œ (ƒ + (  − ý )(2  − ý ) =… ƒ  + 2(ý −  ) + , where Œ = .  ƒ − 2

+ [–2 ô 0

+ 2 ö )ñó ò + ( – 2 õ )ñ = 0.

= 2  − ƒ + (2  − ý )

+ ( 0

ï

+

2

– ö )ñó ò +

J

3

1 2

+ ' + ( P

2

¡

+ 1 + 2 , (  + B )] = 0, + (2  + 2

õï ñ

= 0.

ï 2ñ

= 0.

M

¡

+ 1)( − , ).

 leads to a constant coefficient linear equation:

+  ï + )ñ ó ò + (ù ï + þ )ñ = 0. leads to an equation of the same form:

2

+ ' + 1)¨ 

 + [( ƒ + 2  − 2 ´= ) 2 + (ý + 2  − 2 = ) + 2 − = ]¨ 
+ , [B + (1 − = )( ƒ − ´= )] + + (1 − = )(ý − = ) . ¨ = 0.

 ( 

2

ï (ï

– 1)( ï – ô )ñ ó ò ò ó + , ( 

¡

+  + 1) ï

2

– [  +  + 1 + ô ( + S ) – S ] ï + / ô . ñ óò

+ ( 

ï – )ñ

= 0.

Heun’s equation. 1 b . For |  | ≥ 1 and = ≠ 0, −1, −2, −3,  , a solution can be represented as the power series:

½  ( ,  ; ƒ , ý , = , > ,  ) = ÷ (    ,  =0 where the coefficients are determined by the recurrence formulas:

( 0 = 1,

´=/( 1 =  ,

 (B + 1)(= + B )( 

+1

=  (= + > + B − 1) + ƒ + ý − > + B +

Q

B



SB±( 

− V (B − 1)(B − 2) + (B − 1)( ƒ + ý + 1) + ƒ…ý W ( 

−1 .

A fortiori, the series is convergent for | | ≤ 1.

© 2003 by Chapman & Hall/CRC

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239

2.1. LINEAR EQUATIONS

TABLE 18 Some transformations preserving the form of Heun’s equation No 1* 2 3 4 5 6 7 8 9 10 11 12

Parameters of transformed equation for ¨ = ¨ (ˆ )

New variables

  ƒ ˆ = , ¨ = ý ƒ ˆ =1− , ¨ = ƒ…ý −  ý 1−  1 ƒ −= +1 ý −=  ˆ =  , ¨ = | |; −1   2 ƒ −= +1 ý −= 1 ˆ = , ¨ = | |; −1    1 ¨ 1 3 ƒ ƒ −= ˆ = , = | | 9    1  ˆ = ,¨ = ƒ ý    1  ˆ =1− , ¨ = ƒ ý 1−     ˆ = , ¨ = | | 9  ƒ −= +1 ƒ +=  1  −1 ¨  ƒ ƒ −= ˆ = , = | | 9 1−     ( − 1) ¨  ƒ , = | | 9 ˆ = ƒ −=  −1  (  − 1)    , ¨ = | − 1| 9 ˆ = ƒ ƒ −>  −1  −1  (  − 1) ¨ 1  ƒ , = | − 1| 9 ˆ = 1− ƒ −>  ( − 1)  Notation:  1 =  + ( ƒ − = + 1)(ý − = + 1) − ƒ…ý + > (= − 1),  3 =  −1 + ƒ (ƒ − = + 1) + ƒh −1 (> − ý ) − !>

= > = > +1

2−=

>

+1

2−=

ƒ +ý −= −> +1

+1 ƒ −ý +1

>

=

ƒ +ý −= −> +1

ƒ +ý −= −> +1

=

+1

>

ƒ −ý +1

+1

>

ƒ +ý −= −> +1

−1 ƒ −ý +1 ƒ +ý −= −> +1

+1

=

ƒ −ý +1

+1

=

ƒ +ý −= −> +1

 2 =  1 + !> (1 − = .

* This row corresponds to the original equation, while the others refer to the transformed equation for



=

),

 (~ ).

2 b . If = is not an integer, the general solution of Heun’s equation can be presented as follows: =L

1

 (  ,  ; ƒ , ý , = , > ,  ) + L 2 | |1−;  (  ,  1 ; ƒ − = + 1, ý − = + 1, 2 − = , > ,  ),

where  1 =  + ( ƒ − = + 1)(ý − = + 1) − ƒ…ý + > (= − 1). Table 18 lists some transformations preserving the form of Heun’s equation. (Whenever at least one of the indicated equations is integrable by quadrature with some values of parameters, all the other equations are also integrable for those values of the parameters.)

^_

References: H. Bateman and A. Erd´elyi (1955, Vol. 3), E. Kamke (1977), S. Yu. Slavyanov, W. Lay, and A. Seeger (1955), A. Ronveaux (1995).

205. ( ô ï 3 + õ ï 2 + ö ï + ý )ñ ó ò ò ó – ( ï 2 – Particular solution: 0 =  − Œ . 206. 2( ô

ï

3

+

õï

2

+

öï

+ ý )ñó ò ò ó + (3 ô

The substitution ˆ = X 2
Šª
Š + Œ

= 0.

ï

£ 

3

2

M

+ '

2

 2

)ñ

óò

+ 2õ

+ ( ï + )ñ = 0.

ï

+ (˜ +

+ ö )ñó ò +

Ė

= 0.

leads to a constant coefficient linear equation:

M

© 2003 by Chapman & Hall/CRC

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SECOND-ORDER DIFFERENTIAL EQUATIONS

207. 2(ô ï 3 + õ ï 2 + ö ï + ý )ñó ò ò ó + 3(3 ô ï 2 + 2 õ ï + ö )ñó ò + (6 ô ï + 2 õ + )ñ = 0. This equation is obtained by differentiating the equation 2.1.2.206.

208. ( ô ï 3 + õ ï 2 + ö ï + ý )ñó ò ò ó + [  ï Particular solution: 0 =  + = .

2

+ ( | +  ) ï + C ]ñ„ó ò – ( 

209. ( ô ï 3 + õ ï 2 + ö ï + ý )ñó ò ò ó + ( ï 3 + Particular solution: 0 =  + Œ .

210. 2 ï (ô

ï

2

+

õï

+ [ ô (2 – ü )ï

óò ò ó

+ ö )ñ

3

2

)ñó ò – ( ï

2

–

ï

2

+

+ õ (1 – ü ) ï – ö³ü ]ñ

óò

ï

+  )ñ = 0.

)ñ = 0. +

ï  +1 ñ

= 0.

The substitution ˆ = X  J 2 (  2 + ' + ( )−1 J 2  leads to a constant coefficient linear equation: M 2
Šª
Š + Œ = 0.

2.1.2-7. Equations of the form (  4  211.

212.

ï 4 ñ óò ò ó

– (ô + õ )ï =

™

ï 4 ñ óò ò ó

2

ñóò

+  1  +  0 )

 +  ( )
 + ( ) = 0.

2

leads to a constant coefficient linear equation of the

+ ( ô ï 2 + õ ï + ö )ñ = 0. The transformation H = 1  , ‡ =   H 2 ‡ ‘'

‘ + ( (H 2 + ˜H +  )‡ = 0.

leads to a linear equation of the form 2.1.2.115:

+ [( ô + õ ) ï + ôƒõ ]ñ = 0.

  L 1   − J + L 2   − J  ( L 1  + L 2 ) −  J

if  ≠  , if  =  .

+ 2 ï 2 ( ï + ô )ñó ò + õÿñ = 0. The substitution H = 1  leads to a constant coefficient linear equation:
‘'
‘ − 2  
‘ + 

ï 4 ñ óò ò ó

+

ô ƒï ÷ ñ óò

– (ô

Particular solution:

218.

+  2

ï 4 ñ óò ò ó

215.

217.

3

+ ôñ = 0. The transformation H = 1  , ‡ =   form 2.1.2.1: ‡ ‘'

‘ + ‡ = 0.

Solution:

216.

+  3

ï 4 ñ óò ò ó

213.

214.

4

ï 2(ï

ïƒ÷

–1

+

ôƒõ ïƒ÷

− J  . 0 =   

– ô )2 ñ ó ò ò ó + õÿñ = 0. Solution: = L 1 | | + | −  |1− + + L ( − 1)  2 = −  .

¡

¡

ï 2(ï

– ô )2 ñ Solution:

óò ò ó

+

õÿñ

=

ö ï 2 (ï

2|

–2

+

õ 2 )ñ

= 0.

= 0.

 |1− + | −  | + , where ¡

is a root of the quadratic equation

– ô )2 .

= | | + | −  |1− + OL

(

X | |1− + | −  | +   (2 − 1) M P ¡ ( + + 1− + 1− + + | | | −  | OL 2 −  , X | | | −  |  (2 − 1) M P ¡ where is a root of the quadratic equation ( − 1)  2 = −  . ¡ ¡ ¡ 1

+

ô ï 2 (ï – 1)2 ñóò ò ó + ( õ ï 2 + ö ï + ý )ñ = 0. Let  and  be roots of the quadratic equations D ( − 1) + = 0, ³ ( M

− 1) +  + ( + = 0. M ¨ The substitution =  K ( − 1) & leads to the hypergeometric equation of the form 2.1.2.171:  ( − 1)¨2

 + 2  [( +  ) −  ]¨3
+ (2 D¿ − ( − 2 )¨ = 0. M © 2003 by Chapman & Hall/CRC

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241

2.1. LINEAR EQUATIONS

219.

220.

ï 2(ï

2

ï

+ ô )ñó ò ò ó + ( õ

+ ö )ï

2

ñóò

ýIñ

+

= 0.

The substitution ˆ =  leads to a linear equation of the form 2.1.2.194: 4ˆ 2 (ˆ +  )
Šª
Š + = 0. 2ˆ [(  + 1)ˆ +  + ( ] Š
+ (ï

2

+ 1)

2

ñóò ò ó

2

M

ôñ

+

= 0.

The Halm equation. Solution: = 221. ( ï

2

+1 VL 2+1 VL £  2 + 1 (L

£  £ 



– 1)2 ñó ò ò ó +

Solution: =







cos(ý arctan  ) + L 2 sin(ý arctan  )W 1 cosh( ý arctan  ) + L 2 sinh( ý arctan  )W 1 + L 2 arctan  )

2

ôñ

if  + 1 = ý 2 > 0, if  + 1 = −ý 2 < 0, if  = −1.

1

= 0.

| 2 − 1| V L 1 cos ý ln | H |  + L 2 sin ý ln | H |  W ( + 1) L 1 | H |(2: −1) J 2 + L 2 | H |−(2: +1) J 2  | 2 − 1|  L 1 + L 2 ln | H | 

ü

ü

if  − 1 = 4ý 2 > 0, if  − 1 = −4ý 2 < 0, if  = 1,

where H = ( + 1) ( − 1). 222. ( ï

2

Aô 2 )2 ñó ò ò ó + õ

ñ

2

= 0.

This is the equation of bending of a double-walled compressed bar with a parabolic crosssection. 1 b . For the upper sign (constricted bar), the solution is as follows: =£ 

2

+

2

(L

1

cos ‡ + L

2

sin ‡ ),

where ‡ =

2

ü 1 + (   ) arctan(  ).

2 b . For the lower sign (bar with salients), the solution is given by: =£  223. 4( ï

2

+ 1)2 ñ

óò ò ó

2

−

2

(L

ï

2

+ (ô

cos ‡ + L

1

™

=

224. ( ô

ï

2

+ õ )2 ñ

1 2

+ 2ô

ï (ô ï

The substitution ˆ = X 225. ( ï

2

( (

2 2

£ |  − 1| ln   +

óò ò ó

– 1)2 ñó ò ò ó + 2 ï ( ï



2

sin ‡ ),

where ‡ =

£  2− 2

2

ln

 + ; | | <  .  −

+ ô – 3)ñ = 0.

Solution:

where ˆ =

2

2

M2



+ 1)1 J 4 ( L + 1)1 J 4 ( L |

2

+ õ )ñ

óò

ü

1 1

cos ˆ + L 2 sin ˆ ) if  > 1, cosh ˆ + L 2 sinh ˆ ) if  < 1,

+ 1|  . + ö^ñ = 0.

leads to a constant coefficient linear equation: 
Šª
Š + (

+

– 1)ñó ò – [ b ( b + 1)( ï

2

– 1) + ù

2

= 0.

]ñ = 0.

Here, @ is an arbitrary number and B is a nonnegative integer. This is a special case of equation 2.1.2.226. If B = 0, this equation coincides with the Legendre equation 2.1.2.154. Denote its general solution by Á ( ). If B = 1, 2, 3,  , the general solution of the original equation is given by the formula:

= |

2

− 1| ´J

2

M

 M  

Á ( ).

© 2003 by Chapman & Hall/CRC

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SECOND-ORDER DIFFERENTIAL EQUATIONS

226. (1 – ï 2 )2 ñó ò ò ó – 2 ï (1 – ï 2 )ñó ò + [ b ( b + 1)(1 – ï 2 ) – 2 ]ñ = 0. The Legendre equation, @ and “ are arbitrary parameters. The transformation  = 1 − 2ˆ , = | 2 − 1| žJ 2 ¨ leads to the hypergeometric equation 2.1.2.171: ˆ (ˆ − 1)¨ Šª

Š + (“ + 1)(1 − 2ˆ )¨ Š
+ ( @ − “ )( @ + “ + 1)¨ = 0 with parameters ƒ = “ − @ , ý = “ + @ + 1, = = “ + 1. In particular, the original equation is integrable by quadrature if @ = “ or @ = −“ − 1. In Subsection S.2.9, the Legendre equation are discussed in more detail. See also the books by Abramowitz & Stegun (1964) and Bateman & Erd´elyi (1953, Vol. 1). 227.

– 1)2 ñó ò ò ó + õ ï ( ï 2 – 1)ñó ò + ( ö ï 2 + ý ï +  )ñ = 0. The transformation ˆ = 12 ( + 1), ¨ = | + 1|−K | − 1|− & , where  and  are parameters that are determined by solving the second-order algebraic system

ô (ï

2

4 ³ (  − 1) + 2 U + ( + +  = 0,

( −  )[2 ( +  − 1) +  ] = ,

M

leads to the hypergeometric equation 2.1.2.171 with respect to ¨ = ¨ (ˆ ).

ï

228. ( ô

2

ï

+ õ )2 ñó ò ò ó + (2 ô

The substitution ˆ = X



ï

2

+ õ )2 ñó ò ò ó + ( ô

ï

2

Particular solution: 230. ( ï

2

+ ô )2 ñ

óò ò ó

+

Particular solution:

231. ( ï

2

+ ô )2 ñó ò ò ó +

0

õ ï ÷ (ï õ ïƒ÷ (ï

Particular solution:

M2

+ õ )ñó ò +

2



= 0.

+ ý )ñó ò + 2( õ^ö – ô„ý ) ï (˜ 2 + = exp O − X M  .  2 + 3M P

+ õ )( ö

2

2

2

óò

+ ô )ñ

£  0 = 0

ï

üIñ

leads to a constant coefficient linear equation of the form

+ = 0.

2.1.2.11: Šª

Š + ( 
Š + ,

229. ( ô

ï

+ ö )(ô

2

– (õ

= (

ï ÷

+ .

+ ô )ñó ò – þ 2

M

[õ

+ +  ) J 2.

+1

+ ô )ñ = 0.

ïƒ÷

+1

+ (þ

– 1) ï

ñ

= 0.

2

+ ô ]ñ = 0.

ô ≠õ. 232. ( ï – ô )2 ( ï – õ )2 ñó ò ò ó – ö^ñ = 0,  − , = ( −  )‰ leads to a constant coefficient linear equation: The transformation ˆ = ln û  −  ûû û


û the solution is as follows: (  −  )2 (‰ ŠªŠ − ‰ Š ) − (˜‰ = 0.û Therefore, =L where Œ

2

1|

 −  |(1+  ) J 2 | −  |(1−  ) J 2 + L 2 | −  |(1−  ) J 2 | −  |(1+  ) J 2 ,

= 4 ( (  −  )−2 + 1 ≠ 0.

233. ( ï – ô )2 ( ï – õ )2 ñó ò ò ó + ( ï – ô )( ï – õ )(2 ï + )ñ„ó ò + “ñ = 0. Let , 1 and , 2 be roots of the quadratic equation (  −  )2 , 2 + (  −  )(  +  + Œ ) , + “ = 0. Solution: L |H |  1 + L 2 |H |  2 if , 1 ≠ , 2 , = ™ 1 | H |  L 1 + L 2 ln | H |  if , 1 = , 2 = , , where H = ( −  ) ( −  ).

234. ( ô

ï

2

+

õï

+ ö )2 ñó ò ò ó +


ñ

= 0.



leads to a constant coefficient , ¨ = + ' + ( |  2 + ' + ( | ü  ( − 1  2 )¨ = 0. linear equation of the form 2.1.2.1: ¨uŠª

Š + ( # + ! 4 The transformation ˆ = X



2

M

© 2003 by Chapman & Hall/CRC

Page 242

243

2.1. LINEAR EQUATIONS

235. ( ï

2

– 1)2 ñó ò ò ó + 2 ï ( ï

– 1)ñó ò + [( ï

2

– 1)( ô

2

ï

2 2

– )–þ

2

]ñ = 0.

= 0, 1,  It arises when separating Equation for prolate spheroidal wave functions, ¡ variables in the wave equation written in the system of prolate spheroidal coordinates. 1 b . In applications, one usually looks for eigenvalues Œ = Œ +  and eigenfunctions = +  ( ) that assume finite values at  = A 1. The following functions are solutions of the eigenvalue problem:

8

½ (1) +  ( ,  ) = ÷

8

½ (2) +  ( ,  ) = ÷

Ÿ

Ÿ =−

.

+  ( ). + Ÿ +

M =0,1

+

+  ( ) Æ + Ÿ + M

Ÿ ( +

)

Ÿ (

(prolate angular functions of the first kind), ) (prolate angular functions of the second kind),

÷

where  + ( ) and Æ + ( ) are the associated Legendre Ù . functions of the first and second kind. . For −1 ≤  ≤ 1, we have  + ( ) = (1 −  2 ) + J 2 x x  Ù  ( ). The summation is performed over either even or odd values of & , depending on whether |B − | is even or odd, respectively.

¡

2 b . The following recurrence relations for the coefficients  = +  (  ) hold: M M

ƒ   M where

3 b . For eD

+2

+ (ý  − Œ +  )  + = 

M 

M

−2

= 0,

 2 (2 + , + 1)(2 + , + 2) , ƒ  = ¡ ¡ (2 + 2 , + 3)(2 + 2 , + 5) ¡ ¡ 2( + , )( + , + 1) − 2 2 − 1 , ý  = ( + , )( + , + 1) +  2 ¡ ¡ ¡ ¡ ¡ (2 + 2 , − 1)(2 + 2 , + 3) ¡ ¡  2 , ( , − 1) . =  = (2 + 2 , − 3)(2 + 2 , − 1) ¡ ¡

0, the eigenvalues are defined by: 1

− 1)(2

(2

+ 1)

Œ +  = B (B + 1) + 1 − ¡ ¡ Sž 2 + [   4  . 2Q (2B − 1)(2B + 3) 4 b . For eDGF , we have:

Œ +  = ³ + ¡

^_ 236. ( ï

2

− 18 ( 

2

+ 5) −

1 64

 ( 2 + 11 − 32¡

2

)

−1

+[  

−2

,



= 2(B −

¡

) + 1.

References: H. Bateman and A. Erd´elyi (1955, Vol. 3), M. Abramowitz and I. A. Stegun (1964). 2

+ 1)2 ñó ò ò ó + 2 ï ( ï

2

+ 1)ñó ò + [( ï

2

+ 1)( ô

ï

2 2

– )+þ

2

]ñ = 0.

¶

Equation ¶ of oblate spheroidal wave functions, ¡ = 0, 1,  The transformations  = A"™  ,  = T ™  lead to equation 2.1.2.235. See the books by Bateman & Erd´elyi (1955, Vol. 3) and Abramowitz & Stegun (1964) for more information on this equation. 237. ( ô

ï

2

+

õï

+ ö )2 ñ

óò ò ó

+ (2 ô

ï

+ ü )(ô

ï

2

+

õï

+ ö )ñ

óò

+ þxñ = 0.

 leads to a constant coefficient linear equation of the M 2 + ' + (    = 0. form 2.1.2.11: ªŠ

Š + ( , −  )
Š + ¡

The substitution ˆ = X

© 2003 by Chapman & Hall/CRC

Page 243

244

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.1.2-8. Other equations. 238.

239.

ï 6 ñ óò ò ó

–ï

5

ñóò

ôñ

+

= 0.

The transformation ˆ =  −2 , ¨ =  form 2.1.2.1: 4¨3Šª

Š +  ¨ = 0.

ï 6 ñ óò ò ó

+ (3 ï

2

+ ô )ï

3

The substitution ˆ =  240.

ñóò ò ó + ñ óò where

½

÷

73 

õ÷

3

(1 – 

õ÷ ï

=1

ñ óò

õÿñ

+

−2

leads to a constant coefficient linear equation of the

= 0.

−2

leads to a constant coefficient linear equation: 4
Šª
Š − 2  
Š + 

÷

–

–ô

÷

÷

)

(õ 1 ï

–

ñ

– ô 1 )( õ 2 ï

R R  ÷  ÷ ÷ ï ÷ –1 – ô 3 ) ÷ =1 õ ÷ –ô ÷ ½

– ô 2 )( õ 3 ï

( ƒ  + ý  ) = 1, |   | + |   | > 0, ¢  =    

=1

+1

− 

= 0.

+1

3

  ≠ 0,  

+3

=  ,  

+3

= 0,

=  .

Riemann’s equation. Denote this equation by:



™



1



1





2 2

ƒ û ý

3



3

ƒ

1

û

1

û û

2

ý

2

ƒ ý



3 3

û û

¢

û

= 0.

(1)

û

For  1 =  2 = 0,  3 =  3 = 1, ƒ 1 = ƒ 3 = 0, ƒ 2 = ƒ , ý 1 = 1 − = , ý 2 = ý , and ý equation (1) transforms into the hypergeometric equation 2.1.2.171. À The transformation

− $çL

$

#

#

1

2

$

1

$

2

#

ƒ û ý

3 3

û û û

1 1

+ &ǃ +& ý

2 2

where #  = #2  + $ç  , $  = Lu  + c‚  . In (2), assume & = − ƒ 1 , ° = − ƒ 3 , # =  1 ¢ to obtain the hypergeometric equation (3).

ï ÷ ñ óò ò ó

+ õ )÷

ï

+ ö (ô

–4

ñ

242.

ïƒ÷ ñóò ò ó

 , ¨  + 

ˆ  ¨ = 0.

−2 −

+

ô ï ñóò

– (õ

2

Particular solution: 243.

ïƒ÷ ñóò ò ó

+ (ô

ï

ï ÷ ñ óò ò ó

+ (ô

ï ÷

–1

+

0

ï ÷ ñ óò ò ó

+ (2 ï

+ 2õ = 

$ = − 1 ¢

+° ˆ û +° û ¨

û

¢

= 0,

(3)

û

3,

L = − 2 ¢

2,

and c

=  2 ¢

2





ïƒ÷

–1

=

+

 + 

ôƒõ ï

leads to an equation of the form 2.1.2.7:

+ ô )ñ = 0.

.

÷

–1

+ô

Particular solution:

=  +  .

õ ï )ñ ó ò

Particular solution: 245.

3,

3 3

+ õ )ñó ò – ôñ = 0.

Particular solution: 244.

ïƒ÷ 0

−& −° ƒ −& −Ç ° ý

= 0.

The transformation ˆ =

¨2Šª

Š + (

(2)

≠ 0, brings the original equation into an equation of similar form:

™

241.

== −ƒ −ý ,

À Ÿ ¨ = | 1  −  1| | 3 −  3| , |  2  −  2 |Ÿ +

#  +$ * ˆ = , L3 + c where #2c

3

ï

0

=

2 0

+

+ ( ô – 1) õÿñ = 0.

1−

.

õ ï )ñ ó ò

+ = +  .

õÿñ

= 0.

© 2003 by Chapman & Hall/CRC

Page 244

245

2.1. LINEAR EQUATIONS

246. 247. 248.

+ ( ô ï ÷ + õ )ñ Particular solution:

ï ÷ ñ óò ò ó

+ (ô ï ÷ – ï Particular solution:

ï ÷ ñ óò ò ó ï ÷ ñ óò ò ó

+ (ô

0

÷

ï ÷ +ú

ï ÷

+ õ )ñ

óò ò ó

+ ôƒõ ï + õ )ñ ó ò + −  . 0 = (  + 1) 

óò

+ 1)ñ 0

ï ÷

+ (ö

+ õ ]ñ = 0.

ô 2õ ï ñ

–1

Particular solution: 249. ( ô

÷

+ ö [( ô – ö ) ï  =  −v .

óò

+

= exp O − + ý )ñ

Particular solution:

0

ô ï ú

−

óò 

=  .

ï ÷

(1 + þ



¡

+1

 +

–1

+1

P

= 0. )ñ = 0.

.

+ [( ö – ô ) ï

÷

+ý –

õ ]ñ

= 0.

250. ( ô ï ÷ + õ ï + ö )ñ ó ò ò ó = ôù (ù – 1) ï ÷ –2 ñ . Particular solution: 0 =   + ' + ( . 251.

ï (ïƒ÷

+ 1)ñó ò ò ó + [( ô – õ ) ïƒ÷ + ô – ù ]ñó ò + õ (1 – ô ) ïƒ÷

Particular solution: 252.

ï ( ï 2÷

+ ô )ñó ò ò ó + ( ï

254.

÷

õ 2 ï 2÷ –1 ñ

+ ô – ôù )ñó ò –

J  = L 1    + £  2 +    + L

Solution: 253.

2

 + 1)  J  . 0 = (

–1

ñ

= 0.

= 0.

 + £  2  +   −  J  . 2 

ï 2 ( ô 2 ï 2÷

– 1)ñ ó ò ò ó + ï [ ô 2 (ù + 1) ï 2÷ + ù – 1]ñ ó ò – b ( b + 1) ô 2 ù 2 ï 2÷ ñ = 0. = Á (   ), where Á ( ) is the general solution of the Legendre equation Solution: 2.1.2.154.

ï 2 (ô ïƒ÷

– 1)ñó ò ò ó + ï ( ô' ïƒ÷ + )ñó ò + ( ô/ ïƒ÷ + s)ñ = 0. Find the roots # 1 , # 2 and $ 1 , $ 2 of the quadratic equations 2

#

− (  + 1) # − ° = 0,

2

$

− ( − 1) $ + & = 0

and define parameters ( , ƒ , ý , and = by the relations

= = 1 + ( # 1 − # 2 )B −1 . Then the solution of the original equation has the form =  v ‡ (   ), where ‡ = ‡ ( H ) is the general solution of the hypergeometric equation 2.1.2.171: H ( H − 1)‡ ‘'

‘ + [( ƒ + ý + 1) H − = ]‡ ‘
+ ƒ…ý±‡ = 0. ( = # 1,

255. ( ïƒ÷ + ô )2 ñó ò ò ó –

ƒ = (#

õ ïƒ÷

Particular solution: 256. ( ô

ïƒ÷

+ õ )2 ñó ò ò ó + ( ô

Particular solution: 257. ( ï

÷

+ ô )2 ñ

óò ò ó

+

ïƒ÷

0

ïƒ÷

0

+ õ )2 ñó ò ò ó +

1)

B

−1

ý = (#

,

+$

2)

B

−1

ïƒ÷

,

+ ý )ñó ò + ù ( õ^ö – ô„ý ) ïƒ÷ (˜  + = exp O − X M  .   + 3M P + õ )( ö

+ ô )ñ

óò

–ï

÷

–2

 +  )1 J . 0 = (

0

1

= |  +  |  J  .

ö ïƒú (ô ïƒ÷

Particular solution:

+$

[( õ – 1) ïƒ÷ + ô (ù – 1)]ñ = 0.

õ ï ú (ï ÷

Particular solution: 258. ( ô

–2

1

+ õ )ñó ò + ( ö

= exp O − X



(õ

ï ú

ïƒú

+1

– ôù

–1

ñ

= 0.

+ ôù – ô )ñ = 0.

ïƒ÷

–1

– 1)ñ = 0.

. M   +  P

© 2003 by Chapman & Hall/CRC

Page 245

246 259.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï 2 (ô ïƒ÷

+ õ )2 ñó ò ò ó + (ù + 1) ï (ô 2 ï 2÷ – õ 2 )ñó ò + ö^ñ = 0. 1    ln O leads to a constant coefficient linear equation of the The substitution ˆ = B… 
  +  P

form 2.1.2.11: ŠªŠ −  (B + 2) Š + ( = 0.

260. ( ô

ïƒ÷

+1

ïƒ÷ + ö )ñ óò ò ó

+õ

ïƒ÷ +  ïƒ÷

+( 

Particular solution:

0

–1

+ )ñó ò +[ù (  – ô – ôù )ïw÷

= exp X

( B +  − ƒ )  + ( 'B − ý )    +1 + '  + (

–1

Q

−1

+(ù –1)( – õÿù ) ïw÷

−=

261. ( ô ïƒ÷ + õ ïƒú + ö )ñó ò ò ó + ( – ï )ñó ò + ñ = 0. Particular solution: 0 =  − Œ .

M

–2

]ñ = 0.

 S .

262. ( ô ïƒ÷ + õ ïƒú + ö )ñó ò ò ó + ( 2 – ï 2 )ñó ò + ( ï + )ñ = 0. Particular solution: 0 =  − Œ .

ï ÷

263. 2( ô

+

õï ú

óò ò ó

+ ö )ñ

264. ( ô

ïƒ÷

M

= 0.

+ õ )ú

+1

ñóò ò ó

265.

ïƒ÷

+ (ô

Particular solution:

–1

+

õÿþ ï ú

 £   +M ' + + (

The substitution ˆ = X 2
Šª
Š +

ï ÷

+ ( ôù

0

+ õ )ñó ò – ôùþ



–1

)ñ

óò

+ ýIñ = 0.

leads to a constant coefficient linear equation:

ïƒ÷ –1 ñ

= 0.

S . M (   +  ) +

= exp − X

Q

) + ( ô ï 2 + õ ï ) i ÷ –2 ( ï )]ñ ó ò + õi ÷ –2 ( ï )ñ = 0.  −2 ( ) are arbitrary polynomials of degrees B and B − 2, respectively.  Particular solution: 0 =  +   .

ï g ÷ (ï )ñ óò ò ó + [2 g ÷ (ï . ( ) and Æ Here,

2.1.3. Equations Containing Exponential Functions 2.1.3-1. Equations with exponential functions. 1.

2.

3.

+ ô Solution: functions.

óñ

ñóò ò ó ñóò ò ó

= 0, =L

1

ø

≠ 0. 0 ( H ) + L 2 n 0 ( H ), where H = 2 Œ

ó

+ ( ô – õ )ñ = 0.  = L 1 ø 22  2£   J 2  + L Solution:  Bessel functions.

ñóò ò ó

ó

+ ô ( 

– ô

2

Particular solution:

ñóò ò ó



ó )ñ

0

= 0.

ó

ó



= exp O −  

Œ



n 2

P

2 2

£    J 2 ;

ø

0(

H ) and n 0 ( H ) are the Bessel

  2 £   J 2  , where

ø

Á ( H ) and n Á ( H ) are the

.

4.

– [ ô 2  2 + ô (2 õ + 1) + õ 2 ]ñ = 0.  Particular solution: 0 = exp(   + '  ).

5.

ñóò ò ó

ó



−1

ó

– ( ô 2  + õ + ö )ñ = 0.  The transformation H = ž , ¨ = H −  , where , = £ ( Œ , leads to an equation of the form ' ‘ ‘ 2 ¨

+ Œ 2 (2 , + 1)¨2‘
− ( !H +  )¨ = 0. 2.1.2.108: Œ 2 H

Page 246

247

2.1. LINEAR EQUATIONS

6.

7.

ñ óò ò ó

+ ( ô

ó 

4

+

õ 3 ó

+ ö

2



9.

10.

1 4

–

2

)ñ = 0.

 The transformation ˆ = ž , ¨ =   J ¨2ªŠ

Š + Œ −2 ( ˆ 2 + 'ˆ + ( )¨ = 0.

ñóò ò ó

+ [ ô

2

ó ( õ  ó



+ ö )÷ –

The transformation ˆ =  ž ¨2Šª

Š +  ( Œ )−2 ˆ  ¨ = 0. 8.

ó 

ôñóò

+

õ 2D ó ñ

= 0.

óò

+

õ 2D ó ñ

= 0.

ñóò ò ó

+

ñ óò ò ó

– ôñ

ñóò ò ó

+



1 4

2

2

leads to a linear equation of the form 2.1.2.6:

]ñ = 0.

 =  J

+(, ¨

2

leads to an equation of the form 2.1.2.7:

  The transformation ˆ =   , ‡ =   leads to a constant coefficient linear equation of the

−2 form 2.1.2.1: ‡ ŠªŠ + ˜ ‡ = 0.  The substitution ˆ =   leads to a constant coefficient linear equation of the form 2.1.2.1: 
Šª
Š + ˜ −2 = 0.

ôñóò

ó

+ ( õ

+ ö )ñ = 0.

  n Á± 2 Œ Solution: =   J 2 V L 1 ø Á/ 2 Œ −1 £    J 2  + L 2 ø Á (H ) and n Á (H ) are the Bessel functions. −

11.

ñ óò ò ó

–ñ

óò

+ 

ô 3 ó

õ 2 ó

+

The substitution H =  H 2 
‘'
‘ +  !H 3  + ˜H 2  + 12.

ñóò ò ó

– ñó ò + V ô

ñ óò ò ó

+ 2 ô

2



1 4

–



2

ñ

£     J 2  W , where @ = Œ

−1

£  2 − 4( ;

= 0.

leads to a second-order linear equation of the form 2.1.2.121: − 14 Œ 2  = 0.

1 4

ó ( õ  ó



1 4

+

−1

+ ö )÷ +



1 4

–

1 4

2

Wñ = 0.

leads to a second-order linear equation of the form 2.1.2.122: The substitution H =  H 2 
‘'
‘ + V !H 2  ( ˜H  + ( )  + 14 − 14 Œ 2 W = 0. 13.

ó

 ñ óò

ñóò ò ó

+ ( ô + õ ) 

ó ñóò

Œ

ñ óò ò ó

+

ô  ó ñ óò

ñ óò ò ó

ó ñ óò

+ 2 ü

¨2

 + (   2 

= exp O −  

+ ( ô

17.

ñóò ò ó

+  

– ( ô + 2 õD

ó )ñ óò

Particular solution:

2

0

0



+L

1

Œ

+

= exp O 2



ó

+

 “

exp O

=

+ ( )¨ = 0.

“

+ 0

õ 2  2D ó ñ

= exp O





.

P

õ ó

+ )ñ = 0.

   . P

õ  ó ,

 ).

+ )ñ = 0.



õ ó (ô   ó

–

The substitution ¨



(L

ô ó ( õ  ó

Particular solution: 16.

P

+ )ñ = 0.

+

Particular solution: 15.





= exp O −  

Solution: 14.

ô  ó (ô   ó

+

ü 2  2 ó

+

  

P

+

ü C ó

+ ö )ñ = 0.

leads to a linear equation of the form 2.1.3.5:

= 0.

  .  P

© 2003 by Chapman & Hall/CRC

Page 247

248 18.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó

+ ( ô

2

ó 

óò

+ )ñ

Particular solution: 19.

ñóò ò ó

ó

ñóò ò ó

ó

– ô 0

2



=  

ó

 ñ

= 0.



+ Œ  − .

ó

+ ( ô – )ñó ò + õ 2  ñ = 0.  The substitution ˆ = ž leads to a constant coefficient linear equation: Œ 2 
Šª
Š + Œ 
Š + 

ó

20.

+ ( ô + õ )ñó ò + ö ( ô +  Particular solution: 0 =  − v .

21.

ñ óò ò ó

õ 2 ó )ñ óò

+ (ô +

+ (ô –

Particular solution: 22.

ñ óò ò ó + (ôƒõ ó

õ

+

0

=  

0

ó



−

2

ó )ñ



+   .

(õ – )

2



= (  

– ö )ñ = 0.

– õ

– 3 )ñó ò + ô

Particular solution:

ñóò ò ó



õ

= 0. 2

ó

 ñ

= 0.



+ 1) exp(−    ).

ó

ó

23.

+ (2 ô – )ñó ò + ( ô 2  2  + ö  )ñ = 0. This is a special case of equation 2.1.3.28 with  = , = 0.

24.

ñ óò ò ó

ó

+ (2 ô

+ õ )ñ



óò

+ [ô

 

2 2

= 0.

ó

+ ô (õ + )



ó

+ ö ]ñ = 0.

  leads to a constant coefficient linear equation of the form The substitution ¨ = exp O   Œ P 2 ¨
¨ 3 ¨

2.1.2.11:  +   + ( = 0.

ó

ñóò ò ó

ó

ó

25.

+ ( ô + 2 õ – )ñó ò + ( ö 2   The transformation ˆ = ž  Œ , ¨ ¨2Šª

Š +  ¨2Š
+ ( ¨ = 0.

26.

ñ óò ò ó

+ ( ô + õ )ñ ó ò + [ ö ( ô – ö )  2 + ( ôƒü +  Particular solution: 0 = exp(− (  − ,! ).

27.

ñóò ò ó

+ ( ô + õ )ñó ò + ( q 2  + J + )ñ = 0.  The substitution ˆ =  leads to an equation of the form 2.1.2.146: ˆ 2 
Šª
Š + ( ˆ  +  + 1)ˆ 
Š + ( ƒ…ˆ 2  + ý±ˆ  + = ) = 0.

28.

ñóò ò ó

ó

ó

30.

ó

+ (2 ô

– )ñó ò + ( ô

 

2 2

õ^ö

+ ö – 2ö³ü ) 

ó

+ ü ( õ – ü )]ñ = 0.

ó

ó

õ 2 ó

+



+

ö  ó

+ ü )ñ = 0.

 Œ  leads to a linear equation of the form 2.1.3.5:   − Œ ¨2

 +    2   + (    + , − 1 Œ 2  ¨ = 0. 2 P 4

ñ óò ò ó

+ (2 ô

ó

ñ óò ò ó + (ô ó

+

õ

exp O

=

óò

M

+

ñ óò ò ó

ó

– )ñ

õ  ó )ñ óò ó

0

+ (ô

ó

ó

ó

ó

+ ôƒõ  + ö 2 + ý + ü )ñ = 0.    −Œ  leads to an equation of the form 2.1.3.5: The substitution ¨ = exp O   + Œ ¨2

 + V (  2   +    + , − 1 (  − Œ )2 W ¨ = 20. P 4



Particular solution: 31.

ó

ó

The substitution ¨

29.

+ ôƒõ + õ 2 – õ )ñ = 0.  =   leads to a constant coefficient linear equation:

+ ô

 

2 2

ó ( õ   ó 

= exp O −  

Œ

ó



P

+ )ñ = 0. .

+   ( ô 2 + õ )ñ ó ò + [   ( õ – ô   Particular solution: 0 =    +   −  .

2



ó

) – ]ñ = 0.

© 2003 by Chapman & Hall/CRC

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249

2.1. LINEAR EQUATIONS

32.

ñ óò ò ó

+ ( ô

ó 

+

õ ó

Particular solution: 33.

ñóò ò ó

ó

+ ( ô

+

õ  ó

ñóò ò ó

+ ( ô

ó

+ 2 õ



+ ( ô



= exp O −  

0

Œ

+ ö )ñó ò + [ ôƒõ

ó 



+

−

“

 



( + )

õ C ó )ñ

ó

– )ñó ò + [ ôƒõ

 

= 0.

   − (˜

ó

+ ô„ö

  − (˜ . 0 = exp O −  “ P 

Particular solution: 34.

óò

+ ö )ñ

( + )

ó

ó

.

P

+

õ C  ó ]ñ

+

õ 2  2 ó

ö 2 ó

+

= 0.

+ õ ( – )



ó ]ñ

= 0.

1 b . If Œ = 0, the equation transforms into 2.1.3.24, and if “ = 0, into 2.1.3.25. 1    leads to a constant coefficient 2 b . For Œ “ ≠ 0, the transformation ˆ =   , ¨ = exp O   linear equation: ¨3Šª

Š +  ¨2Š
+ ( ¨ = 0.

35.

ñóò ò ó

+ [ ôƒõ

 

( + )

ó

+

ô ó

Particular solution: 36.

ñóò ò ó

ô

+

v 0 =  −

ó

( ô

+ õ )ñó ò ò ó – ô

2

Particular solution: 38.

(ô

ó

 

2 2

2( ô

ó



+ õ )ñ

=L

óò ò ó

1



  

+



ô 2 õ  (2 + )ó ñ

= 0.

ïƒ÷

) – ö ]ñ = 0.

= 0.



=  

0

– 2 ]ñó ò +

P

.

+ õ )ñó ò ò ó – õƒñó ò – ô

Solution: 39.

ó

 ñ

“

+ 1) exp(−    ).

+ ö [ ô exp( õ

Particular solution: 37.



= (  

0

ïƒ÷ )ñ óò

exp( õ

õ  ó

+

Œ

+ .

ü  óñ

2

2 2 2

= 0.

  + £  2  2 +   + L

ô  ó ñ óò

+



ö^ñ

2

  



+ £  2  2



+ 

−



.

= 0.

The substitution ˆ = X (    +  )−1 J 2  leads to a constant coefficient linear equation of the M form 2.1.2.1: 2 Šª

Š + ( = 0.

40.

ó

( ô

+ õ )ñ



óò ò ó

ó

+ ( ö



( ô

42.

ó (

ó

+ õ )ñ



óò ò ó

+ ( ö

óò

+ ü [( ö – ôƒü ) 

+ ý )ñ

óò

+ (ùJ

 . 0 =  −

Particular solution: 41.

+ ý )ñ

ó





ó

+þ



ý

–õ

ü ]ñ

= 0.



For the case  = 0, see equation 2.1.3.27. For  ≠ 0, the transformation ˆ =  ! , ¨ = ˆ −  , where , is a root of the quadratic equation Œ 2 , 2 + Œ", + = 0, leads to an equation of the form ¡ M 2.1.2.172: Œ 2 ˆ (ˆ +  )¨2Šª

Š + Œ [(2 , Œ + Œ + ( )ˆ +  (2 , Œ + Œ + )]¨3Š
+( , 2 Œ 2 + (, Œ + B )¨ = 0.

ó + ü )ñó ò ò ó + ( ô

ó + õ 

ó

ó ( ˆõ

( ô

+ õ )2 ñó ò ò ó + ö

Particular solution: 44.

+

)ñ = 0.

+ ö ) ñ óò + (ô ó

Integrating yields a first-order linear equation: (  43.

ó

( ô

ó

0

0



ó

ó )ñ



M

–

ó )ñ

= 0.

   + , )
 + (   +    −  + ( ) = L .

= 0.

+ õ )ñó ò + ö

= (  



( +  )  , where , = − . Œ

= (  

+ õ )2 ñó ò ò ó +  ( ô

Particular solution:

– ö

ó + õ C 

ó (

+

ˆõ

– ö

( +  )  , where , = − . Œ

ó )ñ

= 0.

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250 45.

SECOND-ORDER DIFFERENTIAL EQUATIONS ( ô

ó 

óò ò ó

+ õ )2 ñ

+ ( ô

The substitution ˆ = X 2.1.2.11: Šª

Š + ( 
Š +

46.

ó ( ô 

+õ )

2

ñ óò ò ó

4( ô

ó

+ õ )÷

M

  



¡

0

2

+ þxñ = 0.

óò

+ ö



+ ý )÷

–4

–

ó ( ü  ó

– ö

( +  )  , where , = − . Œ

ó (ö  ó



óò

+ õ )ñ

+ õ )ñ



= (  

ó 

leads to a constant coefficient linear equation of the form

+ = 0.

+ [ ü

ñóò ò ó

+ ö )( ô

ü ó (ô   ó

+

Particular solution: 47.

ó 



2

ó

( ô



ó

+

ˆõ )ñ

= 0.

+ õ )÷ ]ñ = 0.

   +  ¨   J 2 , = leads to an equation of the form 2.1.2.7: The transformation ˆ =   (  +  + ¨42Šª

Š + , ( ¢eŒ )−2 ˆ −  ¨ = 0,(  where M ¢ =  − ˜( . M M

2.1.3-2. Equations with power and exponential functions. 48.

ñóò ò ó

+

ô ó ñóò

ïƒ÷  ó

+ õ (ô

Particular solution: 49.

ñóò ò ó

ó ñóò

+ 2 ô

0

+ (ô

The substitution ¨

¨2

 + ( ' 2  + (˜ 

+ ( ô ï + õ )  Particular solution:

51.

ñóò ò ó

ñóò ò ó

+ (ô

ï  ó

53.

ó



55.

+ ( ô ï + õ ) exp( Particular solution:

ñ óò ò ó + ô ïƒ÷

56.

ï ñóò ò ó

– (2 ô ï

2

+1





0

– ( ô = .

ó )ñ óò

.

= exp O −

ïw÷ )ñ óò

ó



B +1

 

0

–ô = .

3

ôù ïƒ÷

+

0

+1

exp(2

57.

ï ñ óò ò ó

+

ô ï   ó ñ óò

Particular solution:

+

ï 2 )ñ

2

ô  ó 0

ïƒ÷

–1

)ñ = 0.

(1 +

ï )ñ 

=  exp O −  

Œ

2

õ 2 )ñ

= 0.

= 0. –1

)ñ = 0.

.

ïƒú )ñ

Solution: = exp    L 1 ø  ( H ) + L 2 Q n Á ( H ) are the Bessel functions.  1 2

P

ïƒ÷ )ñ

exp( õ

–1

+

õ  ó )ñ

+

– ô exp( =  +  .

ïƒ÷

ó

– ô

ïƒ÷  ó

+ ( ôƒõ

ï

+ö

leads to a linear equation of the form 2.1.2.10:

P

ï  ó

= 

+ 1)ñó ò + 4 õ

)ñ = 0.

.

P

õ ï 2÷

+



–1

ó

−

exp( õ ïƒú ) ñ óò

Particular solution:

 

0

0

ñóò ò ó

Œ

ó

 

– ô ñ = 0. 0 =  +  .

ñ óò ò ó + (ô ƒï ÷ + õ ó )ñ óò

Particular solution: 54.

exp O

+ 2 õ )ñó ò + ( ôƒõ

+ ï ( ô + õ Particular solution:

ñóò ò ó

B +1



ïƒ÷

+ù



+ ô

)¨ = 0.

Particular solution: 52.

ó

 

2 2

ó ñóò

50.

= exp O −

=

−1

õ ï 2÷

–

= 0. = 0.

= 0.

n  ( H )S , where H = Œ 2 

−1

£  exp ݌  2  ; ø Á ( H ) and

= 0.



P

.

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251

2.1. LINEAR EQUATIONS

58.

ï ñ óò ò ó

+

ô ï   ó ñ óò

– [ô ( õ

Particular solution: 59.

ï ñóò ò ó

ï  ó

+ (ô

ï ñóò ò ó

ï

+ [ô ( õ

62. 63.

65. 66.

+

0

.

ó

1−

+ ( ô ï ÷ + õ  Particular solution:

ï ñ óò ò ó ï ñóò ò ó

ó + (ô ï   

õï

−

0

70.

+ [( ô ïƒ÷ + 1)  Particular solution:

0

ó

0

73.

= 0.

õ ó ñ

= 0.

ó

ó

+ [ ô ( õ ïƒ÷ – 1)  =  exp(− '  B ).

+ ôù ïƒ÷ + 1 – 2ù ]ñó ò + = (   + 1) exp(−   ).

ó

õÿù ïƒ÷

+

–1

]ñ = 0.

ô 2 ù ï 2÷ –1   ó ñ

= 0.

ó

+ ( ô  + õ C )ñ = 0.   Integrating, we obtain a first-order linear equation:  "
 + (   +    − 1) = L .

ï ñóò ò ó

ïƒ÷

+ [ô

( ï + ô )ñ

óò ò ó

4ï

2

ñóò ò ó

+ [ô

ïƒú

exp( õ

+ ( õ



ï 2÷

) + ö ]ñó ò + ô ( ö – 1) ïƒ÷

v. 0 = 

–1

ó

1−

óò

+ ö )ñ 0

exp( õ

õ  ó ñ

+

ïƒ÷

= 0.

1 − ( −    +

= exp O X

)+1–ù

2



exp( õ

ïƒú )ñ

= 0.

 . M P

]ñ = 0.

= 



−1 2

leads to a linear equation of the form 2.1.3.1:

ó

+ 2 ô ï ñó ò + [( õ 2  2 } – b 2 ) ö 2 ï 2 + ô ( ô – 1)]ñ = 0.   Solution: =  −  V L 1 ø Á (   v )+ L 2 n Á (   v )W , where ø Á ( H ) and n Á ( H ) are the Bessel functions.

ï 2 ñ óò ò ó ï 2 ñ óò ò ó

+

ô ï   ó ñ óò

ï 2 ñ óò ò ó

+ ï ( ô

ó

ï 4 ñ óò ò ó

ó

+ õ ( ô 0

=



−

ó

.

õ

–

+ 2 õ )ñó ò + [ ô ( ö

Particular solution: 72.

ï  ó ñ

+ ô ï ÷ –1 ( õ  + ù – 1)ñ = 0. = exp(−   B ).

ó + õ )ñ ó ò

Particular solution: 71.

2

.

ó )ñ ó ò

ˆ = '  , ¨ The transformation ¨42Šª

Š +  ( 'B )−2  Š ¨ = 0. 69.





+ 2)]ñ = 0.

= 0.

– 1]ñó ò + ôƒõ

= ( ' + 1) 

ï

+ õ ïƒ÷ ) ñ óò

ï ñóò ò ó

ï ñ óò ò ó + (ô  ó

óñ

.

+ [( ô ï 2 + õ ï )  + 2]ñó ò + Particular solution: 0 =  +   .

ï ñóò ò ó

Particular solution: 68.

ó

+ 1)-

Particular solution:

67.

=

0

Particular solution:

64.



+ õ (õ

+ õ )ñó ò + ô ( õ – 1) 

Particular solution: 61.



ó 

+ 1)

= 

0

Particular solution: 60.

ï

– 1)ñ = 0.

ï

− −v  . 0 =   

+ (  2 ) – b 2 )ñ = 0.  Solution: =  V L 1 ø Á (  1 J ) + L

2

+ õ ) 

ó

–ö

2

ï

2

+ õ ( õ – 1)]ñ = 0.

 n Á (  1 J )W , where ø Á ( H ) and n Á ( H ) are the Bessel functions.

+ [ ô exp(2 # ï ) + õ exp( # ï ) + ö ]ñ = 0. ˆ = 1  , ¨ =   leads to a linear equation of the form 2.1.3.5: The transformation ¨2Šª

Š + (   2  Š +   Š + ( )¨ = 0.

ï 4 ñ óò ò ó

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252 74. 75.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ó

ó

+ ô ï 2   ñ óò + [ô ( õ – ï )  – Particular solution: 0 =  exp(   ).

ï 4 ñ óò ò ó

(ï

õ ó (ï

+ ô )2 ñó ò ò ó +

2

ïƒ÷

(ô

( ô



ïƒ÷

+ õ )2 ñó ò ò ó + ö ( ô

ó

õï

+

0

óò ò ó

+ ö )ñ

79.

0

= exp O − X 2

0

ó

  ñ 

=  

+ ö ]ñó ò ò ó – ö

Particular solution:

ó ñóò

+ õ ) 

– ô

Particular solution:

ó   [( ô ï + õ ) -

– ïƒ÷

–2

 +  )1 J . 0 = (

Particular solution: 78.

ó ñóò

( ïƒ÷ + ô )2 ñó ò ò ó + õ ( ïƒ÷ + ô )  Particular solution:

77.

+ .

2

2 −

=(  

ñ

= 0.

ï  ó

+ ô )ñó ò – ( õ

£  0 =

Particular solution:

76.

2

õ 2 ]ñ

+ ô )ñ = 0.

ï  ó

(õ

ó

+ ( ö

ïƒ÷

– ôù



ôù

+

– ô )ñ = 0. – 1)ñ = 0.

–1

. M   +  P = 0.

+ ' + ( . = 0.



+  +  .

2.1.4. Equations Containing Hyperbolic Functions 2.1.4-1. Equations with hyperbolic sine. 1.

ñóò ò ó

+ ( ô sinh2 ï + õ )ñ = 0.

ñ óò ò ó

+

Applying the formula sinh2  = 12 cosh 2 − 12 , we obtain the modified Mathieu equation 2.1.4.9: 

 +   − 12  + 12  cosh 2  = 0. 2.

ô

ï )ñ ó ò

sinh(

Particular solution: 3.

ñóò ò ó

+ [ ô sinh ÷ (

ï

5.

0 =  

) – õ ]ñ = 0.

.

) + ö ]ñó ò + ôƒõ sinh ÷ ( 0

=

−





.

+ ( ô ï + õ ) sinh ÷ ( ï )ñó ò – ô sinh ÷ ( Particular solution: 0 =  +  .

ñóò ò ó

+ ( ô sinh÷

ï ñ óò ò ó

ï

Particular solution: 6.



−

Particular solution: 4.

ï

+ õ [ ô sinh(

sinh2 ( ô

ï )ñ óò ò ó

–

õÿñ

+

õï ú

0

+1

óò

)ñ

= exp O −

= 0.

¡

+



õ ï ú (ô

+1

 +

ï )ñ

= 0.

ï )ñ

= 0.

sinh÷ +1

P

ï

+þ

)ñ = 0.

.

H

( H > 0) leads to a linear equation of the form 2.1.2.190: The substitution  = A ln £ H 2+1 ' ‘ ‘ ‘ 


2 2 −2 H ( H + 1) + (3 H + 1) − 4  ˜H = 0. 7.

ñóò ò ó

sinh2 ï – [ ô

Solution: 8.

sinh2 ï + ù (ù – 1)]ñ = 0,   1 M = sinh   O (L 1   + L sinh   P

ô

2

M

2

≠ 0;

  −  ).

ù

= 1, 2, 3,

uuu

[ ô sinh( ï ) + õ ï + ö ]ñ„ó ò ò ó – ô 2 sinh( ï )ñ = 0. Particular solution: 0 =  sinh( Œ  ) + ' + ( .

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2.1. LINEAR EQUATIONS

2.1.4-2. Equations with hyperbolic cosine. 9.

– ( ô – 2 cosh 2 ï )ñ = 0. The modified Mathieu equation. The substitution  = ™Ýˆ leads to the Mathieu equation 2.1.6.29: Šª

Š + (  − 2  cos 2ˆ ) = 0.

ñóò ò ó

For eigenvalues  =   (  ) and  =   (  ), the corresponding solutions of the modified Mathieu equation are:

½

# 22  ++K cosh[(2 , +  ) ], K  =0 ½ Se2  + ( ,  ) = −™ se2  + (™Ý ,  ) = ÷ $ 22  ++K sinh[(2 , +  ) ], K K K  =0 where  can be either 0 or 1, and the coefficients # 22  ++K and $ 22  ++K are specified in 2.1.6.29. K K The modified Mathieu equation is discussed in the books by Abramowitz & Stegun (1964), Ce2 

+

( ,  ) = ce2 

K

+

K (™Ý ,  ) = ÷

Bateman & Erd´elyi (1955, vol. 3), and McLachlan (1947) in more detail.

10.

ñóò ò ó

+ ( ô cosh2 ï + õ )ñ = 0.

ñ óò ò ó

+

Applying the formula cosh 2 = 2 cosh2  − 1, we obtain the modified Mathieu equation 2.1.4.9: 

 +  12  +  + 12  cosh 2  = 0. 11.

ô

cosh(

ï )ñ ó ò

+ õ [ ô cosh(

Particular solution: 12.

ñóò ò ó

+ [ ô cosh÷ (

ï

0

14. 15.

0

18.

+ ô ïƒ÷ coshú ( Particular solution:

ï )ñ óò

cosh÷ (

ï )ñ óò

ñóò ò ó

ï ñóò ò ó

+

ô ï

ï ñóò ò ó

+ [ô

ï

cosh÷ (

ï )ñ

0

ï

0

.

0

ïƒ÷

–ô = .

– [ô ( õ





= 

= 0.

ï )ñ

ï

–1

coshú (

= 0.

ï )ñ

+ 1) cosh ÷ (

= 0.

ï

) + õ (õ

) + õ ]ñó ò + ô ( õ – 1) cosh ÷ ( =

1−

ï 2 ñ óò ò ó

+

ô ï

cosh÷ (

ï )ñ óò

.

+ õ [ ô cosh÷ (

−

ï

+ 2)]ñ = 0.

.

+ [( ô ï 2 + õ ï ) cosh÷ ( ï ) + 2]ñó ò + õ cosh÷ ( Particular solution: 0 =  +   .

ï ñóò ò ó

Particular solution: 19.

.

+ ( ô ï + õ ) cosh ÷ ( ï )ñó ò – ô cosh÷ ( Particular solution: 0 =  +  .

Particular solution: 17.





) – õ ]ñ = 0.

ñóò ò ó

Particular solution: 16.

=

−



ï

) + ö ]ñó ò + ôƒõ cos÷ (

Particular solution: 13.

=

−



.

0

=

0

=  − .

ï

)–

õ

ï )ñ ï )ñ

= 0.

= 0.

– 1]ñ = 0.

( ô cosh ï + õ )ñ„ó ò ò ó + ( ö cosh ï + ý )ñ„ó ò + [( ö – ô ) cosh ï + ý – õ ]ñ = 0. Particular solution:



© 2003 by Chapman & Hall/CRC

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254

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï )ñ óò ò ó

20.

cosh2 ( ô

õÿñ

21.

[ ô cosh( ï ) + õ ï + ö ]ñ ó ò ò ó – ô 2 cosh( ï )ñ = 0. Particular solution: 0 =  cosh( Œ  ) + ' + ( .

–

= 0. 1 H ln (0 < H < 1) leads to the hypergeometric equation 2.1.2.171: The substitution  = 2  1 − H ' ‘ ‘ ‘


H ( H − 1) + (2 H − 1) +  −2  = 0.

2.1.4-3. Equations with hyperbolic tangent.

ï

22.

ñóò ò ó

+ [ ô tanh(

) + õ ]ñ = 0. 1 − tanh( Œ  ) ¨ , = H − J  , where , is a root of the quadratic The transformation H = 1 + tanh( Œ  ) equation 4 , 2 +  −  = 0, leads to a linear equation of the form 2.1.2.172: 4 Œ 2 H ( H + 1)¨2‘'

‘ + ‘ 4 Œ (2 , + Œ )(H + 1)¨3
+ (4 , 2 +  +  )¨ = 0.

23.

ñóò ò ó

– 4ô

24.

tanh2 (3 ô

ï )ñ

Particular solution:

0

ñ óò ò ó

2

ñóò ò ó

= sinh(3  )[cosh(3  )]−1 J 3 .

ï )]ñ

+ [ ô – ô ( ô + ) tanh2 (

Particular solution: 25.

= 0.

+ [3 ô –

2

0

= 0.

= [cosh( Œ  )]  J  . −

– ô ( ô + ) tanh2 (

Particular solution:

0

ï )]ñ

= sinh( Œ  )[cosh( Œ  )]−  J  .

26.

+ ôñó ò – [ + ô tanh( ï )]ñ = 0. Particular solution: 0 = cosh( Œ  ).

27.

ñóò ò ó

ñóò ò ó

+ 2 tanh ï

Solution: 28.

ñóò ò ó

+

ô

ñ„óò

30. 31.

32.

+ ôñ = 0.

cosh  =

tanh(

ï )„ñ óò

Particular solution: 29.

= 0.

™

L

cos( ' ) + L 2 sin( ' ) if  − 1 =  2 > 0, L 1 cosh( ' ) + L 2 sinh( ' ) if  − 1 = −  2 < 0. 1

+ õ [ ô tanh( 0

=

−





ï

) – õ ]ñ = 0.

.

+ 2 tanh ï ñ„ó ò + ( ô ï 2 + õ ï + ö )ñ = 0. The substitution ¨ = cosh  leads to a second-order linear equation of the form 2.1.2.6: ¨2

 + (  2 + ' + ( − 1)¨ = 0.

ñóò ò ó

+ 2 tanh ï ñ„ó ò + ( ô ïƒ÷ + 1)ñ = 0. ¨ 

 +   ¨ = 0. The substitution ¨ = cosh  leads to a linear equation of the form 2.1.2.7: I

ñóò ò ó

+ 2 tanh ï ñ„ó ò + ( ô ï 2÷ + õ ïƒ÷ –1 + 1)ñ = 0. The substitution ¨ = cosh  leads to a second-order linear equation of the form 2.1.2.10: ¨2

 + (  2  + '  −1 )¨ = 0.

ñóò ò ó

+ (2 tanh ï + ô )ñ„ó ò + ( ô tanh ï + õ )ñ = 0. ¨ 
+ ¨ 

 +  2 The substitution ¨ = cosh  leads to a constant coefficient linear equation: I ¨ (  − 1) = 0.

ñóò ò ó

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255

2.1. LINEAR EQUATIONS

33.

+ ô tanh÷ ( ï )ñó ò – [ + ô tanh÷ Particular solution: 0 = cosh( Œ  ).

34.

ñóò ò ó

ñóò ò ó

+ [ ô tanh÷ (

ï

) + ö ]ñó ò +

Particular solution: 35. 36. 37.

0

40.

+ ô ï ÷ tanhú ( Particular solution:

ï )ñ ó ò

tanh÷ (

ï )ñ óò

ñ óò ò ó

ï ñóò ò ó

+

ô ï

0

ï ñ óò ò ó

ï

+ [ô

tanh÷ (

ï

+ ( ô tanh÷

ï ñ óò ò ó

ï

ï 2 ñ óò ò ó

+

ô ï

+





ï )ñ

=

–1

tanhú (

= 0.

ï )ñ

+ 1) tanh÷ (

ï

= 0.

ï

) + õ (õ

õï ú

+ ô ( õ – 1) tanh÷ (

óò

.

+1

)ñ

óò

= exp O −

ï )ñ ó ò

¡



+1

 +

.

0

=

0

=  − .

tanh÷

õ ï ú (ô

+

+ õ [ ô tanh÷ (

−

ï

+ 2)]ñ = 0.

.

) + õ ]ñ

0

tanh÷ (

ï )ñ

+1

ï

P

ï

= 0.

ï )ñ

= 0.

+þ

)ñ = 0.

. ) – õ – 1]ñ = 0.

( ô tanh ï + õ )ñ„ó ò ò ó + ( ö tanh ï + ý )ñ„ó ò + [( ö – ô ) tanh ï + ý – õ ]ñ = 0. Particular solution:

43.

ï ÷

– [ô ( õ

1−

= 0.

.

–ô = . = 

= 0.

+ [( ô ï 2 + õ ï ) tanh÷ ( ï ) + 2]ñó ò + õ tanh÷ ( Particular solution: 0 =  +   .

ï ñóò ò ó

Particular solution: 42.

0

0

Particular solution: 41.



ï )ñ

+ ( ô ï + õ ) tanh ÷ ( ï )ñó ò – ô tanh÷ ( Particular solution: 0 =  +  .

Particular solution: 39.



tan÷ (

ôƒõ

ï )]ñ

(

ñóò ò ó

Particular solution: 38.

=

−

+1

) + õ ]ñ„ó ò ò ó + [ ö tanh( ï ) + ý ]ñ„ó ò + [ù tanh( ï ) + þ ]ñ = 0. 1 + tanh( Œ  ) ¨ , = H − J  , where , is a root of the quadratic equation The transformation H = 1 − tanh( Œ  ) 4(  −  ) , 2 + 2( ( − ) , + B − = 0, leads to a linear equation of the form 2.1.2.172:

[ ô tanh(

ï



¡

M

4 Œ 2 H [(  +  ) H +  −  ]¨ ‘'

‘ + 2 Œ {[2(2 , + Œ )(  +  ) + ( + ] H + 2(2 , + Œ )(  −  ) + − ( }¨ ‘
M M + [4(  +  ) , 2 + 2( ( + ) , + B + ]¨ = 0.

M

¡

2.1.4-4. Equations with hyperbolic cotangent. 44.

45.

ñóò ò ó

+ [ ô coth(

ï

) + õ ]ñ = 0. 1 − tanh( Œ The transformation H = 1 + tanh( Œ equation 4 , 2 +  −  = 0, leads to 4 Œ (2 , + Œ )(H − 1)¨3‘
+ (4 , 2 +  +  )¨ coth2 (3 ô

ï )ñ

Particular solution:

0

ñóò ò ó

– 4ô

2

 ) ¨ ,  )

= H −  J  , where , is a root of the quadratic

‘'‘

an equation of the form 2.1.2.172: 4 Œ 2 H ( H − 1)¨2

+ = 0.

= 0.

= cosh(3 )[sinh(3  )]−1 J 3 .

© 2003 by Chapman & Hall/CRC

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256 46.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

Particular solution: 47. 48.

ñóò ò ó

ï )]ñ

+ [ ô – ô ( ô + ) coth2 ( + [3 ô –

= [sinh( Œ  )]  J  .

0

– ô ( ô + ) coth2 (

2

0

ñ óò ò ó

+ õ [ ô coth(

ô

coth(

ï )ñ ó ò

Particular solution:

= 0.

=

0





−

.

ï

) – õ ]ñ = 0.

coth÷ (

ï )ñ

= 0.

50.

+ ( ô ï + õ ) coth ÷ ( ï )ñ ó ò – ô coth÷ ( Particular solution: 0 =  +  .

ï )ñ

= 0.

51.

ñóò ò ó

49.

ñóò ò ó

+ [ ô coth÷ (

ï )]ñ

= cosh( Œ  )[sinh( Œ  )]−  J  .

Particular solution: +

= 0.

−

ï

) + ö ]ñó ò +

Particular solution:





.

ñ óò ò ó

+ 2ù coth ï

Solution: 52.

=

0

−

ôƒõ

+ (ù

ñóò

1 = O sinh 



M M

ï )]„ñ óò ò ó

– ô 2 )ñ = 0,

2

3P

ù

uuu

= 1, 2, 3,

  ( L 1   + L 2  −  ).

+ [ ö + ý coth( ï )]ñ„ó ò + [ù + þ coth( ï )]ñ = 0. [ ô + õ coth( Multiply this equation by tanh( Œ  ) to obtain equation 2.1.4.43.

2.1.4-5. Equations containing combinations of hyperbolic functions. 53.

ñóò ò ó

ï

– ô [ ô cosh2 ( õ

) + õ sinh( õ

Particular solution: 54.

ñ óò ò ó

ï

– ô [ ô sinh2 ( õ

ñóò ò ó

Q 

) + õ cosh( õ

Particular solution: 55.

= exp

0



= exp

0



Q 

ï )]ñ

= 0.

sinh( ' )S .

ï )]ñ

= 0.

cosh( ' )S .

+ ( ô cosh2 ï + õ sinh2 ï + ö )ñ = 0.

Apply the formulas 2 sinh2  = cosh(2 ) − 1 and 2 cosh2  = cosh(2 ) + 1 to obtain an  −  + +( + cosh(2 )S = 0. equation of the form 2.1.4.9: 

 + Q 2 2 56.

+ ô sinh( ï )ñ„ó ò – [ + cosh( Particular solution: 0 = sinh( Œ  ).

ñóò ò ó

57.

+ ô cosh( ï )ñ Particular solution:

58.

ñóò ò ó

ñ óò ò ó

–

tanh(

Solution: 59.

ñóò ò ó

– tanh ï

Solution:

=L

óò

Р[ + sinh( 0 = cosh( Π ).

ï )„ñ óò 1

ñóò

–ô

exp

+

ô

2

= £ sinh 

Q Œ

2



ï )]ñ

= 0.

ï )]ñ

= 0.

ï )ñ

cosh2 (

sinh( Π )S + L

= 0. 2

coth2 ï (sinh ï )2ú

Q

L

1

ø

2

1

n Á ( H ) are the Bessel functions.

+

O ¡



exp −



Q Œ

–2

ñ

+ sinh 

sinh( Π )S .

P

= 0. +L

2

n

2

1

+

O ¡



+

sinh 

P

S , where ø Á ( H ) and

© 2003 by Chapman & Hall/CRC

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257

2.1. LINEAR EQUATIONS

60.

61.

sinh ÷ (

ï )ñ óò ò ó

+ [ ô cosh÷

¨2Šª

Š + Œ

−2 −

cosh ÷ (

ï )ñ ó ò ò ó

¨2Šª

Š + Œ

−2 −  ¨ ˆ = 0.

–4

ï

(

The transformation ˆ = tanh( Œ  ), ¨

=

ˆ  ¨ = 0.

+ [ ô sinh÷

–4

ï

(

sinh ÷ (

2

)–

cosh÷ (

The transformation ˆ = coth( Œ  ), ¨

=

= 0.

leads to an equation of the form 2.1.2.7:

cosh( Π )

2

)–

ï )]ñ ï )]ñ

= 0.

leads to an equation of the form 2.1.2.7:

sinh( Π )

2.1.5. Equations Containing Logarithmic Functions 2.1.5-1. Equations of the form  ( )

 + ( ) = 0. 1.

ñóò ò ó

– (ô

2

ï

2

ln2 ï + ô ln ï + ô )ñ = 0.

Particular solution: 2.

ñóò ò ó

– (ô

2

ï 2÷

0

ln2 ï + ôù

Particular solution: 3.

0

7.

8.

ï ñ óò ò ó

– [ô

ï 2 ñ óò ò ó

10.

ln ï + ô

=  −Ž  (

2

ï

ln2÷ ( õ

ï

) + ôù ln÷ 0

+1)

–1

ïƒ÷

ï )]ñ

(õ

= 0.

Q

M

+ ( ô ln2 ï + õ ln ï + ö )ñ = 0.

The transformation ˆ = ln  , ¨ ¨2Šª

Š + ( ˆ 2 + 'ˆ + ( − 1 )¨ = 0. 4

ï 2 ñ óò ò ó

=

+ V ô ( õ ln ï + ö )÷ + 41 UW ñ = 0.

The transformation ˆ =  ln  + ( , ¨ ¨2Šª

Š +  −2 ˆ  ¨ = 0.

ï

2

ln( ô

ï )ñ óò ò ó

ï (ô ï

ln ï

)ñ = 0.

 +1 Ž , where  =   . (B + 1)2

+ õ )ñ = 0. The transformation ˆ =  ln  +  − 14 , ¨ ¨2Šª

Š +  −2 ˆ ¨ = 0.

ï 2 ñ óò ò ó

–1

= exp lX ln  ( ' )  S .

+ ( ô ln ï

Solution: 9.

–1

– ( ô 2 ï ln2 ï + ô )ñ = 0.   Particular solution: 0 =  −    .

Particular solution:

6.

ïƒ÷

ï ñ óò ò ó

4.

5.

2 2 =  − J 4   J 2.

J

= 



J

−1 2

−1 2

= 

leads to an equation of the form 2.1.2.2:

J

leads to an equation of the form 2.1.2.6:

−1 2

leads to an equation of the form 2.1.2.7:

+ ñ = 0. =L

1

ln(  ) + L

+õ ï

+ ö )ñ Particular solution: ln ï + õ ï + ö )ñ Particular solution:

ï 2(ô

óò ò ó 0

óò ò ó 0

2

ln(  ) X [ln(  )]−2  .

M

– ôñ = 0. =  ln  + ' + ( . + ôñ = 0. =  ln  + '  + ( .

© 2003 by Chapman & Hall/CRC

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SECOND-ORDER DIFFERENTIAL EQUATIONS

2.1.5-2. Equations of the form  ( )

 + ( )
 + % ( ) = 0. 11. 12. 13. 14. 15.

+ ô ln ÷ ( õ ï )ñó ò + ö [ ô ln ÷ ( õ  Particular solution: 0 =  − v .

17. 18. 19.

+ [ ô ln÷ ( õ ï ) + ö ]ñó ò + ô„ö ln ÷ ( õ  Particular solution: 0 =  − v . + ( ô ï + õ ) ln ÷ ( ö Particular solution:

ñóò ò ó

0

+ ô ïƒ÷ lnú ( õ ï )ñó ò – ô ïƒ÷ Particular solution: 0 =  .

ï ñóò ò ó

+

ô ï

ln ï

22. 23.

0

25. 26.

= 0.

ï )ñ

= 0.

ln ú ( õ

–1

ï )ñ

= 0.

 =  

1−

 .

+ ( ô ï ln ï + õ )ñ„ó ò + ( ôƒõ ln ï + ô )ñ = 0.   Particular solution: 0 =    −  .

ï ñóò ò ó

+ (2 ô ï ln ï + 1)ñ„ó ò + ( ô   =    − (L 1 + L Solution:

ï ñóò ò ó

ï

ln2 ï + ô ln ï + ô )ñ = 0. 2 ln  ). 2

+ ln ï ( ô ï + õ )ñ„ó ò + ô ( õ ln2 ï + 1)ñ = 0.   Particular solution: 0 =    −  .

ï ñóò ò ó ï ñóò ò ó

+

ln÷ ( õ

ô ï

ï ñ óò ò ó

+

ln÷

ô ï

+ ôù ln÷

ï )ñ óò

0

ï ñ óò

–1

Q

÷

+ ( ô ï ln ÷ ï + 1)ñó ò – ô ln÷ Particular solution: 0 = ln  .

ï ñóò ò ó

ï

ln ÷

ï

–1

ñ

M

–1

–1

=

1−

.

ï )ñ

 . M P

ï ñ

= 0.

ï ñ

= 0.

+ [( ô ï 2 + õ ï ) ln÷ ( ö ï ) + 2]ñó ò + õ ln ÷ ( ö Particular solution: 0 =  +   .

ï ñóò ò ó

+ ( ô ïƒ÷ + õ ln ú Particular solution:

+ ô ïƒ÷ –1 ( õ lnú = exp(−   B ).

ï )ñ óò

ï ñóò ò ó

0

= 0.

= 0.

+ õ )ñó ò + ô ( õ – 1) ln ÷ 0

= 0.

=  exp O − lX ln 

+ ( ô ï ÷ ln ï + 1)ñ ó ò – ô ï Particular solution: 0 = ln  .

+ (ô

ï )ñ

+ ôù ln÷

ï

ï ñ óò ò ó

ï ñóò ò ó

(õ

= exp − lX ln  ( ' )  S .

+ ( ô ln÷ 0

Particular solution:

24.

ï )ñ

+ ô (ln ï + 1)ñ = 0.

ñóò

Particular solution: 21.

– ô ln÷ ( ö =  +  .

ï )ñ óò

ñóò ò ó

Particular solution: 20.

) – ö ]ñ = 0.

ñóò ò ó

Particular solution:

16.

ï

ñóò ò ó

ï

ï )ñ

+ ù – 1)ñ = 0.

+ ( ô ïƒ÷ + õ ï lnú ï )ñó ò + [ õ ( ô ïƒ÷ – 1) lnú Particular solution: 0 =  exp(−   B ).

ï ñóò ò ó

= 0.

ï

+ ôù

ïƒ÷

–1

]ñ = 0.

© 2003 by Chapman & Hall/CRC

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259

2.1. LINEAR EQUATIONS

ï 2 ñ óò ò ó

27.

+

=

Solution: where 28.

29.

30.

31.

32.

ø

+ ô ln÷ ( õ

ï ñóò

ü

ln( ' ) ¬L

+

ln ( ' ) + L

2

•

+ ï (2 ô ln ï + 1)ñ„ó ò + ( ï

n

2

1

+

”

£  ¡

+

ln ( ' ) ® ,

•

¡

=

ï 2 ñ óò ò ó

+ ï (2 ln ï + ô + 1)ñ„ó ò + (ln2 ï +

ï 2 ñ óò ò ó

+ ï (2 ln ï + ô )ñ„ó ò + [ln2 ï + ( ô – 1) ln ï +

2

+ô

2

1 (B + 2), 2

−1

) = 0.

ln2 ï + õ )ñ = 0.

¨ Š
+ ¨ ªŠ

Š +  2 The substitution = ¨ exp(− 21  ln2  ) leads to the Bessel equation 2.1.2.126:  2 2 ¨ ( 2 +  −  ) = 0.

ô

ln ï + õ )ñ = 0.

The transformation ˆ = ln  , ¨ = exp( 21 ln2  ) leads to a constant coefficient linear equation: ¨2Šª

Š +  ¨2Š
+ (  − 1)¨ = 0.

õ ïw÷

+ ö ]ñ = 0.

The substitution ¨ = exp( 12 ln2  ) leads to a linear equation of the form 2.1.2.132:  2 ¨2

 +  ¨2
+ ( '  + ( − 1)¨ = 0. + ô ï ln÷ ( õ ï )ñó ò + ö [ ô ln÷ ( õ Particular solution: 0 =  − v .

ï

ï 2 ñ óò ò ó ï 2 ñ óò ò ó

+ ï (ô

ïƒ÷

ï (ï

ïƒ÷

+ õ ln ï )ñó ò + õ ( ô

+ ô )ñ„ó ò ò ó + ï ( õ ln ï + ö )ñ„ó ò +

+ ô ï 2 ln ÷ ( õ Particular solution:

ï 4 ñ óò ò ó

) – ö – 1]ñ = 0. ln ï – ln ï + 1)ñ = 0.

= exp  − 12  ln2   .

0

Particular solution:

õÿñ

= 0.  ln  + ( − 1  .  + M P

= exp O − X

0

0

+ [ ô ( ö – ï ) ln ÷ ( õ =  exp( (  ).

0

=  − .

ï )ñ ó ò

ï

) – ö 2 ]ñ = 0.

( ô ln ï + õ )ñ„ó ò ò ó + ( ö ln ï + ý )ñ„ó ò + [( ö – ô ) ln ï + ý – õ ]ñ = 0. Particular solution:

ï

ln ï

37.

ñóò ò ó

– ùñó ò – ô

Solution:

ï

ln( ô

38.

ï )„ñ óò ò ó

Solution:

ï

ln2 ï

ñóò ò ó

2



ï

(ln ï )2÷

+1

ñ

= 0.

= L 1   Ž + L 2  −  Ž , where  = X ln  – [ù ln( ô

ï

)+þ

]ñ„ó ò –

õ 2 ï 2÷

+1

ln2ú

+1

M (ô

 .

ï )ñ

= 0.

+ = L 1   Ž + L 2  −  Ž , where  = X   ln (  )  . M + (ô

ï

+ 1) ln ï

The substitution ˆ = X 40.

£ 

”

1

ï 2 ñ óò ò ó

34.

39.

ø

2+ ¡ functions. Á ( H ) and n Á ( H ) are the Bessel

Particular solution:

36.

1

= 0.

+ ï ñ ó ò + ( ô ln2÷ ï + õ ln ÷ –1 ï )ñ = 0. The substitution ˆ = ln  leads to an equation of the form 2.1.2.10:
Šª
Š + ( ˆ 2  + 'ˆ 

ï 2 ñ óò ò ó

33.

35.

ï )ñ

ln ÷ ( ô ï )ñó ò ò ó + ( õ 2 – ï Particular solution:

M



ln 

ñ„óò

+

õï ñ

= 0.

leads to a constant coefficient linear equation: 
Šª
Š +  
Š + 

= 0.

)ñó ò + ( ï + õ )ñ = 0. 0 =  − . 2

© 2003 by Chapman & Hall/CRC

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SECOND-ORDER DIFFERENTIAL EQUATIONS

2.1.6. Equations Containing Trigonometric Functions 2.1.6-1. Equations with sine. 1.

+ ô 2 ñ = õ sin( ï ). Equation of forced oscillations.


ñóò ò ó

=

Solution:

2.

3.

4.






6. 7.

ñóò ò ó

+

ô

sin( õ

ï )„ñ óò

ïƒ÷

+ ö [ô 0

ï

sin( õ

= exp O −

(

)–ö

 

B +1

+ ô sin ÷ ( õ ï )ñó ò + ö [ ô sin÷ ( õ  Particular solution: 0 =  − v .

ï

ñóò ò ó

ï 2÷ +1

= 0.

+ ( ô ï + õ ) sin ÷ ( ö ï )ñó ò – ô sin ÷ ( ö Particular solution: 0 =  +  .

ï )ñ

= 0.

ñóò ò ó

+

ô ïƒ÷

sin ú ( õ

ï )ñ óò

Particular solution:

0

+ ö [ô

sinú ( õ

–1

ïƒ÷ +

= exp O −

(

ï )ñ

sin ú ( õ

, +1

 

+1

= 0.

ï

)–ö

ï ñóò ò ó

+ (ô

ïƒ÷

+1

ô ïƒ÷ ( õ

+ õ sinú'ï )ñó ò +

Particular solution:

0

= exp O −

   B +1

ï )ñ

+ ô ïƒ÷ sinú ( õ Particular solution:

ï ñóò ò ó

ï ) ñ óò 0

– [ô (ö  =  v .

ï

+

ü ï –1 ]ñ

= 0.

= 0.

sinú'ï + ù )ñ = 0.

+1

P

.

+ ( ô ï ÷ + õ ï sin ú ï )ñ ó ò + [ õ ( ô ï ÷ – 1) sin ú Particular solution: 0 =  exp(−   B ).

ï ñ óò ò ó

ï 2

.

P

+ [( ô ï 2 + õ ï ) sin÷ ( ö ï ) + 2]ñó ò + õ sin÷ ( ö Particular solution: 0 =  +   .

ï ñóò ò ó

]ñ = 0.

.

P

ï )ñ

ïƒ÷

–1

) – ö ]ñ = 0.

+ [ ô sin ÷ ( õ ï ) + ö ]ñ ó ò + ô„ö sin÷ ( õ  Particular solution: 0 =  − v .

ñóò ò ó

ïƒ÷

+ù

ñ óò ò ó

ñóò ò ó

13.

, we obtain the Mathieu equation 2.1.6.29:



+ ( ô sin2 ï + õ )ñ = 0. Applying the formula 2 sin2  = 1 − cos 2 , we obtain the Mathieu equation 2.1.6.29: 

 + ( 1  +  − 1  cos 2 ) = 0. 2 2

9.

12.

1 2

2

ñóò ò ó

+ ô ïƒ÷ sin ú ( õ ï )ñó ò – ô Particular solution: 0 =  .

11.

cos(  ) +

+ [ ô sin( ï ) + õ ]ñ = 0. Applying the substitution Œ  = 2ˆ + 
Šª
Š + (4 Œ −2 cos 2ˆ + 4 Œ −2 ) = 0.

8.

10.

2

ñóò ò ó

Particular solution: 5.

1

sin(  ) + L



sin( Œ  ) if  ≠ Œ ,  −Œ 2  L 1 sin(  ) + L 2 cos(  ) −  cos(  ) if  = Œ . 2

L

+ 1)ïw÷

–1

sinú ( õ

ï ï

+

ôù ï ÷

)+ö

2

ï

–1

]ñ = 0.

+ 2 ö ]ñ = 0.

© 2003 by Chapman & Hall/CRC

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261

2.1. LINEAR EQUATIONS

14.

ï ñóò ò ó

+ [ô

ïƒ÷

ï

sin ú ( õ

ï 2 ñ óò ò ó

+ ï ( ô sin÷

ï

ï 2 ñ óò ò ó

+ ï ( ô sin÷

0

ï

ï 2 ñ óò ò ó

+

ô ïƒ÷

0

19.

ï 4 ñ óò ò ó

ï

+ [ ô sin( #

ï 4 ñ óò ò ó

+

ô ï

2

sin ÷ ( õ

−

ï ) ñ óò 0

sin2 ï

ñóò ò ó

1

ï

.

ïƒ÷

+ ö [ô

–1

−

) + õ ]ñ = 0.

sin(2 ï ) ñ„ó ò ò ó – ñó ò + 2 ô =L

sin ú ( õ

ï )ñ

= 0.

– õ )ñ = 0.

.

The transformation ˆ = 1  , ¨ ¨2Šª

Š + [  sin( Œ ˆ ) +  ]¨ = 0.

Solution: 21.

−

v. 0 = 

Particular solution: 20.

=

ï )ñ óò

sin ú ( õ

Particular solution: 18.

=

ï

+ õ )ñó ò + õ ( ô sin÷

Particular solution: 17.

v. 0 = 

+ 1)ñó ò + õ ( ô sin÷

Particular solution: 16.

–1

1−

Particular solution: 15.

) + ö ]ñó ò + ô ( ö – 1) ïƒ÷

– 1)ñ = 0.

ï

sinú ( õ

=  

) – ö – 1]ñ = 0.

leads to a linear equation of the form 2.1.6.2:

+ [ ô ( ö – ï ) sin ÷ ( õ

ï

=  exp( (  ).

) – ö 2 ]ñ = 0.

2

sin2 ï

ñ

sin( ‡ ) + L

2

= 0.

cos( ‡ ), where ‡ = X

£ tan  M

 .

+ ôñ = 0.

This is a special case of equation 2.1.6.23. 22.

sin2 ï

ñóò ò ó

– [ ô sin2 ï + ù (ù – 1)]ñ = 0, 1 = sin  ‹  O sin 

Solution: 23.

sin2 ï

ñóò ò ó

M M



3P

ù

= 1, 2, 3,

 2  2 Ý L 1   + L 2  −   .

uuu

+ ( ô sin2 ï + õ )ñ = 0.

Set  = 2ˆ . Applying the trigonometric formulas sin 2ˆ = 2 sin ˆ cos ˆ and  =  (sin 2 ˆ + cos2 ˆ )2 and dividing both sides of the equation by sin 2  , we arrive at an equation of the form 2.1.6.131: Šª

Š + (  tan2 ˆ +  cot2 ˆ + 4  + 2  ) = 0. 24.

sin2 ï

ñóò ò ó

– {[( ô

õ

2 2

Particular solution: 25.

ï

[ ô sin(

)+õ

ï

0

sin ÷ ( ô

ï ) ñ óò ò ó

+ (ï

Particular solution: 27.

( ô sin ÷

ï

 =    sin   (cos  +  sin  ).

+ ö ]ñ„ó ò ò ó + ô

Particular solution: 26.

– ( ô + 1)2 ] sin2 ï + ô ( ô + 1) õ sin 2 ï + ô ( ô – 1)}ñ = 0.

0 2

sin(

ï )ñ

= 0.

=  sin( Π ) + ' + ( .

– õ 2 )ñó ò – ( ï + õ )ñ = 0. 0

= − .

+ õ )ñó ò ò ó + ( ö sin ÷

Particular solution:

2

0

ï −



+ ý )ñó ò + [( ö – ô ) sin ÷

ï

+ý –

õ ]ñ

= 0.

=  .

© 2003 by Chapman & Hall/CRC

Page 261

262

SECOND-ORDER DIFFERENTIAL EQUATIONS TABLE 19 The general solution of the Mathieu equation 2.1.6.29 expressed in terms of auxiliary periodical functions Y 1 ( ) and Y 2 ( )

Constraint 1( 1(

)>1

 L 1  2  Y 1 ( ) + L 2 

−2

Y

2(

) < −1

 L 1  2 ~ Y 1 ( ) + L 2 

−2

~Y

2(





General solution = ( )

| 1 ( )| < 1



Y

Period of 1 and Y 2

“ is a real number

 )



 )

2

L 1 Y 1 ( ) + L 2 "Y 2 ( )



“ =  + 12 ™ , ™ 2 = −1,  is the real part of “





“ = ™›@ is a pure imaginary number, cos(2 @ ) = 1 ( )



“ =0

( L 1 cos @! + L 2 sin @! ) Y 1 ( ) + + ( L 1 cos @! − L 2 sin @! ) Y 2 ( )

1( ) = A 1

Index





2.1.6-2. Equations with cosine. 28.

+ ô 2 ñ = õ cos( ï ). Equation of forced oscillations.


ñóò ò ó

Solution: 29.

=






 cos( Œ  ) if  ≠ Œ ,  2−Œ 2  L 1 sin(  ) + L 2 cos(  ) +  sin(  ) if  = Œ . 2 L

1

sin(  ) + L

2

cos(  ) +

+ ( ô – 2 cos 2 ï )ñ = 0. The Mathieu equation.

ñóò ò ó

1 b . Given numbers  and  , there exists a general solution ( ) and a characteristic index “ such that ( + ) =  2 zf ( ).



For small values of  , an approximate value of “ can be found from the equation: cosh( “ ) = 1 + 2 sin2 



1 2



£   +

  (1 −  ) £ 2



sin  £   + [ (  4 ).



1 (  ) is the solution of the Mathieu equation satisfying the initial conditions 
(0) = 0, the characteristic index can be determined from the relation: 1

If

cosh(2 “ ) =



1(

= 1 and

1 (0)

).



The solution 1 ( ), and hence “ , can be determined with any degree of accuracy by means of numerical or approximate methods. The general solution differs depending on the value of 1 ( ) and can be expressed in  terms of two auxiliary periodical functions Y 1 ( ) and Y 2 ( ) (see Table 19). 2 b . In applications, of major interest are periodical solutions of the Mathieu equation that exist for certain values of the parameters  and  (those values of  are referred to as eigenvalues). The most important solutions are listed in Table 20. The Mathieu functions possess the following properties: ce2  ( , −  ) = (−1)  ce2  O −  ,  , P 2 se2  ( , −  ) = (−1) 

−1

se2  O −  ,  , 2 P

 , −  ) = (−1)  se2 

+1

O −  ,



 , −  ) = (−1)  ce2 

+1

O −  ,



ce2 

+1 (

se2 

+1 (

2 2

,

P

P

.

© 2003 by Chapman & Hall/CRC

Page 262

263

2.1. LINEAR EQUATIONS

TABLE 20 Periodical solutions of the Mathieu equation ce  = ce  ( ,  ) and se  = se  ( ,  ) (for odd B , the functions ce  and se  are 2 -periodical, and for even B , they are -periodical); certain eigenvalues   =   (  ) and  =   (  ) correspond to each value of the parameter  ; B = 0, 1, 2,  Recurrence relations for coefficients

Mathieu functions

7

# 22 =  2 # 20 ;  # =( 2 −4)# 22 −2# 20 ; # 22+ +2 =( 2 −4¡ 2)# 22+ − # 22 + −2, ≥2 ¡ 2  +1 # 3 =( 2 +1−1−  )# 12 +1; # 22+ +1+3 =[ 2 +1−(2¡ +1)2] 2  +1 ≥1 × # 22 + +1 +1 − # 2 + −1, ¡ $ 42 =(  2 −4)$ 22 ; $ 22+  +2 =(  2 −4¡ 2)$ 22+  − $ 22+  −2, ≥2 ¡ $ 32 +1 =(  2 +1−1−  )$ 12 +1; $ 22+  +1+3 =[  2 +1−(2¡ +1)2] 2  +1 ≥1 × $ 22+  +1 +1 − $ 2 + −1, ¡

+1(

7

se2  ( ,  )= ÷

+

se2 

2 2

7  ,  )= ÷ # 22 + + =0

+1(

+1 +1 cos

V (2 +1) W ¡

$ 22+  sin(2  ), ¡ =0

se0 =0

7  ,  )= ÷ $ 22+  + =0

+1 +1 sin

(#

2 4

ce2  ( ,  )= ÷ # + cos(2  ) ¡ + =0

ce2 

Normalization conditions

V (2 +1) W ¡

 )2 + 7  # + ÷ =0 2 if B =¦ 1 if B 2 0

7

+÷

=0

# 22 +

2 2

  +

2

=0 ≥1

+1 2 +1 =1



7

2 2 ÷+ $ 2 +  =1 =0

7

$ 22+  +÷ =0

+1 2 +1 =1



Selecting a sufficiently large and omitting the term with the maximum number in the ¡ recurrence relations (indicated in Table 20), we can obtain approximate relations for the eigenvalues   (or   ) with respect to parameter  . Then, equating the determinant of the corresponding homogeneous linear system of equations for coefficients # + (or $ +  ) to zero, we obtain an algebraic equation for finding   (  ) (or   (  )). For fixed real  ≠ 0, the eigenvalues   and   are all real and different, while:



if if

> 0 then < 0 then





0



0

0, #

≠ 0:

{ = , ( L   * − L  −  * ) +  ·L cos  ÑU , L 1   * + L 2  −  * + L 3 sin  ÑU , 2 1 2 3 2 2 1J 2 1J 2 2 1J 2 1J 2 # 1 + 3 # 2 ) + # 1 ]} ,   = { 3 [( # 1 + 3 # 2 ) − # 1 ]} ; the constants L 1 , L 2 , where , = and L 3 are related by the constraint 4 , 2 L 1 L 2 +   2 L 32 = 0. 2 b . For − # 21 < 3 # 2 < 0, # 1 > 0: { = L U  1 + L U −  1 + L U  2 + L U −  2 , { = , ( L U  1 − L U −  1 ) + , ( L U  2 − L U −  2 ), 1 1 2 3 4 2 1 1 2 2 3 4 where , 1 = { 23 [ # 1 + ( # 21 + 3 # 2 )1 J 2 ]}1 J 2 , , 2 = { 32 [ # 1 − ( # 21 + 3 # 2 )1 J 2 ]}1 J 2 ; the constants L 1 , L 2 , and L 3 are related by the constraint ( L 1 L 2 + L 3 L 4 )( # 21 + 3 # 2 )1 J 2 + ( L 1 L 2 − L 3 L 4 ) # 1 = 0. 3 b . For − # 21 < 3 # 2 < 0, # 1 < 0: { = L sin   U + L cos   U + L sin   U , { =   ( L cos   U − L sin   U ) +   L cos   U , 1 1 1 2 1 3 2 2 1 1 1 2 1 2 3 2 2 2 2 1J 2 1J 2 2 1J 2 1J 2 where   1 = {− 3 [ # 1 + ( # 1 + 3 # 2 ) ]} ,   2 = {− 3 [ # 1 − ( # 1 + 3 # 2 ) ]} ; the constants L 1 , L 2 , and L 3 are related by the constraint   12 ( L 12 + L 22 ) −   22 L 32 = 0. 4 b . For # 21 + 3 # 2 = 0, # 1 > 0: { = ( L + L U )  * + ( L + L U ) −  * , { = ( , L + L + , L U )  * − ( , L − L + , L U ) −  * , 1 1 2 3 4 2 1 2 2 3 4 4 where , = ( 32 # 1 )1 J 2 ; the constants L 1 , L 2 , and L 3 are related by the constraint L 2 L 4 + ( L 1 L 4 − L 2 L 3 )( 61 # 1 )1 J 2 = 0. 5 b . For # 21 + 3 # 2 = 0, # 1 < 0: { = ( L + L U ) sin  ÑU + L U cos  ÑU , { = ( ·L + L +  ·L U ) cos  ÑU + ( L −  ·L U ) sin  ÑU , 1 1 2 3 2 1 3 2 2 3 2 1J 2 where   = (− 3 # 1 ) ; the constants L 1 , L 2 , and L 3 are related by the constraint L 22 + L 32 + L 1 L 3 (− 32 # 1 )1 J 2 = 0. 6 b . For 3 # 2 < − # 21 : { =   * ( L sin  ÑU + L cos  ÑU ) + L  −  * sin  ÑU , 1 1 2 3 { =   * [( , L +  ·L ) cos  ÑU + ( , L −  ·L ) sin  ÑU )] + L  −  * (  cos  ÑU − , sin  ÑU ), 2 2 1 1 2 3 1 1 1J 2 1J 2 1J 2 1J 2 where , = { 3 [ # 1 + (−3 # 2 ) ]} ,   = { 3 [− # 1 + (−3 # 2 ) ]} ; the constants L 1 , L 2 , and L 3 are related by the constraint L 2 # 1 + L 1 (− # 2 − 3 # 2 )1 J 2 = 0. 2

1

1 = { 32 [(

1

© 2003 by Chapman & Hall/CRC

Page 327

328 54.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

=


1ñ

)

–1 3

+


2 ï 2ñ

)

–5 3

.

Solution in parametric form:

{

 = 55.

ñóò ò ó

=


1ï

) ñ

)

–8 3 –1 3

+


2ï

) ñ

)

–10 3 –5 3

Solution in parametric form:

{

]

{ ñóò ò ó

1

=

™

2

=

™

=


1ñ

L 1 Ÿ* + L 2  −Ÿ* + L 3 U L 1 sin  ÑU + L 2 cos  ÑU + L 3 U   ( L 1 Ÿ * − L 2  −Ÿ* ) + L 3   ( L 1 cos  ÑU − L 2 sin  ÑU ) + L

)

–1 3

+


2ñ

)

–5 3

if # if #

J

where the constants L

ñ óò ò ó

=


1ï

L 2 , and L

1,

3( # 3( #

) ñ

)

–8 3 –1 3

+

L

1

L

2 3 2 3

1


2ï

3

2)

) ñ

)

–4 3 –5 3

L 2 , and L

3( # 3( #

ñóò ò ó

=


1ñ

)

–1 3

+


2ï ñ

1

L

1

L

2 3 2 3

)

–5 3

{

1

=

1,

+ 16 # 21 L 2 2) + 4# 1(L

+# +#

 = 1,

1

where

  = | 43 # 1 |1 J 2 ,

> 0, < 0,

where

  = | 43 # 1 |1 J 2 .

3

J

3 2 2 ,

L

1 2 =0 2 2 1 + 2) =

L

if # 0 if #

1 1

> 0, < 0.

.

{

−1 1 ,

{

=

{

J

−1 3 2 1 2 ,

are related by the constraint

+ 16 # 21 L 2 2) + 4# 1(L

+# +#

{

are related by the constraint

Solution in parametric form: where the constants L

1

> 0, < 0,

.

 =

58.

{

−1 3 2 1 2 .

1

if # if #

3

Solution in parametric form:

57.

{

=

In the solutions of equations 56–59, the following notation is used:

{

56.

J

3 2 2 .

. −1 1 ,

 =

{

=

1,

L

1 2 =0 2 2 1 + 2) =

2)

L

if # 0 if #

1 1

> 0, < 0.

.

Solution in parametric form:

 = where the constants L 3# 59.

ñóò ò ó

=


1ï

) ñ

)

–8 3 –1 3

+

2 3 2 3

L

1 1

−

1

L 2 , and L

1,

3#

{

L


2ï

2

3

1

L 1L

2 1(

2 1

L

) ñ

U 2,

{

= O

−

2

#

2#

2 1

J

3 2

U P

,

are related by the constraint

2 1

+ 16 # + 4#

#

4#

)

2

+L

–7 3 –5 3

+

9 16

2 2)

#

+

−2 2 1 2 =0 −2 2 9 2 = 16 1

#

#

#

if #

1

> 0,

0 if #

1

< 0.

.

Solution in parametric form:

 = O

{

1

−

#

4#

2 1

U

2

P

−1

,

= O

{

1

−

#

4#

2 1

U

2

P

−1

O

{

2

−

#

2#

2 1

U

J

P

3 2

,

© 2003 by Chapman & Hall/CRC

Page 328

Ï Ò˜Í Í Ò

2.4. EQUATIONS OF THE FORM

where the constants L

]

L 2 , and L

1,

3#

1

L

3#

1

L

2 3 2 3

+ 16 # + 4#

 1Ë E Ï ±

=

1

1

 2Ë E Ï ±

+

2

329

2

are related by the constraint

3 2 1

L 1L

2

2 1(

2 1

+L

L

9 16

+

2 2)

#

+

−2 2 1 2 =0 −2 2 9 2 = 16 1

#

#

#

if #

1

> 0,

0 if #

1

< 0.

In the solutions of equations 60–67, the following notation is used:

 =

A (4 < ü

U =X

− 1),

3

ü

M A (4
0:

 = L 1 [cosh U − sin(U + L 2 )]−1 J 2 [sinh U − cos(U + L 2 )],  2J 5 . where # = 5  −2  2 J 3 O 6 P 33.

= L

6 1 [sinh

U + cos(U + L 2 )]3 ,

=
ï –5 ) 2 ñ –1 ) 2 (ñ ó ò )1 ) 2 . Solution in parametric form:

ñ óò ò ó

 = L

−1 1 [cosh(

U + L 2 ) cos U ]−1 ,

= L

1

cosh(U + L

2 ) cos

U [tanh(U + L 2 ) − tan U ]2 ,

where # = −4 . 34.

=
ï –5 ) 3 ñ 2 (ñó ò )3 . 1 b . Solution in parametric form with # > 0:

ñóò ò ó

 = L 13 [cosh(U + L 2 ) cos U ]3 J 2 , = L 12 cosh(U + L 2 ) cos U [tanh(U + L 2 ) + tan U ], 3 8 J 3 −4   . where # = 16 2 b . Solution in parametric form with # < 0:  = L 3 [cosh U − sin(U + L 2 )]3 J 2 , = L 2 [sinh U + cos(U + L 2 )], 1

where # = 35.

1

 J 

− 43 8 3 −4 .

=
ï –2 ) 3 ñ (ñó ò )8 ) 5 . 1 b . Solution in parametric form with # > 0:

ñóò ò ó

 = L 16 cosh3 (U + L 2 ) cos3 U [tanh(U + L 2 ) + tan U ]3 , = L 1 [cosh(U + L 2 ) cos U ]1 J 2 [tanh(U + L 2 ) − tan U ], where # = 5  2 J 3 

−2

O



J

2 5

. 12 rP 2 b . Solution in parametric form with # < 0:

U + cos(U + L 2 )]3 ,  2J 5 . where # = −5 2 J 3  −2 O 6 ·P  = L

36.

6 1 [sinh

= L

1 [cosh

U − sin(U + L 2 )]−1 J 2 [sinh U − cos(U + L 2 )],

=
ï –1 ) 2 ñ –5 ) 2 (ñó ò )5 ) 2 . Solution in parametric form:

ñóò ò ó

 = L

1

cosh(U + L

2 ) cos

U [tanh(U + L 2 ) − tan U ]2 ,

= L

−1 1 [cosh(

U + L 2 ) cos U ]−1 ,

where # = 4  .

© 2003 by Chapman & Hall/CRC

Page 345

346

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

37.

=


ï

) ñ

) (ñ óò

–5 3 –10 3

)3 .

1 b . Solution in parametric form with # > 0:

 = L 1 [cosh(U + L 2 ) cos U ]1 J 2 [tanh(U + L 2 ) + tan U ]−1 , = L 1−2 [cosh(U + L 2 ) cos U ]−1 [tanh(U + L 2 ) + tan U ]−1 ,  J  J

3 8 3 4 3 . 16

where # =

2 b . Solution in parametric form with # < 0:

 = L 1 [cosh U − sin(U + L 2 )]3 J 2 [sinh U + cos(U + L 2 )]−1 , = L 1−2 [sinh U + cos(U + L 2 )]−1 , where # = − 43  8 J 3  4 J 3 .

]

In the solutions of equations 38–45, the following notation is used:

8

« = exp(3U ),

8

38.

3

=2

ñ óò ò ó

8

8

1( 2)

=


ï ñ

1

=« +L


− ( 8 )
8 1

*

*

2

2

−

7

.

) ( ñ ó ò )9 )

–1 2

sin  £ 3 U  ,

8 8

8

1 2,

4

=2

8

8

2

8

= 2« − L

1( 3)

2

sin  £ 3 U  + £ 3 L


− 5( 8 )
8 1

*

*

3

+

2

8 8

cos  £ 3 U  ,

1 3.

Solution in parametric form:

J 8

−1 6

 = L 1 « 39.

ñóò ò ó

=


ï ñ

J 8

−1 2 2, 1

) (ñ óò )9 )

–13 8

7

8 1

= L

J 8 2, 3

−4 3

«

where

# = 7

J

  J O

−2 1 2

2 7

64 "P

.

.

Solution in parametric form:

 = L 40.

ñóò ò ó

=

25 1


ï

J 8

J 8

−1 2 4, 1

−5 6

«

) ñ

) (ñ óò

)18 )

–7 2 –1 2

= L 13

32 1

−16 15

8 8J 5 ,

J 8

8

J

«

3

where

# = 7

J

25   J O

−2 13 8

2 7

256 P

.

.

Solution in parametric form:

 = L 41.

ñóò ò ó

=

−1 1


ï

J

1 15

«

8

8 J

−1 2 5 1 3 ,

) ñ (ñ óò

–7 5

25 1

= L

«

−5 3

= L

4 1

−1 2 1 4,

where

J

 # = −208 5 J 2  1 J 2 O

5 13

25 "P

.

)3 .

Solution in parametric form:

 = L 42.

ñóò ò ó

=


ï

5 1

«

) ñ (ñ óò

–1 2

J 8 5J 2, 1

−5 6

«

J 8 3,

−2 3

where

5 # = − 1024 

J 

12 5 −3

.

)12 ) 7 .

Solution in parametric form:

 = L

8 1

«

J 8 2, 3

−4 3

= L

1

«

J 8

−1 6

J 8

−1 2 2, 1

where

# = −7 1 J 2 

−2

O



64 rP

J

2 7

.

© 2003 by Chapman & Hall/CRC

Page 346

2.5. GENERALIZED EMDEN–FOWLER EQUATION

43.

=
ï –7 ) 5 ñ –13 ) 5 (ñó ò )3 . Solution in parametric form:

J 8 5J 2 8

−1 6

{

−1 3 ,

1

J 8

−4 1

= L

=
ï –13 ) 8 ñ (ñ ó ò )12 ) 7 . Solution in parametric form:

(Ï

ÒÍ

)

Ë

347

2 3

«

−1 3 ,

5 # = − 1024 

where

J  J .

12 5 3 5

32 1

8 8J 5 ,

J

−16 15

«

= L

3

25 1

«

J 8

25  2 J 7 . 256  P

J 8

where

# = −7

8 J

where

# = 208 1 J 2  5 J 2 O

−1 2 4, 1

−5 6

J 

13 8 −2

O

=
ï –1 ) 2 ñ –7 ) 2 (ñ ó ò )21 ) 13 . Solution in parametric form:

ñ óò ò ó

 = L ]

 ËEϱ

ñ óò ò ó

 = L 45.

=

ñóò ò ó

 = L 1 «

44.

Ï Ò˜Í Í Ò

25 1

«

J 8

8

−5 3

−1 2 1 4,

−1 1

= L

«

8

J

1 15

−1 2 5 1 3 ,

In the solutions of equations 46–49, the following notation is used:

{

{

J



5 13

25 rP

{

.

{ {

U + L 2) cos U , 2 = tanh( U + L 2) + tan U , 3 = tanh( U + L 2) − tan U , 4 = 3 2 3 − 4, 9 9 9 9 9 9 92 1 = cosh U − sin(U + L 2), 2 = sinh U + cos(U + L 2), 3 = sinh U − cos(U + L 2), 4 = 3 2 3 − 2 1.

46.

1 = cosh(

=
ï ñ –10 ) 7 (ñó ò )7 ) 5 . 1 b . Solution in parametric form with # < 0:

ñóò ò ó

 = L

{

{

J

9 3 2 4, 1 1

{

{

J

J

14 7 3 7 3 1 1 2 ,

= L

where

5 9

J

9  J O

# =− 

−2 10 7

5  9

9  J O

2 5

28  P

.

2 b . Solution in parametric form with # > 0:

9 J 9

9 −1 2 4, 1 1

 = L 47.

9 J

14 7 3 1 2 ,

# =

where

=
ï –5 ) 2 ñ –1 ) 2 (ñó ò )13 ) 10 . Solution in parametric form:

−2 10 7

J

2 5

14 "P

.

ñóò ò ó

 = L 48.

= L

{

{

J

J

−1 −1 3 2 3 1 1 2 ,

= L

{ {

9 3 2 1 1 4,

 # = −20 ( )1 J 2 O

where

J

3 10

9  P

=
ï –10 ) 7 ñ (ñó ò )8 ) 5 . 1 b . Solution in parametric form with # > 0:

.

ñóò ò ó

 = L

{

J

{

J

14 7 3 7 3 1 1 2 ,

= L

{

J

{

9 3 2 4, 1 1

where

# =

5  9

J 

9 2J 5 . 28 rP

10 7 −2

O

J 

9 2J 5 . 14 rP

2 b . Solution in parametric form with # < 0:

 = L 49.

9 J

14 7 3 1 2 ,

= L

9 J 9

9 −1 2 4, 1 1

where

5 9

# =− 

=
ï –1 ) 2 ñ –5 ) 2 (ñó ò )17 ) 10 . Solution in parametric form:

10 7 −2

O

ñóò ò ó

 = L

{ {

9 3 2 1 1 4,

= L

{

J

{

J

−1 −1 3 2 3 1 1 2 ,

where

 # = 20  ( )1 J 2 O

9 ·P

J

3 10

.

© 2003 by Chapman & Hall/CRC

Page 347

348

]

SECOND-ORDER DIFFERENTIAL EQUATIONS

In the solutions of equations 50–65, the following notation is used:

² =

A (4U 3 − 1), ü

U

where ® (U ) = X 50.

²

U M



1

=U

−1

( ²u

1



− 1),

−3 1

² ,

= L

5 −1 1, 1

U



where

2 −2 ( 3

# =T



=
ï ñ –1 ) 2 (ñ ó ò )7 ) 4 . Solution in parametric form:

3

= 4U 

2 1

Tç

2 2,

T 6   )3 J 4 .

−1 2, 1



= L

5 2 −2 1 1 ,

where

# =T

J 

where

# =T

U 

=
ï –1 ) 2 ñ (ñ ó ò )8 ) 7 . Solution in parametric form:

 J A 3   )3 J 4 .

2 −2 1 2 ( 3



ñ óò ò ó

−16 2 1 3,



5 −3 2 2, 1 1

= L



=
ï –7 ) 6 ñ –1 ) 2 (ñ ó ò )2 ) 3 . Solution in parametric form:

7  16

J 

−1 2 −1

16  1 J 7 . 3 þP

O

ñ óò ò ó

5 −3 6 1 1 3,





= L

=
ï –3 ) 4 ñ (ñ ó ò )8 ) 7 . Solution in parametric form:

−3 1 1 ( 2 3



 

2 2 1) ,

− 8

where

J  J (   )2 J 3 .

−5 6 3 2

# = T 4

ñ óò ò ó

 = L 55.

2

ñ óò ò ó

 = L

54.



=
ï ( ñ ó ò )7 ) 4 . Solution in parametric form:

 = L 53.

U2Tç² ,

ñ óò ò ó

 = L 52.

2

is the incomplete elliptic integral of the second kind in the form of Weierstrass.

 = L 51.

= 2UÈ® (U ) + L

ñóò ò ó

=


ï

−32 −4 1 3 ,



–4

J  

3 −3 2 ( 2 3 1 1

= L



7  32

2 1 ),

where

# =T

5 −1 1, 1

where

# = A 6 5 

− 8

J 

−1 4 −1

O

32  1 J 7 . 3 þP

(ñó ò )3 .

Solution in parametric form:

 = L 56.

U

= L

U



−2

.

=
ñ (ñ ó ò )5 ) 4 . Solution in parametric form:

ñ óò ò ó

5 −1 1, 1

 = L 57.

2 −1 , 1

ñóò ò ó

=


ï

–4

ñ (ñ óò

U



−3 1

= L

² ,

where

# =A

2 −2 ( 3



T 6   )3 J 4 .

)3 .

Solution in parametric form:

 = L

58.

3 −1 1 1 ,



= L

=
ï –1 ) 2 ñ (ñ ó ò )5 ) 4 . Solution in parametric form:

5 1

U 

−1 1 ,

where

# = A 6 5 

−3

.

ñ óò ò ó

 = L

5 2 −2 1 1 ,

U 

= L

−1 2, 1



where

# =A

 J 

2 1 2 −2 ( 3

A 3   )3 J 4 .

© 2003 by Chapman & Hall/CRC

Page 348

2.5. GENERALIZED EMDEN–FOWLER EQUATION

59.

=
ï –1 ) 2 ñ –4 ) 3 (ñó ò )3 . Solution in parametric form: 4 2 1 2,



5 −3 2 1 1 2,





Ë

349

 J 

J

4 3 2 −2 3 . 3

# =T

where

9 −3 1 1 ,

= L



 J 

J

4 3 2 −5 6 . 3

# =T

where

J 

5 −3 2 2, 1 1



= L

−16 2 1 3,



# =A

where

=
ï –7 ñ (ñó ò )3 . Solution in parametric form:

7  16

J O 16  1 J 7 . 3 P

−1 −1 2



J

3 1 2 1 1 ,



8 1 3,

= L



where

# =T

=
ï –7 ñ 3 (ñ ó ò )3 . Solution in parametric form:

3 8 −3 . 64

 

ñ óò ò ó

J 

5 1 2 −1 1 1 3 ,



8 −1 1 3 ,

= L



3 8 −5 . 64

# =T

where

 

=
ï –1 ) 2 ñ –7 ) 6 (ñó ò )7 ) 3 . Solution in parametric form:

ñóò ò ó

−3 1 ( 2 3

 

− 8

2 2 1) ,

J  

− 8

2 1 ),

5 −3 6 1 1 3,

= L

=
ï ñ –3 ) 4 (ñ ó ò )13 ) 7 . Solution in parametric form:





# = A 4 3 J 2 

where

J (   )2 J 3 .

−5 6

ñ óò ò ó

 = L ]

)

ñóò ò ó

 = L 1 

65.



=
ï ñ –1 ) 2 (ñó ò )13 ) 7 . Solution in parametric form:

 = L 64.

ÒÍ

(Ï

ñóò ò ó

 = L 63.

9 3 1 1,

= L

=
ï –1 ) 2 ñ –7 ) 6 (ñ ó ò )3 . Solution in parametric form:

 = L 62.

 ËEϱ

ñ óò ò ó

 = L 61.

=

ñóò ò ó

 = L 60.

Ï Ò˜Í Í Ò

3 −3 2 ( 2 3 1 1



−32 −4 1 3 ,

= L



where

7  32

# =A

J O 32  1 J 7 . 3 P

−1 −1 4



In the solutions of equations 66–95, the following notation is used:

U =X

M

ü

A (4
( + 1) −1

.

(16)



If we pass on to the new independent variable H =  , the solutions are single-valued functions of . Solutions of the fifth Painlev´e transcendent (16) corresponding to different values of parameters are related by: ( H , ƒ , ý , = , > ) = (− H , ƒ , ý , −= , > ), 1 . (H , ƒ , ý , = , > ) = ( H , − ý , − ƒ , −= , > ) 2 b . On setting H = 

g in (16), we obtain

3 −1 2

( g
) + ( − 1)2 Oƒ gg =

+

2 ( − 1)

ý

+=

P

> ( + 1) 2g  g +  . −1

(17)

If = = > = 0, equation (17) is reduced, by integration, to a first-order autonomous equation:


g = ( − 1) ü 2 ƒ which is readily integrable by quadrature. If the condition = = £ −2 >3 1 + is satisfied, any solution of the Riccati equation

H ‘
= £ 2ƒ

2

ü

2

+L

− 2ý ,

−2ý − £ 2 ƒ 

+  £ −2 >hH − £ 2 ƒ −

ü

−2ý 

+

ü

−2ý

(18)

is simultaneously a solution of the fifth Painlev´e transcendent (16). Equation (18) can be reduced to the degenerate hypergeometric equation 2.1.2.70.

© 2003 by Chapman & Hall/CRC

Page 436

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

437

2.8.2-7. Sixth Painlev´e transcendent. 1 b . The sixth Painlev´e transcendent has the form ‘
‘'

‘ = 1 O 1 + 1 + 1 ( ‘
)2 − O 1 + 1 + 1 2 −1 − HrP H H −1 − HrP H H −1 H ( H − 1) ( − 1)( − H ) ƒ +ý 2 += + +> S . (19) H 2 ( H − 1)2 Q ( − 1)2 ( − H )2 In equation (19), the points H = 0, H = 1, and H = F are fixed logarithmic branch points. Painlev´e found two integrable cases of the equation. First, if ƒ = ý = = = > = 0, the general solution of equation (19) has the form: = « (L

 

1 1

+L

 

2 2,

H ),

where « (‡ , H ) is the elliptic function, defined by the integral

‡ = X•” £ 0

M

( − 1)( − H )

,

(20)

with periods 2  1 and 2  2 , which are functions of H . Secondly, if ƒ = ý = = = 0, > = 12 , the general solution of equation (19) has the form: = « (¨ + L

1   1 + L 2   2 , H ), ¨ where 1 0 is any particular solution of the linear equation 1 ¨ ‘'

‘ − 2 H − 1 ¨ ‘
+ ¨ =0 H ( H − 1) 4 H ( H − 1) and « (‡ , H ) is the elliptic function defined by formula (20). 2 b . Solutions of the sixth Painlev´e transcendent (19) corresponding to different values of parameters

are related by:

( H , −ý , − ƒ , = , > ) =

O

( H , − ý , −= , ƒ , > ) = 1 −

1

H

1 , ƒ , ý , = , > 1

,

P

, 1 , ƒ , ý , = , > 1−H P 1 1 H . = OH , −ý , − ƒ , −> + , −= + 2 2P (H , ƒ , ý , = , > ) The successive application of these relations yields 24 equations of the form (19) with different values of parameters related by known transformations. 3 b . Every solution of the Riccati equation

‘
= £ 2ƒ H ( H − 1)

where

2

£ 2ƒ − (ƒ + ý + = + > ) £ 2 ƒ − £ −2ý − 1 ,

+

O

£ −2ý ŒH + “ , + H ( H − 1) H −1

£ −2ý − ( ƒ + ý − = − > ) £ 2 ƒ − £ −2ý − 1 , is simultaneously a solution of equation (19) if £ 2 ƒ − £ −2ý ≠ 1 and the condition 2 £ 2 ƒ (3ý − ƒ + = − > ) + 2 −2ý (3 ƒ − ý − = + > ) + 4 − ƒ…ý (ý − ƒ + = − > − 1) ü ü + ( ƒ + ý + = + > )2 + 2( ƒ − ý − = − 4 ƒ…ý − 2 ƒ±= − 2ý…> ) = 0 £ £ £ ^is_ satisfied (one should take the value of − ƒ…ý that coincides with ƒ −ý ). Œ =

“ =

References for Subsection 2.8.2: P. Painlev´e (1900), B. Gambier (1910), G. M. Murphy (1960), E. L. Ince (1964), A. S. Fokas and M. J. Ablowitz (1982), V. I. Gromak and N. A. Lukashevich (1990), R. Conte (1999), A. R. Chowdhury (2000).

© 2003 by Chapman & Hall/CRC

Page 437

438

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.8.3. Equations Containing Exponential Functions 2.8.3-1. Equations of the form  ( , )

 + ( , ) = 0. 1.

2.

ñóò ò ó

ô ó +_ ]

=

+õ .

¨ The substitution ? ¨2

 =   : +  .

ô ó ñ ÷

.

ô ï ÷  ]

.

ñóò ò ó

=

ñ óò ò ó

=

ñóò ò ó

=

ñóò ò ó

=

2

ñóò ò ó

=

2

+ ( Œ ý )

=

 

The transformation H =   !H − ¨ 2 . 3.

4.

The transformation H =   !H + ¨ .

õ ó ñ

+

ôñ

–3

+2

−1

leads to an autonomous equation of the form 2.9.1.1:

‘ , ¨ = 
 leads to a first-order equation: H [(B −1)¨ + Œ ]¨
=

 , ¨ =  
 leads to a first-order equation: H ( Œ ¨ + B + 2)¨u‘
=

.



This is a special case of equation 2.9.1.2 with  ( ) = −   . 5. 6.

ñ

+ ô exp[ (ù + 3) ï ]ñ

ñ

+ ô

.

This is a special case of equation 2.9.1.29 with  (ˆ ) = ˆ  .

ó  ñ ú

,

≠ 0.



  The transformation ˆ =   , ‡ =  leads to the Emden–Fowler equation ‡ Šª

Š = 4Œ “ − 3Œ − Œ ¡ , whose special cases are given in Section 2.3. where B = 2Œ 2

7.

÷

ñóò ò ó

=

ñóò ò ó

=

ñ

+ ô

ñ

+

2

(

ú

+3)

ó ( õ  2 ó



+ ö )÷

ñ ú

+ (, ¨

,



2

ˆ  ‡ + ,

≠ 0.

=  leads to the Emden–Fowler equation The transformation ˆ =    ¨ Šª

Š =  ˆ  ¨ + , whose special cases are given in Section 2.3. 4 2 Œ 2 8.

2

2


B (ú

+3)

ó (ô  2 ó 

+ õ )÷ ( ö

2



ó

+ý )

÷ ú

– –

–3



ñ ú

.

   + ¨  , = 2  leads to the Emden–Fowler equation  2 (  + (  + ¨ Šª

Š = # (2 ¢eŒ )−2 ˆ  ¨ + , where M M ¢ =  − ˜( (see Section 2.3). M 2

The transformation ˆ =

9.

10.

ñ óò ò ó

=

ô

exp( 

ï

2

+

ï

) exp( ƒñ ) + õ .

¨ = + ( ƒ… The substitution ? ¨2

 =   ; +  + 2 ƒ±= −1 .

ï 2 ñ óò ò ó

=

ô ïƒ÷ +2  ]

2

+ ý± ) = leads to an autonomous equation of the form 2.9.1.1:

+ù .

This is a special case of equation 2.9.1.31 with  (ˆ ) = ˆ . 2.8.3-2. Equations of the form  ( , )

 + ( , )
 + % ( , ) = 0. 11.

ñóò ò ó

=

ôñóò

+

õ 2D ó ñ ÷

. This is a special case of equation 2.9.2.17 with  ( ) =   .

© 2003 by Chapman & Hall/CRC

Page 438

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

12.

13.

ñóò ò ó

= – ôñó ò +

õD ÷ ó ñ ÷

15.

.

This is a special case of equation 2.9.2.36 with  (ˆ ) = 'ˆ 

ôñóò

+

ñóò ò ó

= –( + b )ñó ò – b “ñ +

+

õÿñ

ö ó ñ ú

ñóò ò ó

=

The substitution ˆ =  14.

–1



−1

=

ñ óò ò ó

=

ƒñóò

+

ƒñ óò

+

õï ñ

.

.

leads to an equation of the form 2.8.1.39: ˆ 2 
Šª
Š +(  +1)ˆ 
Š + 

ô (÷  –2 )ó ñ ÷

–1

ô 2 ó ñ

+

–3

= (˜ˆ  + .

.

This is a special case of equation 2.9.2.37 with  (ˆ ) = ˆ 

ñóò ò ó

439

−1

.

.

This is a special case of equation 2.9.2.14 with  ( ) = − ' . 16.

õ ó ñ

+ ô

2

ó

 ñ

–3

.



This is a special case of equation 2.9.2.14 with  ( ) = −   17.

18.

ñóò ò ó

=

ôñóò

ñóò ò ó

+ 3ññó ò + ñ

ñ óò ò ó

=

ï

+ õ exp(2 ô

÷

+ ö^ñ

.

).

This is a special case of equation 2.9.2.17 with  ( ) =  exp( (  ). 3

óñ

+ ô

= 0.



This is a special case of equation 2.9.2.1 with  ( ) =   . 19.

ô ï  ] ñ óò =L

Solution: 20.

ñóò ò ó

+ ô 1

ó ññóò

= 2 ô

]

.

  − ln O −  X   k 1  + L 2 . P M

óñ

+ ô

2

.

Solution in parametric form:

U

 = ln O

2

2L

:

1

= L 1 ø 1 (U ) + L 2 n 1 (U ) or Here, Bessel functions, and ® 1 (U ) and ; 21.

ñ óò ò ó

=

ô ï ÷  ] ñ óò =L

Solution: 22.

ñóò ò ó

=

+ ôù 1

ï ÷ –1  ]

 − ln L Q

ô ó ñ –1 ) 2 ñ óò

2

P

:

,

−2

L 1U

:

−1

(U

:
*

:

+ ).

= L 1 ® 1 (U ) + L 2 ; 1 (U ), where ø 1 (U ) and n 1 (U ) are the ( U 1 ) are the modified Bessel functions.

.

−  X   exp( L

ó ñ 1) 2.

+ 2 ô

−1

= −

1

 )  S . M

Solution in parametric form:

 = ln OA 23.

24.

L 

1



ñóò ò ó

+ (2 ôñ +

ñ óò ò ó

= 

P

TCU 2 ,

=L

õ ó )ñ óò

+

2 1

V 2U3A exp(T"U 2 )  W 2 ,

ˆõ ó ñ

= 0.

Integrating yields a Riccati equation: 
 + 

ô _ ó ñ

Solution: 2 X

+ 



M

2

ñ óò

+L

2

+  



where

 = X exp(T"U 2 ) U + L 2 S Q M

−1

.

=L .

. 1

=L

2

+

ý

1 :   .

© 2003 by Chapman & Hall/CRC

Page 439

440 25.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó

=

ô ó +] (ñ óò

Solution: = − − ln( L

26.

+ 1).

= − ln O L 1

1



−

k 2 −

−  ).





where

 (a ) =

28. 29.

30. 31. 32. 33. 34.

2



 P

. To the limiting case L

2

D

−1 there corresponds

+ ñ ó ò = ô ï ÷  ] . This equation is encountered in combustion theory and hydrodynamics. The transformation ? ˆ = ln  , ¨ = Œ +(B +1) ln  leads to an autonomous equation of the form 2.7.3.1: ¨IŠª

Š = Œ  . Solution in parametric form:

ï ñ óò ò ó

 = exp[ L

27.



1+ L






1

Aç (a )],

=

Œ

a

−

B +1 [L Œ

£ L 2 + 2 Œ  g − £ ln £ L 2 £ L 2 + 2 Œ  g + £ 2 −£ 2 Œ  g £ L 2 + 2 Œ 2 £ − L 2 arctan £ −L 2

L

1

















2

L

2

 g

1

Aç (a )],

if L

2

> 0,

if L

2

= 0,

if L

2

< 0.

= ùñó ò + ô ï 2÷ +1 ] . This is a special case of equation 2.9.2.20 with  ( ) =   .

ï ñóò ò ó

= ùñ ó ò + ô ï 2÷ +1 exp( ƒñ ú ). This is a special case of equation 2.9.2.20 with  ( ) =  exp( Œ "+ ).

ï ñ óò ò ó

ï 2 ñ óò ò ó

+ ï ñó ò = ô] . The substitution a = ln | | leads to an equation of the form 2.7.3.1: Solution: 1 Œ = − ln ¬ sin2 ( L 1 ln | | + L 2 )® if Œ Œ 2 L 12 1 Œ = − ln ¬ sinh2 ( L 1 ln | | + L 2 )® if Œ Œ 2 L 12 1 Œ = − ln ¬ − 2 cosh2 ( L 1 ln | | + L 2 )® if Œ Œ 2L 1


g
g =   . > 0, > 0, < 0.

ï 2 ñ óò ò ó

+ ï ñó ò = ô] + õ . This is a special case of equation 2.9.2.23 with  ( ) =   +  .

+ ï ñ ó ò = ü ï ÷  Df] + õ . This is a special case of equation 2.9.2.40 with  (ˆ ) = ,!ˆ +  .

ï 2 ñ óò ò ó

( ô ï 2 + õ )ñó ò ò ó + ô ï ñó ò + ö] = 0. This is a special case of equation 2.9.2.24 with  ( ) = (   .

ó

ó

ó

ó

( ô 2 + õ )ñó ò ò ó + ô 2 ñó ò + ö^ñ ÷ = 0.  This is a special case of equation 2.9.2.34 with ( ) =   2 +  and  ( ) = − (  .

( ô 2 + õ )ñó ò ò ó + ô 2 ñó ò + ö] = 0.  This is a special case of equation 2.9.2.34 with ( ) =   2 +  and  ( ) = − ( ž .

© 2003 by Chapman & Hall/CRC

Page 440

441

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

2.8.3-3. Equations of the form  ( , )

 + ( , )(
 )2 + % ( , )
 + & ( , ) = 0. 35.

36.

ïƒ÷

ñóò ò ó

= ô (ñó ò )2 –

õ 4fD ]

+ö

ñóò ò ó

= ô (ñó ò )2 –

õ 4fD ]

+ ö

ñóò ò ó

= ô (ñó ò )2 +

õ ƒï ÷  Df]

+

ö ïƒú

.

ñóò ò ó

= ô (ñó ò )2 +

õDf] +} ó

+

ü ïƒú

.

ñóò ò ó

= ô (ñó ò )2 +

õDf] + ó

+ ö

.

This is a special case of equation 2.9.3.18 with  ( ) = − (˜  .

ó

.



This is a special case of equation 2.9.3.18 with  ( ) = − (  . 37.

38.

This is a special case of equation 2.9.3.17 with  ( ) = − '  and ( ) = −(˜ + .

v

This is a special case of equation 2.9.3.17 with  ( ) = −   39.



ó

.

This is a special case of equation 2.9.3.17 with  ( ) = −   40.

ôñ ÷ (ñ óò

ñóò ò ó

+

õ]

+

ñóò ò ó

+

ô] (ñ óò

)2 +

õÿñ ÷

+

ñóò ò ó

+

ô] (ñ óò

)2 +

õ  ]

ñóò ò ó

=

ôñ ÷ (ñ óò

)2 +

õ ó ñóò

.

ñóò ò ó

=

ô] (ñ óò

)2 +

õ ƒï ÷ ñ óò

.

ñóò ò ó

=

ô] (ñ óò

)2 +

õ  ó ñóò

)2 +

ö

= 0.

ö

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  41.

42.

43.

45.



and ( ) = −( 

+

ö

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =   and ( ) =     + ( .

 and ( ) =    .

This is a special case of equation 2.9.3.38 with  ( ) =    and ( ) = '  . .

ñóò ò ó

=

ô ó (ï ñ óò

– ñ )2 +

õ  ó

.

This is a special case of equation 2.9.3.2 with  ( ) =   47.

ññóò ò ó

– (ñó ò )2 =

ô ó

The substitutions ? ¨2

 = 2   − . 48.

.

 and ( ) =   + ( .

This is a special case of equation 2.9.3.38 with  ( ) =    and ( ) =   46.



This is a special case of equation 2.9.3.25 with  ( ) =   and ( ) =   + ( .

This is a special case of equation 2.9.3.38 with  ( ) =  44.

and ( ) = − ,! + .

2ññó ò ò ó = (ñó ò )2 +

.

= A exp 

õ ó ñ

2

1 2

  , (



.



) = 0, and % ( ) =    .

¨ + 1 Œ   leads to an autonomous equation of the form 2.7.3.1: 2

–ô .



This is a special case of equation 2.9.3.5 with  ( ) = −   .

49.

ññóò ò ó

=

ù (ñ óò

)2 – ôñ

÷

4 –2

+

õ ó ñ

2

.



This is a special case of equation 2.9.3.8 with  ( ) = −   .

© 2003 by Chapman & Hall/CRC

Page 441

442 50.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ññ óò ò ó

– (ñ

óò

ô  ó ñ 

)2 =

.

1 b . For , ≠ 2, the substitution

= exp O ¨ +

?



the form 2.7.3.1: ¨  =   (  −2) . 2 b . Solution for , = 2: ln | | = L 1  + L 2

+

õ ó ñ ÷

+1

 ñ

2

+

õï ú ñ ÷

+1

óñ

2

õ  ó ñ ÷

+1

ïƒú ñ

+ Œ

2

Œ

2−, −2

 P

leads to an autonomous equation of

  .

51.

ññóò ò ó

=

ù (ñ óò

)2 + ô

.  This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  ž .

52.

ññ óò ò ó

=

ù (ñ ó ò

)2 + ô

ññóò ò ó

=

ù (ñ óò

)2 + ô

ó

.



and ( ) = − ' + .

This is a special case of equation 2.9.3.9 with  ( ) = −   53.

+

.

This is a special case of equation 2.9.3.9 with  ( ) = −   54.

ññóò ò ó

– (ñó ò )2 =

ô

 ï

ï



and ( ) = −  



.

. ¨ + 12 ý± 2 + 12 Œ   leads to an autonomous equation of the form The substitutions =? A exp  3 ¨

− − 2ý . 2.9.1.1:  = 2   exp 

2

+ 1 2

55.

ññ óò ò ó

– (ñ

óò

ôñ

õ

ï

exp –

ï

. ¨ 1 2 + 2 ý± + 12 Œ   leads to an autonomous equation of the form The substitutions =? A exp  3 ¨

− − 2(  + ý ). 2.9.1.1:  = 2   )2 +

2

=

2

+

1 2

56.

ññóò ò ó

– (ñó ò )2 =

ô

 ï

exp 

2

ï ñ 

+

.

1 ý± = exp ¨ + Q 2−, ? 2ý

. of the form 2.9.1.1: ¨   =   (  −2) − 2−, 2 b . Solution for , = 2: 1 b . For , ≠ 2, the substitution

ln | | = L 57.

ññóò ò ó

– (ñó ò )2 +

ôñ

2

=

õ

1

2

+ Π  S leads to an autonomous equation



 +L

 ï

exp 

2

2

+

+  X  ( − a ) exp  ý±a 2 + Œ a  a . M 0

ï ñ 

.

1  ý± 2 + Œ   S leads to an autonomous 2−, 2ý . equation of the form 2.9.1.1: ¨ 

 =   (  −2) −  − 2 −, ? 2 b . For , = 2, the substitutions = A  leads to a linear equation: ¨ 

 =  exp  ý± 2 + Œ   −  . Solution: = exp ¨ +

1 b . For , ≠ 2, the substitution



ln | | = −  2 58.

59.

ñóò ò ó

+ ô (ñó ò )2 –

1 2

ñóò

=



2

ó (õ ñ 2

Q?

+L

1

2

õ 1ñ

+



2

+  X  ( − a ) exp  ý±a 2 + Œ a  a . M 0

õ

0 ).

 +L +

 The substitution ¨ ( ) =  − (
 )2 leads to a first-order linear equation: ¨u
+ 2  ¨ = 2   2  1 + 2  0.

ññóò ò ó

= (ñó ò )2 +

ô ƒï ÷ ñ ñóò

+

õ ó ñ

2

2

+

2

.  This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  ž .

© 2003 by Chapman & Hall/CRC

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443

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

60.

ññ óò ò ó

= (ñ ó ò )2 + ô  ññ ó ò + õ ï ÷ ñ 2 .  This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = − '  .

ó

61.

ññóò ò ó

ó

62.

ó

= (ñó ò )2 + ô ññó ò + õ  ñ 2 .   This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −    .

ó

+  (ñó ò )2 =  õ _ +` ] +   1 b . Solution with = ≠ − ƒ :

ñóò ò ó

 9  M  ( )+ L 2 b . Solution with = = − ƒ : ù

=L

1

ñóò 2

. +

ó

= ô _ (ñó ò + õÿñ )2 + õ The substitution ¨ = 
+ 

ñóò ò ó

2

ñ

1 : 



,

 9  M  +L ù

63.

ý

=L

1

ó

2

+

ý

  ( 9 +; )  . ƒ +=

 ( )=

where

1 : 



.

+ ö . ¨ 
=   :  ¨ leads to a Riccati equation: 3

2

 +  ¨ + (  .

2.8.3-4. Other equations. 64.

ñóò ò ó

+ õD ñ ú (ñó ò )3 + ôñó ò = 0. This is a special case of equation 2.9.3.35 with  ( ) =  "+ .

ó

65.

ñóò ò ó

+ õD + ] (ñó ò )3 + ôñó ò = 0. This is a special case of equation 2.9.3.35 with  ( ) =   .

ó

66.

ñóò ò ó

ô ó (ñ óò

=

)3 + ô

Solution:  = L 67.

1

ó ñ (ñ óò

)2 .

 k 1

− ln O X

+L

M

= ô ï  ] (ñ ó ò )3 + ô ] (ñ ó ò )2 . Solution in parametric form:

 *

(U

:
*

M

= ô ï 2 ] (ñó ò )3 + 2 ô ï ] (ñó ò )2 . Solution in parametric form:

 =

:

−1

L 1U

= 2 ô ï 1 ) 2 ] (ñó ò )3 + ô ï –1 ) Solution in parametric form:

ñóò ò ó

 =L 70.

−1

ñóò ò ó

= L 1 ø 1 (U ) + L 2 n 1 (U ) or Here, Bessel functions, and ® 1 (U ) and ; 69.

.

P

ñ óò ò ó

 = L 1  −  * O XZU 68.

2

2 1

V 2U2A exp(T"U 2 ) W 2 ,

ôùJ ó ñ ÷

–1

Solution:  = L

1

ñóò ò ó

=

(ñó ò )3 +

2

−2

:

:

−1

:

+ ),

2

P

= ln U .

,

= ln O

U

2L

2 1

P

.

= L 1 ® 1 (U ) + L 2 ; 1 (U ), where ø 1 (U ) and n 1 (U ) are the ( U 1 ) are the modified Bessel functions.

] (ñ óò

  k 1

)2 .

L

= ln O˜T

ô ó ñ ÷ (ñ óò

− ln O X

U +L



1



TCU 2 , P

where

 = X exp(T"U 2 ) U + L 2 S Q M

−1

.

M

)2 . +L

2

P

.

© 2003 by Chapman & Hall/CRC

Page 443

444 71.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

=

ô ó +] [(ñ óò

)3 + (ñó ò )2 ].

Solution:  = − ln OL

 = − − ln( L 72.

ñóò ò ó

=

1

+

1



 k 2 +

).

1−L

2

  . To the limiting case L P

D

2

1 there corresponds

ô ó (ñ óò )3 ) 2 + ô ó ñ (ñ óò )1 ) 2 .

Solution in parametric form:

 = ln U 2 , where

:

= −2 

−2 −4

U

V:

−1

(U

:
*

:

+2 )T

1 2

U 2W ,

ø

L

ø

U ) + L 2 n 2 (U ) for the upper sign, L 1 ® 2 (U ) + L 2 ; 2 (U ) for the lower sign, 2 ( U ) and n 2 ( U ) are the Bessel functions, and ® 2 ( U ) and ; 2 ( U ) are the modified Bessel 1 2(

™

=

functions. 73.

ñóò ò ó

=

ô ï ] (ñ óò )5 ) 2 + ô] (ñ óò )3 ) 2 .

Solution in parametric form:

 = −2  where

ø

:

U

L

V:

−1

(U

:
*

:

+2 )T

1 2

U 2W ,

ñóò ò ó

= – ôñó ò +

=

™

ø

1 2(

õD ú ó ñ  (ñ óò )ú

+2

.

This is a special case of equation 2.9.4.17 with  ( ) = −   and B =

75.

ñóò ò ó

=–

2–ü ñóò + õD ó ñ ú – +1 (ñóò ) . þ 1–ü This is a special case of equation 2.9.4.31 with  (ˆ ) = 'ˆ .

76.

ñóò ò ó

=

ñóò ò ó

= –(ñó ò )2 +

77.

= ln U 2 ,

U ) + L 2 n 2 (U ) for the upper sign, L 1 ® 2 (U ) + L 2 ; 2 (U ) for the lower sign, 2 ( U ) and n 2 ( U ) are the Bessel functions, and ® 2 ( U ) and ; 2 ( U ) are the modified Bessel

functions. 74.

−2 −4

ñóò

ô

+




exp[(ù + 2 – @ ) ï ]ñ



¡

+ 2.

ú (ñ óò )A .

The substitution ˆ =  leads to the generalized Emden–Fowler equation 
Šª
Š = #*ˆ  + (
Š )- , which is discussed in Section 2.5.


ïƒ÷

exp[(þ

+ @ – 1)ñ ] (ñ„ó ò )A .

The substitution ¨ =   leads to the generalized Emden–Fowler equation ¨ 

 = #*  ¨ + (¨ 
)- ,

which is discussed in Section 2.5. 78.

ñóò ò ó

=

ô

1–ü (ñó ò )2 + õ ïƒ÷ +  –2 Df] (ñó ò )  . ù 2–ü This is a special case of equation 2.9.4.30 with  (ˆ ) = 'ˆ .

79.

ñóò ò ó

=

ñóò ò ó

=

ôñ ÷ (ñ óò

)2 +

õ] (ñ óò )

.

This is a special case of equation 2.9.4.13 with  ( ) =  80.

ô] (ñ óò

)2 +

õÿñ ÷ (ñ óò )

 and ( ) =   .

. This is a special case of equation 2.9.4.13 with  ( ) =   and ( ) =   .

© 2003 by Chapman & Hall/CRC

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445

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

81. 82.

83.

ñ óò ò ó

=

ô ] (ñ óò

ñóò ò ó

=

ô 2ñ

ñóò ò ó

=

ô ó +] [(ñ óò )

)2 +

õ ] (ñ óò )

.

This is a special case of equation 2.9.4.13 with  ( ) =   and ( ) =     . +

õ _ ó (ñ óò

+

ôñ )

The substitution ¨ = 
+ 

+ (ñó ò ) 

.

¨ 
=  ¨ +   :  ¨  . leads to a Bernoulli equation: u

–1

ü

],

≠ 2.

Solution in parametric form:

 =U −X 84. 85.

M 

U

−L

=X

2,

M

õ  ó (ï ñ óò



ñ óò ò ó

=

ô ï ÷ (ï ñ óò

ñóò ò ó

=

ô ó (ï ñ óò

–ñ )+

õ ïƒ÷ (ï ñ óò

ñóò ò ó

=

ô ó (ï ñ óò

–ñ )+

õ  ó (ï ñ óò

–ñ )+

U

+L

2,

where

 = V  (2 − , )  * + L 1 W 

 This is a special case of equation 2.9.4.4 with  ( ) =   and ( ) =   . – ñ ) .

– ñ ) .

This is a special case of equation 2.9.4.4 with  ( ) =    87.

ï ñ óò ò ó + ñóò = ô ïƒ÷  ] (ñóò )ú .  The transformation =  
, ¨ Œ

88.

+ 1.

– ñ ) .



and ( ) = '  .

This is a special case of equation 2.9.4.4 with  ( ) =   86.

1 −2

+B −

ï ñóò ò ó

+ 1.

¡

+ þxñó ò + ô

ïƒ÷¿ú

–2

ú

+1

=   −+



and ( ) =  



.



+1

] (ñ óò )÷

 leads to a first-order linear equation:  +2¨2û
=

= 0.

This is a special case of equation 2.9.4.14 with  ( ) =   .

89.

90. 91.

ï ñ óò ò ó + ñóò = (ô ïƒ÷  ] + õ ïƒú –1 )(ñóò )ú .  =  
, ¨ =   −+ +1   The transformation + (  ¨ +  )¨2û
= ( Œ + B − + 1)¨ . ¡ ó ÷ 2 ññ óò ò ó = (ñ óò ) + õ D ñ (ñ óò ) .

leads to a first-order separable equation:





This is a special case of equation 2.9.4.68 with  (ˆ ) = 'ˆ , ( ) =  , and B =

ññóò ò ó

= (ñó ò )2 + ( ô

óñ ÷

+

õÿñ 2–ú

)(ñó ò )ú .

The transformation ˆ = 
 , ¨ =  ˆ + (  ¨ +  )¨2Š
= [(B + − 2)ˆ + Œ ]¨ .

  ++

−2

¡

− , + 2.

leads to a first-order separable equation:

¡

2.8.4. Equations Containing Hyperbolic Functions 2.8.4-1. Equations with hyperbolic sine. 1.

2.

ñóò ò ó

=

ñ óò ò ó

=

2

ñ

+ ô (sinh

ï )–÷ –3 ñ ÷

.

This is a special case of equation 2.9.1.34 with  (ˆ ) = ˆ  .

õ

sinh(

ï )ñ

+ ôñ

–3

.

This is a special case of equation 2.9.1.2 with  ( ) = −  sinh( Œ  ).

© 2003 by Chapman & Hall/CRC

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446 3.

4.

5.

SECOND-ORDER DIFFERENTIAL EQUATIONS sinh÷ ( ôñ +

õï

ñ óò ò ó

=

ñóò ò ó

= ô (ñ + õ sinh ï ) ÷ – õ sinh ï + ö .



)+ .

This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ sinh  ¨ + ý and ( = 0. The substitution ¨ ¨2

 =  ¨  + ( .

+  sinh 

=

leads to an autonomous equation of the form 2.9.1.1:

+ 3ññó ò + ñ

ñóò ò ó

+

ñ óò ò ó

+

ô

sinh ÷

ñ (ñ ó ò

)2 +

ñóò ò ó

+

ô

sinh ÷

ñ (ñ óò

)2 + õ sinh ú ( ƒñ ) +

ñ óò ò ó

=

3

+ ô sinh(

ï )ñ

ñóò ò ó

= 0.

This is a special case of equation 2.9.2.1 with  ( ) =  sinh( Π ). 6.

)2 + õ sinh ú

ôñ ÷ (ñ óò

ñ

+ ö = 0.

This is a special case of equation 2.9.3.25 with  ( ) =  7.

8.

9.

õÿñ ú

+

ö

= 0.

ö

= 0.

11.

+

This is a special case of equation 2.9.3.25 with  ( ) =  sinh 

ôñ ÷ (ñ óò

)2 + õ sinhú

ñ (ñ ó ò ) 

ñóò ò ó

=

ô

sinh ÷

=

ùñóò

ñ (ñ óò

õÿñ ú (ñ óò )

)2 +

.

and ( ) =  sinh ( Œ

 and ( ) =  sinh+

.

ï 2÷

+ô

+1

sinhú ( ƒñ ).

+

This is a special case of equation 2.9.2.20 with  ( ) =  sinh ( Π12.

2ññ

óò ò ó

= (ñ

óò

)2 + õ sinhú (

ï )ñ

2

)+ ( .

.

and ( ) =  + .

This is a special case of equation 2.9.4.13 with  ( ) =  sinh 

ï ñóò ò ó

+( .

and ( ) =  + + ( .

This is a special case of equation 2.9.3.25 with  ( ) =  sinh 

This is a special case of equation 2.9.4.13 with  ( ) =  10.

 and ( ) =  sinh+

–ô .

).

+

This is a special case of equation 2.9.3.5 with  ( ) = −  sinh ( Œ  ).

13.

ù (ñ óò

)2 – ôñ

ù (ñ óò

)2 + ô

÷

ññóò ò ó

=

ññóò ò ó

=

ññóò ò ó

= (ñó ò )2 +

ô ƒï ÷ ñ ñóò

ññóò ò ó

= (ñó ò )2 +

ô

4 –2

+ õ sinh ú (

ï )ñ

+ õ sinh  (

ï )ñ ÷

+ õ sinh ú (

ï )ñ

2

õ ïƒú ñ

2

2

.

+

This is a special case of equation 2.9.3.8 with  ( ) = −  sinh ( Œ  ). 14.

15.

16.

17.

18.

ïƒú ñ

2

+1

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  sinh  ( Œ  ). .

+ This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  sinh ( Œ  ). sinh ÷ (

ï ) ñ ñóò

+

.

This is a special case of equation 2.9.3.7 with  ( ) = −  sinh  ( Œ  ) and ( ) = − ' + .

ï 2 ñ óò ò ó

+

ï ñ óò

=

ô

sinh ÷ ( ƒñ ) + õ .

This is a special case of equation 2.9.2.23 with  ( ) =  sinh  ( Œ (ô

ï

2

+ õ )ñó ò ò ó +

ô ï ñóò

+ sinh÷ ( ƒñ ) + ö = 0.

This is a special case of equation 2.9.2.24 with  ( ) = sinh  ( Œ

)+  . )+( .

© 2003 by Chapman & Hall/CRC

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447

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

2.8.4-2. Equations with hyperbolic cosine. 19. 20. 21. 22.

23. 24. 25. 26. 27. 28. 29. 30.

= 2 ñ + ô (cosh ï )–÷ –3 ñ ÷ . This is a special case of equation 2.9.1.35 with  (ˆ ) = ˆ  .

ñóò ò ó

= õ cosh( ï ) ñ + ôñ –3 . This is a special case of equation 2.9.1.2 with  ( ) = −  cosh( Œ  ).

ñóò ò ó

=  cosh÷ ( ôñ + õ ï ) +  . This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ cosh  ¨ + ý and ( = 0.

ñóò ò ó

= ô (ñ + õ cosh ï ) ÷ – õ cosh ï + ö . The substitution ¨ = +  cosh  leads to an autonomous equation of the form 2.9.1.1: ¨2

 =  ¨  + ( .

ñóò ò ó

+ 3ññó ò + ñ 3 + ô cosh( ï ) ñ = 0. This is a special case of equation 2.9.2.1 with  ( ) =  cosh( Œ  ).

ñóò ò ó

+ ôñ ÷ (ñó ò )2 + õ coshú ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) = 

ñóò ò ó

 and ( ) =  cosh+

+ ô cosh÷ ñ (ñó ò )2 + õÿñ ú + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  cosh 

ñóò ò ó

+ ô cosh÷ ñ (ñó ò )2 + õ coshú ( ƒñ ) + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  cosh 

ñóò ò ó

= ôñ ÷ (ñó ò )2 + õ coshú ñ (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) = 

ñóò ò ó

= ô cosh÷ ñ (ñó ò )2 + õÿñ ú (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) =  cosh 

= ùñó ò + ô ï 2÷ +1 coshú ( ƒñ ). + This is a special case of equation 2.9.2.20 with  ( ) =  cosh ( Œ

ï ñóò ò ó

and ( ) =  + + ( .

+

and ( ) =  cosh ( Œ

 and ( ) =  cosh+

ñóò ò ó

.

).

2ññó ò ò ó = (ñó ò )2 + õ coshú ( ï ) ñ 2 – ô . + This is a special case of equation 2.9.3.5 with  ( ) = −  cosh ( Œ  ). = ù (ñó ò )2 – ôñ 4÷ –2 + õ coshú ( ï ) ñ 2 . + This is a special case of equation 2.9.3.8 with  ( ) = −  cosh ( Œ  ).

32.

ññóò ò ó

34.

)+( .

and ( ) =  + .

31.

33.

+( .

ññóò ò ó

=

ù (ñ óò

)2 + ô

ïƒú ñ

2

+ õ cosh  (

ï )ñ ÷

+1

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  cosh  ( Œ  ).

= (ñó ò )2 + ô ïƒ÷ ññó ò + õ coshú ( ï ) ñ 2 . + This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  cosh ( Œ  ).

ññóò ò ó

= (ñó ò )2 + ô cosh÷ ( ï ) ññó ò + õ ïƒú ñ 2 . This is a special case of equation 2.9.3.7 with  ( ) = −  cosh  ( Œ  ) and ( ) = − ' + .

ññóò ò ó

© 2003 by Chapman & Hall/CRC

Page 447

448 35. 36.

SECOND-ORDER DIFFERENTIAL EQUATIONS + ï ñó ò = ô cosh÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.23 with  ( ) =  cosh  ( Œ

ï 2 ñ óò ò ó

( ô ï 2 + õ )ñó ò ò ó + ô ï ñó ò + cosh÷ ( ƒñ ) + ö = 0. This is a special case of equation 2.9.2.24 with  ( ) = cosh  ( Œ

)+ . )+( .

2.8.4-3. Equations with hyperbolic tangent. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

= õ tanh( ï ) ñ + ôñ –3 . This is a special case of equation 2.9.1.2 with  ( ) = −  tanh( Œ  ).

ñ óò ò ó

=  tanh÷ ( ôñ + õ ï ) +  . This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ tanh  ¨ + ý and ( = 0.

ñ óò ò ó

+ 3ññ ó ò + ñ 3 + ô tanh( ï ) ñ = 0. This is a special case of equation 2.9.2.1 with  ( ) =  tanh( Œ  ).

ñ óò ò ó

+ ôñ ÷ (ñó ò )2 + õ tanhú ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) = 

ñóò ò ó

 and ( ) =  tanh+

+ ô tanh÷ ñ (ñó ò )2 + õÿñ ú + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  tanh 

ñóò ò ó

+ ô tanh÷ ñ (ñó ò )2 + õ tanhú ( ƒñ ) + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  tanh 

ñóò ò ó

= ôñ ÷ (ñó ò )2 + õ tanhú ñ (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) = 

ñóò ò ó

= ô tanh÷ ñ (ñó ò )2 + õÿñ ú (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) =  tanh 

= ùñó ò + ô ï 2÷ +1 tanhú ( ƒñ ). + This is a special case of equation 2.9.2.20 with  ( ) =  tanh ( Œ

ï ñóò ò ó

and ( ) =  + + ( .

+

and ( ) =  tanh ( Œ

 and ( ) =  tanh+

ñóò ò ó

.

).

2ññó ò ò ó = (ñó ò )2 + õ tanhú ( ï ) ñ 2 – ô . + This is a special case of equation 2.9.3.5 with  ( ) = −  tanh ( Œ  ). = ù (ñó ò )2 – ôñ 4÷ –2 + õ tanhú ( ï ) ñ 2 . + This is a special case of equation 2.9.3.8 with  ( ) = −  tanh ( Œ  ).

48.

ññ óò ò ó

50.

)+( .

and ( ) =  + .

47.

49.

+( .

ññóò ò ó

=

ù (ñ ó ò

)2 + ô

ï ú ñ

2

+ õ tanh  (

ï )ñ ÷

+1

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  tanh  ( Œ  ).

= (ñ ó ò )2 + ô ï ÷ ññ ó ò + õ tanhú ( ï ) ñ 2 . + This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  tanh ( Œ  ).

ññ óò ò ó

= (ñ ó ò )2 + ô tanh÷ ( ï ) ññ ó ò + õ ï ú ñ 2 . This is a special case of equation 2.9.3.7 with  ( ) = −  tanh  ( Œ  ) and ( ) = − ' + .

ññ óò ò ó

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449

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

51. 52.

+ ï ñó ò = ô tanh÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.23 with  ( ) =  tanh  ( Œ

ï 2 ñ óò ò ó

( ô ï 2 + õ )ñó ò ò ó + ô ï ñó ò + tanh÷ ( ƒñ ) + ö = 0. This is a special case of equation 2.9.2.24 with  ( ) = tanh  ( Œ

)+ . )+( .

2.8.4-4. Equations with hyperbolic cotangent. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62.

= õ coth( ï ) ñ + ôñ –3 . This is a special case of equation 2.9.1.2 with  ( ) = −  coth( Œ  ).

ñ óò ò ó

=  coth÷ ( ôñ + õ ï ) +  . This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ coth  ¨ + ý and ( = 0.

ñ óò ò ó

+ 3ññ ó ò + ñ 3 + ô coth( ï ) ñ = 0. This is a special case of equation 2.9.2.1 with  ( ) =  coth( Œ  ).

ñ óò ò ó

+ ôñ ÷ (ñó ò )2 + õ cothú ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) = 

ñóò ò ó

 and ( ) =  coth+

+ ô coth÷ ñ (ñó ò )2 + õÿñ ú + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  coth 

ñóò ò ó

+ ô coth÷ ñ (ñó ò )2 + õ cothú ( ƒñ ) + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  coth 

ñóò ò ó

= ôñ ÷ (ñó ò )2 + õ cothú ñ (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) = 

ñóò ò ó

= ô coth÷ ñ (ñó ò )2 + õÿñ ú (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) =  coth 

= ùñó ò + ô ï 2÷ +1 cothú ( ƒñ ). + This is a special case of equation 2.9.2.20 with  ( ) =  coth ( Œ

ï ñóò ò ó

and ( ) =  + + ( .

+

and ( ) =  coth ( Œ

 and ( ) =  coth+

ñóò ò ó

.

).

2ññó ò ò ó = (ñó ò )2 + õ cothú ( ï ) ñ 2 – ô . + This is a special case of equation 2.9.3.5 with  ( ) = −  coth ( Œ  ). = ù (ñó ò )2 – ôñ 4÷ –2 + õ cothú ( ï ) ñ 2 . + This is a special case of equation 2.9.3.8 with  ( ) = −  coth ( Œ  ).

64.

ññ óò ò ó

66.

)+( .

and ( ) =  + .

63.

65.

+( .

ññóò ò ó

=

ù (ñ ó ò

)2 + ô

ï ú ñ

2

+ õ coth (

ï )ñ ÷

+1

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  coth  ( Œ  ).

= (ñ ó ò )2 + ô ï ÷ ññ ó ò + õ cothú ( ï ) ñ 2 . + This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  coth ( Œ  ).

ññ óò ò ó

= (ñ ó ò )2 + ô coth÷ ( ï ) ññ ó ò + õ ï ú ñ 2 . This is a special case of equation 2.9.3.7 with  ( ) = −  coth  ( Œ  ) and ( ) = − ' + .

ññ óò ò ó

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450 67. 68.

SECOND-ORDER DIFFERENTIAL EQUATIONS + ï ñó ò = ô coth÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.23 with  ( ) =  coth  ( Œ

ï 2 ñ óò ò ó

( ô ï 2 + õ )ñó ò ò ó + ô ï ñó ò + coth÷ ( ƒñ ) + ö = 0. This is a special case of equation 2.9.2.24 with  ( ) = coth  ( Œ

)+ . )+( .

2.8.4-5. Equations containing combinations of hyperbolic functions. 69.

70.

ñóò ò ó

=

72.

73. 74. 75. 76.

) cosh–÷

ï

ú

–

–3

The transformation ˆ = tanh( Œ  ), ¨

¨2Šª

Š = Œ

−2

ñóò ò ó

2

=

ñ

−2

(

ï )ñ ú

.

=

cosh( Œ  ) ˆ  ¨2+ , which is discussed in Section 2.3. + ô cosh÷ (

ï

) sinh–÷

–

ú

The transformation ˆ = coth( Œ  ), ¨

¨2Šª

Š = Œ

71.

+ ô sinh÷ (

ñ

2

–3

( =

ï )ñ ú

leads to the Emden–Fowler equation

.

sinh( Œ  ) ˆ  ¨2+ , which is discussed in Section 2.3.

leads to the Emden–Fowler equation

= ô (ñó ò sinh ï – ñ cosh ï )  + ñ . The substitution ¨ = 
sinh  − cosh  leads to a first-order separable equation: ¨ 
=  sinh  ¨  .

ñóò ò ó

= ô (ñó ò cosh ï – ñ sinh ï )  + ñ . ¨ 
= The substitution ¨ = 
cosh  − sinh  leads to a first-order separable equation: p ¨  cosh   .

ñóò ò ó

sinh ï ñ„ó ò ò ó + 12 cosh ï ñó ò = ô sinh( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sinh  and  ( ) =  sinh( Œ

cosh ï ñó ò ò ó + 21 sinh ï ñó ò = ô cosh( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = cosh  and  ( ) =  cosh( Œ

)+ . )+ .

+ (ñó ò )2 + ô sinh( ï )ññ„ó ò + õ cosh( ï ) + ö = 0. This is a special case of equation 2.9.3.6 with  ( ) =  sinh(ý± ) and ( ) =  cosh( Œ  ) + ( .

ññóò ò ó

– (ñó ò )2 + ô sinh( ï )ññ„ó ò + õ cosh( ï )ñ 2 = 0. This is a special case of equation 2.9.3.7 with  ( ) =  sinh(ý± ) and ( ) =  cosh( Œ  ).

ññóò ò ó

2.8.5. Equations Containing Logarithmic Functions 2.8.5-1. Equations of the form  ( , )

 + ( , )
 + % ( , ) = 0. 1. 2. 3.

= õÿñ ( ô ï + þ ln ñ ). This is a special case of equation 2.9.1.27 with  ( H ) =  ln H .

ñ óò ò ó

= õ ï –2 ( ôñ + ù ln ï ). This is a special case of equation 2.9.1.28 with  ( H ) =  ln H .

ñóò ò ó

= ô ï –3 (ln ñ – ln ï ). This is a special case of equation 2.9.1.8 with  (ˆ ) =  ln ˆ .

ñ óò ò ó

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451

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

4.

ñóò ò ó

=

ñóò ò ó

=

ñ óò ò ó

=

ô ï –3 ) 2(2 ln ñ

– ln ï ).

This is a special case of equation 2.9.1.9 with  (ˆ ) = 2  ln ˆ . 5. 6. 7. 8.

ln÷ ( ôñ +

ü

õï

) + s.

This is a special case of equation 2.9.1.4 with  (¨ ) = , ln  ¨ + ° and ( = 0.

ñ

–3

ï

V 2 ln ñ – ln(ô

2

+ ö )W .

This is a special case of equation 2.9.1.21 with  (¨ ) = 2 ln ¨ and  = 0.

ï 2 ñ óò ò ó

=

ï 2 (ñ

+

ln ï + õ )÷ + ô .

ï 2 ñ óò ò ó

=

ù (ù

+ 1)ñ + ô

ï 2 ñ óò ò ó

+

ô

This is a special case of equation 2.9.1.36 with  (ˆ ) = ˆ  .

ï 3÷

(ln ñ + ù ln ï ).

+2

This is a special case of equation 2.9.1.12 with  (ˆ ) =  ln ˆ . 9.

10.

1 4

ñ

+

ú


ï

( ô ln ï + õ )÷

1– 2

ñ ú

= 0.

The transformation ˆ =  ln  +  , ¨ =  ¨2Šª

Š + #2 −2 ˆ  ¨2+ = 0 (see Section 2.3).

ï ñóò ò ó

=

ùñóò

ï 2÷

+ô

+1

J

−1 2

leads to the Emden–Fowler equation:

lnú ( ƒñ ).

+

This is a special case of equation 2.9.2.20 with  ( ) =  ln ( Π11.

ï ÷

ï ñ óò ò ó

= –(ù + 1)ñ

ï ñóò ò ó

= ( ôñ + ù ln ï )ñ„ó ò .

ï ñóò ò ó

+ ï (2 ôñ + ln ï + õ )ñ„ó ò + ñ = 0.

ï ñ óò ò ó

=

óò

+ô

–1

).

(ln ñ + ù ln ï ).

This is a special case of equation 2.9.2.30 with  (ˆ ) =  ln ˆ . 12.

This is a special case of equation 2.9.2.39 with  (ˆ ) = ln ˆ . 13. 14.

15. 16.

Integrating yields a Riccati equation: 
 +  ln  ( õÿñ )ñ

ô

óò

2

+ (ln  +  ) = L .

.

This is a special case of equation 2.9.2.21 with  ( ) =  ln  (  ).

ï 2 ñ óò ò ó

+

ï ñóò

=

ï

+ õ )ñ

óò ò ó

+

ô

ln ÷ ( ƒñ ) + õ .

This is a special case of equation 2.9.2.23 with  ( ) =  ln  ( Œ (ô

2

ô ï ñ óò

+ ö ln ÷ ( ƒñ ) = 0.

This is a special case of equation 2.9.2.24 with  ( ) = ( ln  ( Œ

)+ . ).

2.8.5-2. Other equations. 17.

ñóò ò ó

+

ñ óò ò ó

=

ôñ ÷ (ñ óò

)2 + õ lnú

ôñ ÷ (ñ óò

)2 + õ lnú (

ñ

+

ö

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  18.

ï ) ñ óò

.

This is a special case of equation 2.9.3.38 with  ( ) = 

 and ( ) =  ln+

+( .

 and ( ) =  ln+ ( Π ).

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452 19. 20. 21. 22.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

+

ñ óò ò ó

=

ñóò ò ó

=

ñóò ò ó

=

ln ÷

ñ (ñ óò

)2 +

õÿñ ú

ô

ln÷

ñ (ñ ó ò

)2 +

õ ï ú ñ óò

ô

ln÷

ñ (ñ óò

)2 + õ lnú (

ô

+

ö

= 0.

and ( ) =  + + ( .

This is a special case of equation 2.9.3.25 with  ( ) =  ln  .

and ( ) = ' + .

This is a special case of equation 2.9.3.38 with  ( ) =  ln 

ï ) ñ óò

.

+

This is a special case of equation 2.9.3.38 with  ( ) =  ln 

ô ï –2 ñ

–1

ï

(ñó ò )4 – 2 ô

–3

and ( ) =  ln ( Π ).

ln ñ (ñó ò )3 .

Solution in parametric form: 1J 2  = ΠV ( wA 2U )2 A 4 ln( L 1  )W ,

= exp(T"U 2 ) X exp(T"U 2 ) U + L

where  23.

ñóò ò ó

Q

= 2 ô ln ï

ñ

–3

ï –1 ñ

–ô

M

–2

−1

S

2

, Π= A

1 2

=L

 ,

1

 J

2 1 4 . 1

L

(ñó ò )–1 .

Solution in parametric form: = ΠV ( wA 2U )2 A 4 ln( L

 = L 1 ,

= exp(T"U 2 ) X exp(T"U 2 ) U + L

where  24.

ñóò ò ó

ôñ ÷ (ñ óò

=

Q

)2 + õ lnú

ñ (ñ óò )

M

2

S

−1

, Π= A

1 2

.

This is a special case of equation 2.9.4.13 with  ( ) =  25. 26.

27.

ô

ln÷

ññóò ò ó

=

ù (ñ óò

)2 + ô

ññ óò ò ó

= (ñ

óò

)2 +

ô

óò

)2 +

ô ï ÷ ñ ñ óò

ñóò ò ó

=

ñ (ñ óò

õÿñ ú (ñ óò )

)2 +

1

2

ï )ñ ÷

+1

,

 J

 and ( ) =  ln+

.

+ õ ln  (

J

1 2

2 1 4 . 1

L

.

and ( ) =  + .

This is a special case of equation 2.9.4.13 with  ( ) =  ln 

ïƒú ñ

 )W

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  ln  ( Œ  ). ln ÷ (

ï ) ñ ñ óò

+

õï ú ñ

2

.

This is a special case of equation 2.9.3.7 with  ( ) = −  ln  ( Œ  ) and ( ) = − ' + . + õ lnú (

ï )ñ

28.

ññ óò ò ó

= (ñ

. + This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  ln ( Œ  ).

29.

ññóò ò ó

= (ô

ï

2

+ ù ln ñ )(ñ„ó ò )2 .

This is a special case of equation 2.9.3.40 with  (ˆ ) = ln ˆ .

2.8.6. Equations Containing Trigonometric Functions 2.8.6-1. Equations with sine. 1.

ñóò ò ó

=–

2

ñ

+ ô (sin

ï )÷ ñ –÷

–3

. This is a special case of equation 2.9.1.40 with  (ˆ ) = ˆ − 

−3

.

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453

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

2.

3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

+ ô ) sin ú ( ï + õ ) ñ –÷ –ú –3 . sin( Œ  +  ) ¨ , = leads to the Emden–Fowler equation: The transformation ˆ = sin( Œ  +  ) sin( Œ  +  ) ¨2Šª

Š = # [ Œ sin(  −  )]−2 ˆ  ¨ −  − + −3 (see Section 2.3).

ñóò ò ó

=–

2

ñ

+




sin÷ (

ï

= õ sin( ï ) ñ + ôñ –3 . This is a special case of equation 2.9.1.2 with  ( ) = −  sin( Œ  ).

ñóò ò ó

=  sin÷ ( ôñ + õ ï ) +  . This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ sin  ¨ + ý and ( = 0.

ñóò ò ó

= ô (ñ + õ sin ï ) ÷ + õ sin ï + ö . The substitution ¨ = +  sin  leads to an autonomous equation of the form 2.9.1.1: ¨2

 =  ¨  + ( .

ñóò ò ó

+ 3ññó ò + ñ 3 + ô sin( ï ) ñ = 0. This is a special case of equation 2.9.2.1 with  ( ) =  sin( Œ  ).

ñóò ò ó

+ ôñ ÷ (ñ ó ò )2 + õ sin ú ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) = 

ñ óò ò ó

= ôñ ÷ (ñó ò )2 + õ sinú ( ï ) ñó ò . This is a special case of equation 2.9.3.38 with  ( ) = 

ñóò ò ó

= ôñ ÷ (ñó ò )2 + õ sinú ñ (ñó ò )  . This is a special case of equation 2.9.4.13 with  ( ) = 

ñóò ò ó

 and ( ) =  sin+

 and ( ) =  sin + ( Œ  ).  and ( ) =  sin+­ .

+ ô sin ÷ ñ (ñó ò )2 + õÿñ ú + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  sin 

ñóò ò ó

+ ô sin ÷ ñ (ñó ò )2 + õ sin ú ( ƒñ ) + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =  sin 

ñóò ò ó

= ô sin ÷ ñ (ñó ò )2 + õ ïƒú ñó ò . This is a special case of equation 2.9.3.38 with  ( ) =  sin 

ñóò ò ó

= ô sin ÷ ñ (ñó ò )2 + õ sinú ( ï ) ñó ò . This is a special case of equation 2.9.3.38 with  ( ) =  sin 

ñóò ò ó

= ô sin ÷ ñ (ñ ó ò )2 + õÿñ ú (ñ ó ò )  . This is a special case of equation 2.9.4.13 with  ( ) =  sin 

ñ óò ò ó

= ùñó ò + ô ï 2÷ +1 sinú ( ƒñ ). This is a special case of equation 2.9.2.20 with  ( ) =  sin + ( Œ

ï ñóò ò ó

+( .

and ( ) =  + + ( . and ( ) =  sin + ( Œ

)+( .

and ( ) = ' + . and ( ) =  sin + ( Π ). and ( ) =  + . ).

2ññ ó ò ò ó = (ñ ó ò )2 + õ sin( ï ) ñ 2 – ô . This is a special case of equation 2.9.3.5 with  ( ) = −  sin( Œ  ). = ù (ñó ò )2 – ôñ 4÷ –2 + õ sin( ï ) ñ 2 . This is a special case of equation 2.9.3.8 with  ( ) = −  sin( Œ  ).

ññóò ò ó

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Page 453

454 18. 19. 20. 21. 22. 23. 24.

SECOND-ORDER DIFFERENTIAL EQUATIONS = ù (ñ ó ò )2 + ô ï ú ñ 2 + õ sin  ( ï ) ñ ÷ +1 . This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  sin  ( Œ  ).

ññ óò ò ó

= ù (ñó ò )2 + ô sin( ï ) ñ 2 + õ sin( ï ) ñ ÷ +1 . This is a special case of equation 2.9.3.9 with  ( ) = −  sin( Œ  ) and ( ) = −  sin(“/ ).

ññóò ò ó

= (ñó ò )2 + ô ïƒ÷ ññó ò + õ sin ú ( ï ) ñ 2 . This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  sin + ( Œ  ).

ññóò ò ó

= (ñó ò )2 + ô sin ÷ ( ï ) ññó ò + õ ïƒú ñ 2 . This is a special case of equation 2.9.3.7 with  ( ) = −  sin  ( Œ  ) and ( ) = − ' + .

ññóò ò ó

ï 2 ñ óò ò ó

+ ï ñó ò = ô sin ÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.23 with  ( ) =  sin  ( Œ

( ô ï 2 + õ )ñó ò ò ó + ô ï ñó ò + sin÷ ( ƒñ ) + ö = 0. This is a special case of equation 2.9.2.24 with  ( ) = sin  ( Œ

)+ . )+( .

sin2 ï ñó ò ò ó = ù (ù + 1 – ù sin2 ï )ñ + ô (sin ï ) ÷¿ú +3÷ +2 ñ ú . This is a special case of equation 2.9.1.44 with  (ˆ ) = ˆ + .

2.8.6-2. Equations with cosine. 25. 26.

27. 28. 29.

30. 31. 32.

= – 2 ñ + ô (cos ï )÷ ñ –÷ –3 . This is a special case of equation 2.9.1.41 with  (ˆ ) = ˆ − 

ñóò ò ó

−3

.

+ ô ) cosú ( ï + õ ) ñ –÷ –ú –3 . cos( Œ  +  ) ¨ , = leads to the Emden–Fowler equation: The transformation ˆ = cos( Œ  +  ) cos( Œ  +  ) ¨ Šª

Š = # [ Œ sin(  −  )]−2 ˆ  ¨ −  − + −3 (see Section 2.3).

ñóò ò ó

=–

2

ñ

+




cos÷ (

ï

= õ cos( ï ) ñ + ôñ –3 . This is a special case of equation 2.9.1.2 with  ( ) = −  cos( Œ  ).

ñ óò ò ó

=  cos÷ ( ôñ + õ ï ) +  . This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ cos  ¨ + ý and ( = 0.

ñ óò ò ó

= ô (ñ + õ cos ï )÷ + õ cos ï + ö . The substitution ¨ = +  cos  leads to an autonomous equation of the form 2.9.1.1: ¨2

 =  ¨  + ( .

ñóò ò ó

+ 3ññ ó ò + ñ 3 + ô cos( ï ) ñ = 0. This is a special case of equation 2.9.2.1 with  ( ) =  cos( Œ  ).

ñ óò ò ó

+ ôñ ÷ (ñ ó ò )2 + õ cosú ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) = 

ñ óò ò ó

= ôñ ÷ (ñ ó ò )2 + õ cosú ( ï ) ñ ó ò . This is a special case of equation 2.9.3.38 with  ( ) = 

ñ óò ò ó

 and ( ) =  cos+­ + ( .  and ( ) =  cos + ( Œ  ).

© 2003 by Chapman & Hall/CRC

Page 454

455

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

33.

ôñ ÷ (ñ óò

)2 + õ cosú

ñ (ñ ó ò ) 

ñ óò ò ó

=

ñ óò ò ó

+

ô

cos÷

ñ (ñ ó ò

)2 +

ñ óò ò ó

+

ô

cos÷

ñ (ñ ó ò

)2 + õ cosú ( ƒñ ) + ö = 0.

ñ óò ò ó

=

ô

cos÷

ñ (ñ ó ò

)2 +

ñ óò ò ó

=

ô

cos÷

ñ (ñ ó ò

)2 + õ cosú (

ñ óò ò ó

=

ô

cos÷

ñ (ñ ó ò

)2 +

=

ùñ óò

+ô

.

This is a special case of equation 2.9.4.13 with  ( ) =  34.

35.

36.

37.

38.

39.

40.

õÿñ ú

ö

+

 and ( ) =  cos+­ .

= 0.

and ( ) =  + + ( .

This is a special case of equation 2.9.3.25 with  ( ) =  cos 

and ( ) =  cos + ( Œ

This is a special case of equation 2.9.3.25 with  ( ) =  cos 

õ ï ú ñ óò

.

and ( ) = ' + .

This is a special case of equation 2.9.3.38 with  ( ) =  cos 

ï ) ñ óò

.

and ( ) =  cos + ( Π ).

This is a special case of equation 2.9.3.38 with  ( ) =  cos 

õÿñ ú (ñ óò )

.

and ( ) =  + .

This is a special case of equation 2.9.4.13 with  ( ) =  cos 

ï ñ óò ò ó

ï 2÷

+1

cosú ( ƒñ ).

This is a special case of equation 2.9.2.20 with  ( ) =  cos + ( Œ 2ññ

óò ò ó

= (ñ

óò

)+( .

ï )ñ

)2 + õ cos(

2

).

–ô .

This is a special case of equation 2.9.3.5 with  ( ) = −  cos( Œ  ).

41.

ññ óò ò ó

=

ññóò ò ó

÷

ù (ñ ó ò

)2 – ôñ

=

ù (ñ óò

)2 + ô

ññóò ò ó

=

ù (ñ óò

)2 + ô cos(

ññóò ò ó

= (ñó ò )2 +

ô ƒï ÷ ñ ñóò

ññóò ò ó

= (ñó ò )2 +

ô

4 –2

+ õ cos(

ï )ñ

2

.

This is a special case of equation 2.9.3.8 with  ( ) = −  cos( Œ  ). 42.

43.

ïƒú ñ

2

+ õ cos  (

ï )ñ ÷

+1

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  cos  ( Œ  ).

ï )ñ

2

+ õ cos(

ï )ñ ÷

+1

.

This is a special case of equation 2.9.3.9 with  ( ) = −  cos( Œ  ) and ( ) = −  cos(“/ ). 44.

45.

46.

47.

48.

ï )ñ

2

õ ïƒú ñ

2

+ õ cosú (

.

This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  cos + ( Œ  ). cos÷ (

ï ) ñ ñóò

+

.

This is a special case of equation 2.9.3.7 with  ( ) = −  cos  ( Œ  ) and ( ) = − ' + .

ï 2 ñ óò ò ó

+

ï ñóò

=

ô

cos÷ ( ƒñ ) + õ .

This is a special case of equation 2.9.2.23 with  ( ) =  cos  ( Œ (ô

ï

2

+ õ )ñó ò ò ó +

ô ï ñóò

+ cos÷ ( ƒñ ) + ö = 0.

This is a special case of equation 2.9.2.24 with  ( ) = cos  ( Œ

cos2 ï

ñóò ò ó

=

ù (ù

+ 1 – ù cos2 ï )ñ + ô (cos ï )÷¿ú

÷ ñ ú

+3 +2

 ˆ + . This is a special case of equation 2.9.1.45 with  (ˆ ) = 

)+ . )+( .

.

© 2003 by Chapman & Hall/CRC

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456

SECOND-ORDER DIFFERENTIAL EQUATIONS

2.8.6-3. Equations with tangent. 49.

ñóò ò ó

õ

=

ï )ñ

tan(

+ ôñ

–3

.

This is a special case of equation 2.9.1.2 with  ( ) = −  tan( Œ  ). 50.

51.

tan÷ ( ôñ +

ñóò ò ó

=



ñ óò ò ó

+

ôñ ÷ (ñ óò

õï

)+ .

This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ tan  ¨ + ý and ( = 0. )2 + õ tanú

ñ

+

ö

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  52.

ñóò ò ó

ôñ ÷ (ñ óò

=

)2 + õ tanú (

ï ) ñ óò

.

This is a special case of equation 2.9.3.38 with  ( ) =  53.

ñ óò ò ó

ôñ ÷ (ñ óò

=

)2 + õ tanú

ñ (ñ ó ò ) 

.

This is a special case of equation 2.9.4.13 with  ( ) =  54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

õÿñ ú

ñóò ò ó

+

ô

tan÷

ñ (ñ óò

)2 +

ñóò ò ó

+

ô

tan÷

ñ (ñ óò

)2 + õ tanú ( ƒñ ) +

+

ö

 and ( ) =  tan+

 and ( ) =  tan+ ( Π ).  and ( ) =  tan+

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  tan 

ö

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  tan 

õ ï ú ñ óò

ñ óò ò ó

=

ô

tan÷

ñ (ñ ó ò

)2 +

ñóò ò ó

=

ô

tan÷

ñ (ñ óò

)2 + õ tanú (

ñ óò ò ó

=

ô

tan÷

ñ (ñ ó ò

)2 +

=

ùñóò

+ô

.

This is a special case of equation 2.9.3.38 with  ( ) =  tan 

ï ) ñ óò

.

This is a special case of equation 2.9.3.38 with  ( ) =  tan 

õÿñ ú (ñ óò )

.

This is a special case of equation 2.9.4.13 with  ( ) =  tan 

ï ñóò ò ó

ï 2÷

+1

=

ù (ñ óò

ññ óò ò ó

= (ñ

ï 2 ñ óò ò ó

+

)2 + ô

.

and ( ) =  + + ( . and ( ) =  tan + ( Œ

)+( .

and ( ) = ' + . and ( ) =  tan+ ( Π ). and ( ) =  + .

tanú ( ƒñ ).

This is a special case of equation 2.9.2.20 with  ( ) =  tan + ( Œ

ññóò ò ó

+( .

ïƒú ñ

2

+ õ tan  (

ï )ñ ÷

+ õ tanú (

ï )ñ

+1

).

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  tan  ( Œ  ).

óò

)2 +

ô ï ÷ ñ ñ óò

2

.

This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  tan+ ( Œ  ).

ï ñóò

=

ô

tan÷ ( ƒñ ) + õ .

This is a special case of equation 2.9.2.23 with  ( ) =  tan  ( Œ (ô

ï

2

+ õ )ñ

óò ò ó

+

ô ï ñ óò

+ tan÷ ( ƒñ ) +

ö

)+  .

= 0.

This is a special case of equation 2.9.2.24 with  ( ) = tan  ( Œ

)+( .

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2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

2.8.6-4. Equations with cotangent. 64.

ñóò ò ó

õ

=

ï )ñ

cot(

+ ôñ

–3

.

This is a special case of equation 2.9.1.2 with  ( ) = −  cot( Œ  ). 65.

66.

cot÷ ( ôñ +

ñóò ò ó

=



ñ óò ò ó

+

ôñ ÷ (ñ óò

õï

)+ .

This is a special case of equation 2.9.1.4 with  (¨ ) = ƒ cot  ¨ + ý and ( = 0. )2 + õ cotú

+ ö = 0.

ñ

This is a special case of equation 2.9.3.25 with  ( ) =  67.

ñóò ò ó

ôñ ÷ (ñ óò

=

)2 + õ cotú (

ï ) ñ óò

.

This is a special case of equation 2.9.3.38 with  ( ) =  68.

ñ óò ò ó

ôñ ÷ (ñ óò

=

)2 + õ cotú

ñ (ñ ó ò ) 

.

This is a special case of equation 2.9.4.13 with  ( ) =  69.

70.

71.

72.

73.

74.

75.

76.

77.

78.

õÿñ ú

ñóò ò ó

+

ô

cot÷

ñ (ñ óò

)2 +

ñóò ò ó

+

ô

cot÷

ñ (ñ óò

)2 + õ cotú ( ƒñ ) + ö = 0.

 and ( ) =  cot+­ + ( .  and ( ) =  cot+ ( Œ  ).  and ( ) =  cot+­ .

+ ö = 0.

This is a special case of equation 2.9.3.25 with  ( ) =  cot  This is a special case of equation 2.9.3.25 with  ( ) =  cot 

õ ï ú ñ óò

ñ óò ò ó

=

ô

cot÷

ñ (ñ ó ò

)2 +

ñóò ò ó

=

ô

cot÷

ñ (ñ óò

)2 + õ cotú (

ñ óò ò ó

=

ô

cot÷

ñ (ñ ó ò

)2 +

=

ùñóò

+ô

.

This is a special case of equation 2.9.3.38 with  ( ) =  cot 

ï ) ñ óò

.

This is a special case of equation 2.9.3.38 with  ( ) =  cot 

õÿñ ú (ñ óò )

.

This is a special case of equation 2.9.4.13 with  ( ) =  cot 

ï ñóò ò ó

ï 2÷

+1

=

ù (ñ óò

ññ óò ò ó

= (ñ

ï 2 ñ óò ò ó

+

)2 + ô

and ( ) =  cot + ( Œ

)+( .

and ( ) = ' + . and ( ) =  cot + ( Π ). and ( ) =  + .

cotú ( ƒñ ).

This is a special case of equation 2.9.2.20 with  ( ) =  cot + ( Œ

ññóò ò ó

and ( ) =  + + ( .

ïƒú ñ

2

+ õ cot (

ï )ñ ÷

+ õ cotú (

ï )ñ

+1

).

.

This is a special case of equation 2.9.3.9 with  ( ) = −  + and ( ) = −  cot  ( Œ  ).

óò

)2 +

ô ï ÷ ñ ñ óò

2

.

This is a special case of equation 2.9.3.7 with  ( ) = −   and ( ) = −  cot + ( Œ  ).

ï ñóò

=

ô

cot÷ ( ƒñ ) + õ .

This is a special case of equation 2.9.2.23 with  ( ) =  cot  ( Œ (ô

ï

2

+ õ )ñ

óò ò ó

+

ô ï ñ óò

+ cot÷ ( ƒñ ) +

ö

)+ .

= 0.

This is a special case of equation 2.9.2.24 with  ( ) = cot  ( Œ

)+( .

© 2003 by Chapman & Hall/CRC

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SECOND-ORDER DIFFERENTIAL EQUATIONS

2.8.6-5. Equations containing combinations of trigonometric functions. 79.

80.

81.

82.

83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93.

ñ óò ò ó

ñ

=–

2

¨2Šª

Š = Œ

−2

cos÷ (

ô

+

ï

) sinú (

ï ) ñ –÷ –ú

The transformation ˆ = cot( Œ  ), ¨

ñ óò ò ó

=–

2

ñ

ˆ  ¨ − −+ +

ô

−3

sin ÷ (

=

(see Section 2.3).

ï

)[sin(

ï

–3

.

sin( Π )

) + õ cos(

leads to the Emden–Fowler equation:

ï )]ú ñ –÷ –ú

–3

.

leads to the Emden–Fowler equation: The transformation ˆ = 1 +  cot( Œ  ), ¨ = sin( Œ  ) ¨2Šª

Š =  ( Œ )−2 ˆ +3¨ −  − + −3 (see Section 2.3). = ô (ñ ó ò sin ï – ñ cos ï )  – ñ . The substitution ¨ = 
sin  − cos   sin  ¨  .

ñ óò ò ó

= ô (ñ ó ò cos ï + ñ sin ï )  – ñ . The substitution ¨ = 
cos  + sin   cos  ¨  .

ñ óò ò ó

leads to a first-order separable equation: ¨ 
=

leads to a first-order separable equation: ¨ 
=

= 2(cos ï )–2 ñ + ô (cot ï )÷ +3 ñ ÷ . This is a special case of equation 2.9.1.43 with  (ˆ ) = ˆ  .

ñóò ò ó

= 2(sin ï )–2 ñ + ô (tan ï )÷ +3 ñ ÷ . This is a special case of equation 2.9.1.42 with  (ˆ ) = ˆ  .

ñóò ò ó

= (ù + 1)(tan ï )ñ ó ò + ùñ + ô (cos ï )÷¿ú –2 ñ ú –1 . This is a special case of equation 2.9.2.44 with  (ˆ ) = ˆ +

ñ óò ò ó

+ (2 ôñ + õ sin ï )ñ„ó ò + õ (cos ï )ñ = 0. Integrating yields a Riccati equation: 
 + 

ñóò ò ó

2

.

+  (sin  ) = L .

+ ô ï 2 tan ï ñ ó ò + ù ( ô ï tan ï – ù – 1)ñ = õ ï ÷¿ú This is a special case of equation 2.9.2.47 with  (ˆ ) = 'ˆ +

ï 2 ñ óò ò ó

−1

(cos ï )2 D −3 .

+2

ñ ú

sin ï ñó ò ò ó + 12 cos ï ñó ò = ôñ ÷ + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) = 

–3

.

 + .

sin ï ñó ò ò ó + 12 cos ï ñó ò = ô sin ÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  sin  ( Œ

sin ï ñó ò ò ó + 12 cos ï ñó ò = ô cos÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  cos  ( Œ sin ï ñó ò ò ó + 12 cos ï ñó ò = ô tan÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  tan  ( Œ

)+  . )+ . )+ .

+ (ñ ó ò )2 + ô sin( ï )ññ ó ò + õ cos( ï ) + ö = 0. This is a special case of equation 2.9.3.6 with  ( ) =  sin(ý± ) and ( ) =  cos( Œ  ) + ( .

ññ óò ò ó

– (ñó ò )2 + ô sin( ï )ññ„ó ò + õ cos( ï )ñ 2 = 0. This is a special case of equation 2.9.3.7 with  ( ) =  sin(ý± ) and ( ) =  cos( Œ  ).

ññóò ò ó

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459

2.8. SOME NONLINEAR EQUATIONS WITH ARBITRARY PARAMETERS

2.8.7. Equations Containing the Combinations of Exponential, Hyperbolic, Logarithmic, and Trigonometric Functions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

ó

= 2 ñ + ô 3  (ln ñ + ï ). This is a special case of equation 2.9.1.29 with  (ˆ ) =  ln ˆ .

ñóò ò ó

ó

= – ôñó ò + õD (ln ñ + ô ï ). This is a special case of equation 2.9.2.36 with  (ˆ ) =  ln ˆ .

ñóò ò ó

ó

= ôñó ò + õ 2 D ln÷ ( ƒñ ). This is a special case of equation 2.9.2.17 with  ( ) =  ln  ( Œ

ñóò ò ó

).

ó

= ôñó ò + õ 2 D sin ÷ ( ƒñ ). This is a special case of equation 2.9.2.17 with  ( ) =  sin  ( Œ

ñóò ò ó

).

ó

= ôñó ò + õ 2 D tan÷ ( ƒñ ). This is a special case of equation 2.9.2.17 with  ( ) =  tan  ( Œ

ñóò ò ó

ó

+ ô tan ï ñó ò + õ ( ô tan ï – õ )ñ = ö  ú (cos ï )2 D³ñ This is a special case of equation 2.9.2.46 with  (ˆ ) = (˜ˆ +

ñóò ò ó

ú

–3 −3

). .

.

= ô (ñó ò )2 – õ 4 Df] + ö sinh( ï ). This is a special case of equation 2.9.3.18 with  ( ) = − ( sinh( Œ  ).

ñóò ò ó

= ô (ñó ò )2 + õ cosh÷ ( ï ) Df] + ö ïƒú . This is a special case of equation 2.9.3.17 with  ( ) = −  cosh  ( Œ  ) and ( ) = − (˜ + .

ñóò ò ó

= ô (ñó ò )2 + õ ln÷ ( ï ) Df] + ö ïƒú . This is a special case of equation 2.9.3.17 with  ( ) = −  ln  ( Œ  ) and ( ) = − (˜ + .

ñóò ò ó

ó

= ô (ñó ò )2 + õ ln÷ ( ï ) Df] + ö . Á This is a special case of equation 2.9.3.17 with  ( ) = −  ln  ( Œ  ) and ( ) = − (  .

ñóò ò ó

= ô (ñó ò )2 – õ 4 Df] + ö sin( ï ). This is a special case of equation 2.9.3.18 with  ( ) = − ( sin( Œ  ).

ñóò ò ó

= ô (ñó ò )2 + õ sin÷ ( ï ) Df] + ö ïƒú . This is a special case of equation 2.9.3.17 with  ( ) = −  sin  ( Œ  ) and ( ) = − (˜ + .

ñóò ò ó

ó

= ô (ñó ò )2 + õ sin÷ ( ï ) Df] + ö . Á This is a special case of equation 2.9.3.17 with  ( ) = −  sin  ( Œ  ) and ( ) = − (  .

ñóò ò ó

+ ô] (ñó ò )2 + õ ln÷ ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =   and ( ) =  ln 

ñóò ò ó

+ ô] (ñó ò )2 + õ sin÷ ñ + ö = 0. This is a special case of equation 2.9.3.25 with  ( ) =   and ( ) =  sin 

ñóò ò ó

+( . +( .

= ô ] (ñ ó ò )2 + õ ln ÷ ( ï )ñ ó ò . This is a special case of equation 2.9.3.38 with  ( ) =   and ( ) =  ln  (“/ ).

ñ óò ò ó

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460 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó

=

ô ] (ñ óò

)2 + õ sin ÷ (

ï )ñ ó ò

.

ñóò ò ó

=

ô] (ñ óò

)2 + õ tan÷ (

ï )ñ óò

.

ñóò ò ó

+

ñ óò ò ó

=

ñóò ò ó

=

ñóò ò ó

+

ñ óò ò ó

=

ñóò ò ó

=

ñóò ò ó

=

ñ óò ò ó

+

This is a special case of equation 2.9.3.38 with  ( ) =   and ( ) =  sin  (“/ ). This is a special case of equation 2.9.3.38 with  ( ) =   and ( ) =  tan  (“/ ). ln ÷

ô

ñ (ñ óò

)2 +

ñ (ñ ó ò

)2 +

õ]

+

ö

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  ln 

õ  ó ñ óò

ô

ln÷

ô

ln÷

ñ (ñ óò

)2 + õ sinú (

sin ÷

ñ (ñ óò

)2 +

ñ (ñ ó ò

)2 +

ñ (ñ óò

)2 + õ lnú (

ñ (ñ óò

)2 +

and ( ) =   + ( .

.



This is a special case of equation 2.9.3.38 with  ( ) =  ln 

ï )ñ óò

and ( ) =   .

.

and ( ) =  sin + ( Π ).

This is a special case of equation 2.9.3.38 with  ( ) =  ln 

ô

õ]

ö

+

= 0.

This is a special case of equation 2.9.3.25 with  ( ) =  sin 

ô

sin ÷

ô

sin ÷

ô

tan÷

õ  ó ñ óò

and ( ) =   + ( .

.



This is a special case of equation 2.9.3.38 with  ( ) =  sin 

ï )ñ óò

and ( ) =   .

.

+

This is a special case of equation 2.9.3.38 with  ( ) =  sin 

õ ó ñóò

.

)3 + ôñ

óò

and ( ) =  ln ( Π ).



This is a special case of equation 2.9.3.38 with  ( ) =  tan 

õ D ó

óò

cosh( ƒñ )(ñ

and ( ) =    .

= 0.

This is a special case of equation 2.9.3.35 with  ( ) =  cosh( Π27.

ñóò ò ó

+

õD ó

sin( ƒñ )(ñ„ó ò )3 +

ôñóò

= 0.

This is a special case of equation 2.9.3.35 with  ( ) =  sin( Π28.

ï ñóò ò ó

=

ô ï

Solution: 29.

ññóò ò ó

=

ln ï

]Èñóò

ô ó (ñ óò

)3 + ô

Solution:  = − ln 

Q

30.

ññóò ò ó

+

ô]

.

2

−

k  = − ln  1 O L Q

= (ñó ò )2 +

óñ

k 1  O L

ô ó ññóò

 L

1

X 

  k 1

−1 −

ln ñ (ñó ò )2 . 2

+

L



1

X

+ õ sin÷ ( b

 k 1

−1 −

ï )ñ

2



 + L M P

M P

−

L

( ô

2

( ô

2

ó

+ õ )ñó ò ò ó +



ln S .

1

.

ô 2ó ñ óò

= cosh÷ ( ƒñ ) + ö .

ô 2ó ñ óò

= tanh÷ ( ƒñ ) + ö .

This is a special case of equation 2.9.2.34 with ( ) =  

32.

ó

+ õ )ñ

óò ò ó

+

This is a special case of equation 2.9.2.34 with ( ) =  

).

ln  S .

1



This is a special case of equation 2.9.3.7 with  ( ) = −   31.

).

and ( ) = −  sin  ( @! ).

2



+  and  ( ) = cosh  ( Œ

)+( .

2



+  and  ( ) = tanh  ( Œ

)+( .

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461

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

33. 34. 35. 36.

ó

ó

( ô 2 + õ )ñó ò ò ó + ô 2 ñó ò = ln÷ ( ƒñ ) + ö .  This is a special case of equation 2.9.2.34 with ( ) =   2 +  and  ( ) = ln  ( Œ

ó

ó

( ô 2 + õ )ñó ò ò ó + ô 2 ñó ò = sin÷ ( ƒñ ) + ö .  This is a special case of equation 2.9.2.34 with ( ) =   2 +  and  ( ) = sin  ( Œ

ó

)+( . )+( .

ó

( ô 2 + õ )ñó ò ò ó + ô 2 ñó ò = tan÷ ( ƒñ ) + ö .  This is a special case of equation 2.9.2.34 with ( ) =   2 +  and  ( ) = tan  ( Œ sin ï

ñóò ò ó

+

1 2

cos ï

ñóò

=

ô]

)+( .

+õ .

This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  ! +  .

37. 38. 39. 40.

sin ï ñó ò ò ó + 12 cos ï ñó ò = ô cosh÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  cosh  ( Œ

sin ï ñó ò ò ó + 12 cos ï ñó ò = ô sinh ÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  sinh  ( Œ

sin ï ñó ò ò ó + 12 cos ï ñó ò = ô tanh÷ ( ƒñ ) + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  tanh  ( Œ

sin ï ñ ó ò ò ó + 12 cos ï ñ ó ò = ô ln ÷ ñ + õ . This is a special case of equation 2.9.2.34 with ( ) = sin  and  ( ) =  ln 

)+ . )+ . )+ .

+ .

2.9. Equations Containing Arbitrary Functions ]

Notation:  , , % , Y , and j are arbitrary composite functions of their arguments indicated in parentheses just after the function name (the arguments can depend on  , , 
).

2.9.1. Equations of the Form — ( æ , â )â$äãÑã ä + ˜ ( æ , â ) = 0 2.9.1-1. Arguments of arbitrary functions are algebraic and power functions of  and . 1.

ñóò ò ó

= ! (ñ ). This autonomous equation arises in mechanics, combustion theory, and the theory of mass transfer with chemical reactions. The substitution ‡ ( ) = "
 leads to a first-order separated equation: ‡"‡
=  ( ).



J

−1 2

= L 2 AI . Solution: X L 1 + 2 Xv ( ) S M M Q = #  , where #  are roots of the algebraic (transcendental) Particular solutions: equation  ( #  ) = 0. 2.

+ ! ( ï )ñ = ôñ –3 . Yermakov’s equation. Let ¨

ñ óò ò ó

= ¨ ( ) be a nontrivial solution of the second-order linear

¨ 

 +  ( )¨ = 0. The transformation ˆ = X equation 2

equation of the form 2.9.1.1: H Šª

Š = !H −3 . Solution: L 1 2 =  ¨ 2 + ¨ 2 O L 2 + L 1 X ^_ Reference: V. P. Yermakov (1880).

¨M



2 2

P

¨M



2

, H = ¨

leads to an autonomous

.

© 2003 by Chapman & Hall/CRC

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462 3.

SECOND-ORDER DIFFERENTIAL EQUATIONS + ! ( ï )ñ = "Ió ò ( ï )ñ –1 – " 2 ( ï )ñ –3 . Generalized Yermakov’s equation. ( )  1 J 2 ¨ = ¨ L +2X Solution: ¨ 2M S , where

ñóò ò ó

Q

¨ 

 −  ( )¨ = 0. linear equation 3 4. 5. 6. 7.

8. 9.

= ! ( ôñ + õ ï + ö ). ¨ 

 =  (¨ ). The substitution ¨ =  + ' + ( leads to an equation of the form 2.9.1.1: I

ñóò ò ó

= ! (ñ + ô ï 2 + õ ï + ö ). ¨ 

 =  (¨ )+2  . The substitution ¨ = +  2 + ' + ( leads to an equation of the form 2.9.1.1: p

ñóò ò ó

= ! (ñ + ô ï ÷ + õ ï 2 + ö ï ) – ôù (ù – 1) ï ÷ –2 . ¨ 

 =  (¨ )+2  . The substitution ¨ = +   + ' 2 + (˜ leads to an equation of the form 2.9.1.1: u

ñ óò ò ó

= ï –1 ! (ñ ï –1 ). Homogeneous equation. The transformation a = − ln | |, H =   leads to an autonomous equation: H g

g − H g
=  ( H ). Reducing its order with the substitution ¨ ( H ) = ¨pg
, we arrive at ‘ the Abel equation ¨*¨3
− ¨ =  (¨ ), which is discussed in Subsection 1.3.1.

ñóò ò ó

= ï –3 ! (ñ ï –1 ). ¨ ªŠ

Š =  (¨ ). The transformation ˆ = 1  , ¨ =   leads to an equation of the form 2.9.1.1: I

ñóò ò ó

= ï –3 ) 2 ! (ñ ï –1 ) 2 ). Having set ¨ =  −1 J 2 , we obtain x x  ( ¨2
)2 = equation, we arrive at a separable equation.

ñóò ò ó

Solution: X 10.

= ¨ ( ) is the general solution of the

Q

L

1

+ 14 ¨

2

+2X

 (¨ ) ¨ S M

= ï –2 ! ( ï –  ñ ). Generalized homogeneous equation.

ñóò ò ó

1 b . The 2.9.6.2:

J

−1 2

M

1 2

¨*¨2
+ 2  (¨ )¨2
. Integrating the latter

¨ =L

2

A ln  .

transformation a = ln  , H =  −  leads to an autonomous equation of the form H g

g + (2 , − 1)H g
+ , ( , − 1) H =  (H ).

2 b . The transformation H =  −  , ¨ =  
  leads to a first-order equation: H (¨ − , )¨u‘
= H −1  ( H ) + ¨ − ¨ 2 .

11.

12. 13. 14.

= ñ ï –2 ! ( ïƒ÷ ñ ú ). Generalized homogeneous equation. The transformation H =   + , ¨ ‘ first-order equation: H ( ¨ + B )¨u
=  ( H ) + ¨ − ¨ 2 .

ñóò ò ó

=  
  leads to a

¡

= ù (ù + 1) ï –2 ñ + ï 3÷ ! (ñ ïƒ÷ ). This is a special case of equation 2.9.1.46 with j =  −  .

ñóò ò ó

= ï –3 ) 2 ! ( ôñ ï –1 ) 2 + õ ï 1 ) 2 ). ¨ 

 =  The substitution ¨ =  + ' leads to an equation of the form 2.9.1.9: I

ñ óò ò ó ï (ï

+ ô )2 ñó ò ò ó = ! (ñ

ï

J  (¨ 

−3 2

J ).

−1 2

).

 + û , H =  leads to an autonomous equation of the form 2.9.6.2: û  û
H Šª
Š − H Š
=  −2  ( H ). Reducing û itsû order with the substitution ¨ ( H ) = H Š
, we arrive at an Abel * ¨ equation of the second kind: ¨u‘
− ¨ =  −2  ( H ). See Subsection 1.3.1 for information about The transformation ˆ = ln û

an Abel equation of the second kind.

© 2003 by Chapman & Hall/CRC

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463

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

15.

16.

ñ óò ò ó

=

1

ï 3 !IO ï

ñ

ï

+

ô

+

ï

õ

+ö

.

P ¨ ªŠ

Š = The transformation ˆ = 1  , ¨ =   leads to an equation of the form 2.9.1.5: I  (¨ + ˆ 2 + 'ˆ + ( ). 2

ô ï  ï

+ õÿñ + ö . + õÿñ + ö + ñ + P This is a special case of equation 2.9.1.18.

ñóò ò ó

=

1

ô ï

!IO

1 b . For ý − ˜ƒ = 0, we have an equation of the form 2.9.1.4. 2 b . For ý − ˜ƒ ≠ 0, the transformation

¨ = − , 0

H =  −  0, where 

0

and

0

are the constants determined by the linear algebraic system of equations



0

+

0

+ ( = 0,

ƒ…

0

+ý

0

+ = = 0,

leads to a homogeneous equation of the form 2.9.1.7:

¨ ¨ '‘

‘ = 1   O , H HCP

17.

 + 'ˆ 1 *O .  + 'ˆ ƒ + ý±ˆÑP

 (ˆ ) =

where

ô ï  ï

+ õÿñ + ö . + ñ + P + õÿñ + ö ) This is a special case of equation 2.9.1.20.

ñ óò ò ó

=

1

(ô ï

!IO 3

1 b . For ý − ˜ƒ = 0, we have an equation of the form 2.9.1.4. 2 b . For ý − ˜ƒ ≠ 0, the transformation

¨ = − , 0

H =  −  0, where 

0

and

0

are the constants determined by the linear algebraic system of equations



0

+

0

+ ( = 0,

ƒ…

0

+ý

0

+ = = 0,

leads to a solvable equation of the form 2.9.1.8:

¨ ¨ '‘

‘ = 1  O , H 3 H P

18.

ñóò ò ó

=

ô 1ï

1

+

õ 1ñ

+ö

1

!IO

ô 2ï

ô 3ï

+ +

õ 2ñ

+ö +ö

õ 3ñ

Let the following condition be satisfied: For  2  3 −  3 

2

2 3

û û



P

û û

û ≠ 0, the transformation H =  −  0,

where 

0

and

0

1  + 'ˆ  O . 3 (  + 'ˆ ) ƒ +± ý ˆ P

 (ˆ ) =

where .



1 2 3

(

1

 

1

( 2 ûû = 0. ( 3 ûû û

2 3

¨ = − , 0

are the constants determined by the linear algebraic system of equations

 2

0

+

2 0

+(

2

= 0,

 3

0

+

3 0

+(

3

= 0,

leads to a homogeneous equation of the form 2.9.1.7:

¨ ¨ '‘

‘ = 1   O , H HCP

where

 (ˆ ) =

1

 1 +  1ˆ

*O

 2 +  2ˆ .  3 +  3Ñ ˆ P

© 2003 by Chapman & Hall/CRC

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464 19.

SECOND-ORDER DIFFERENTIAL EQUATIONS 1 ô ï + õÿñ + ö !IO ï .  + ñ + P + õÿñ + ö ) For ý − ˜ƒ ≠ 0, the transformation

ñ óò ò ó

=

ï 2 (ô ï

¨ = − , 0

H =  −  0, where 

0

and

are the constants determined by the linear algebraic system of equations

0



+

0

0

+ ( = 0,

ƒ…

+ý

0

0

+ = = 0,

leads to an equation of the form 2.9.1.14:

¨

H ( H +  0 )2 ¨ '‘

‘ = O 20.

1

,

HCP

ô 2ï

ö

õ 3ñ

Let the following condition be satisfied:

û û

=

(ô 1 ï

+

õ 1ñ

ö

+

For  2  3 −  3 

2

1)

!IO 3

+ +

õ 2ñ

+ +

ñóò ò ó

ô 3ï



ö

û û

û ≠ 0, the transformation

2 3

P

2 3

0

and

1

 + 'ˆ

*O

 + 'ˆ . ƒ + ý±ˆÑP

.



1

(

1

 

1

( 2 ûû = 0. ( 3 ûû û

2 3

¨ = − , 0

H =  −  0, where 

 (ˆ ) =

where

are the constants determined by the linear algebraic system of equations

0

 2

0

+

2 0

+(

2

 3

= 0,

0

+

3 0

+(

(

1   O +  1 ˆ )3 

3

= 0,

leads to a solvable equation of the form 2.9.1.8:

¨ ¨ '‘

‘ = 1  O , 3 H H P 21.

22.

ñ óò ò ó

ñ –3 e ! ” 7

ï

2

+

õï

ô ï

2

+

( 

2


+ ' + ( )2 (‡  )2 = ( 14 

+ ö )3 )

Setting ¨ = ƒ

2

2 3

+  2ˆ . +  3ˆ P

(ô

ï ÷

+ õ )ñ

óò ò ó

ñóò ò ó

õï

=e ! ”

ñ 7

ô ï

+

ï

2 + õ ï

2

=

−3

YIO £ 

ôù (ù

– 1) ï

¨

2

+ ' + ( P

÷ –2 ñ

+ñ

–2

2

− !( )‡

+

.

! O ï ÷ ô

+ 2 Xȇ

−3

 (‡ ) ‡ + L 1 . M

.

+ö •

+ ý± + = and denoting  ( H ) =

2.9.1.21: ¨ 

 = ¨ 23.

ñ

1

. +ö • Setting ‡ ( ) = (  2 + '  + ( )−1 J 2 and integrating the equation, we obtain a first-order separable equation:

(ô

=

 (ˆ ) =

where

1

ƒhH

ñ

3

Y ( H ), we obtain an equation of the form

+õ P

.

 , ¨ = leads to an autonomous equation of the M  (   +  )2   +  3 ¨

¨ ¨ −2  ( ). form 2.9.1.1: ŠªŠ =

The transformation ˆ = X

© 2003 by Chapman & Hall/CRC

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2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

24.

ñóò ò ó

ï

= (ô

) ! (ñ 1 )

+ õ )–2 ñ

–1 2

2

öï

+

) + 2ö 2.

The solution is determined by the first-order equation


(  +  )(  + 2 ( £

) = Y ( (˜ + £

, L ),

(1)

where the function Y = (‡ , L ) is the general solution of the Abel equation of the second kind:

YsY
 − 2( ‡ + ˜( ) Y = 2  (‡ ). Abel equations of this type are discussed in Subsection 1.3.3. By the change of variable 1J 2 = ‡ − (˜ , equation (1) is reduced to the form 2(  +  )(‡ − (˜ )‡
 = Y (‡ , L ). 25.

ñóò ò ó

=

ï

(ö

+ ý )÷

(ô

–1

!e” ï ( ô ï + õ )÷ +2 (ö

ï

+ õ )÷

ñ

+ ý )÷

+1

•

.

 +  (  +  )  , ¨ = leads to an autonomous equation of (˜ + P ( (˜ + )  +1 M M

The transformation ˆ = ln O the form 2.9.6.2:

¨ Šª

Š − (2B + 1)¨ Š
+ B (B + 1)¨ = ¢

−2

 (¨ ),

where

¢ = M

− ˜( .

2.9.1-2. Arguments of the arbitrary functions are other functions. 26.

27.

ó

ó

ñ óò ò ó

=

 –D ! ( D ñ

ñóò ò ó

=

ñ! ( D ó ñ ú

).

 1 b . The substitution H =  

leads to an autonomous equation: H 

 − 2 !H 
+  2 H =  ( H ).  ‘ , ¨ = 
  leads to a first-order equation: H (¨ +  )¨u
= 2 b . The transformation H =  ¨ −1 2 H  (H ) − . ).

Equation invariant under “translation–dilatation” transformation. The transformation  H =   + , ¨ = 
  leads to a first-order equation: H ( ¨ +  )¨u‘
=  ( H ) − ¨ 2 . 28.

¡

ñóò ò ó = ï –2 ! (ƒï ÷  Df]

).

Equation invariant under “dilatation–translation” transformation. The transformation H =      , ¨ =  
leads to a first-order equation: H (  ¨ + B )¨ ‘
=  ( H ) + ¨ . 29.

ñóò ò ó

=

ñóò ò ó

= ! (ñ + ô

2

ñ

+

3

ó

 ! (ñ 

ó

).



This is a special case of equation 2.9.1.46 with j =  −  . 30. 31.

33.

+ õ ) – ô

2



ó

.

 ¨ 

 =  (¨ ). The substitution ¨ = +  ž +  leads to an equation of the form 2.9.1.1: u

ï 2 ñ óò ò ó

=

ï 2 ! (ƒï ÷  ]

The substitution 32.

ó

)+ù . ¨ 

 =  (  = ¨ − B ln  leads to an equation of the form 2.9.1.1: I

ñóò ò ó

= ! (ñ + ô sinh ï + õ ) – ô sinh ï .

ñóò ò ó

= ! (ñ + ô cosh ï + õ ) – ô cosh ï .

The substitution ¨ = The substitution ¨ =

?

).

¨ 

 =  (¨ ). +  sinh  +  leads to an equation of the form 2.9.1.1: I

¨ 

 =  (¨ ). +  cosh  +  leads to an equation of the form 2.9.1.1: I

© 2003 by Chapman & Hall/CRC

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466

SECOND-ORDER DIFFERENTIAL EQUATIONS =

. sinh ï P This is a special case of equation 2.9.1.46 with j = sinh Œ  .

35.

ñ óò ò ó

=

36.

+ (sinh

ñ

+ (cosh

ñ

ñóò ò ó

2

ñ

ï

34.

)–3 !IO

ï

ñ

)–3 ! O

. cosh ï P This is a special case of equation 2.9.1.46 with j = cosh Œ  .

ï 2 ñ óò ò ó

2

ï 2 ! (ñ

=

+ ô ln ï + õ ) + ô .

The substitution ¨ =

¨ 

 =  (¨ ). +  ln  +  leads to an equation of the form 2.9.1.1: I

37.

ñóò ò ó

= –ï

1 ñ + !IO ï . ln ï (ln ï )3 ln P This is a special case of equation 2.9.1.46 with j = ln  .

38.

ñóò ò ó

= ! (ñ + ô sin ï + õ ) + ô sin ï .

ñóò ò ó

= ! (ñ + ô cos ï + õ ) + ô cos ï .

39.

ñ

2

The substitution ¨ =

¨ 

 =  (¨ ). +  sin  +  leads to an equation of the form 2.9.1.1: I

The substitution ¨ = + (sin

ñ

+ (cos

ï

ñ

40.

ñóò ò ó

=–

. sin ï P This is a special case of equation 2.9.1.46 with j = sin Œ  .

41.

ñ óò ò ó

=–

. cos ï P This is a special case of equation 2.9.1.46 with j = cos Œ  .

42.

ñ óò ò ó

= 2(sin ï )–2 ñ + (tan ï )3 ! (ñ tan ï ).

ñóò ò ó

= 2(cos ï )–2 ñ + (cot ï )3 ! (ñ cot ï ).

2

ñ

¨ 

 =  (¨ ). +  cos  +  leads to an equation of the form 2.9.1.1: I

2

)–3 !IO

ï

ñ

)–3 ! O

This is a special case of equation 2.9.1.46 with j = cot  . 43.

This is a special case of equation 2.9.1.46 with j = tan  .

44.

sin2 ï

ñóò ò ó

=

+ 1 – ù sin2 ï )ñ + sin3÷

ù (ù

+ z2

The substitution  = ˆ cos2 ï

ñóò ò ó

=

The transformation ˆ = X cos2    , ¨ M form 2.9.1.1: ¨3Šª

Š =  (¨ ). 46.

ñóò ò ó

=

s s

óò ò ó ñ

+

–3

s

!IO

The transformation ˆ = X Solution: X

Q

L

1

sin ÷

ï

).

leads to an equation of the form 2.9.1.45:

+ 1 – ù cos2 ï )ñ + cos3÷

ù (ù

ï ! (ñ

Šª

Š = B (B + 1 − B cos2 ˆ ) + cos3 

cos2 ˆ 45.

+2

ñ s

P

+2X

j

M

,



2

, ¨ =

 (¨ ) ¨ S M

j

ï ! (ñ

cos÷

ï

ˆ´ ( cos ˆ ).

).

= cos   leads to an autonomous equation of the

( ï ).

=

s

+2

+2

s

¨ ªŠ

Š =  (¨ ). leads to an equation of the form 2.9.1.1: u J

−1 2

M

¨ =L

2

A X

 . M j 2 ( )

© 2003 by Chapman & Hall/CRC

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467

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

47.

ñóò ò ó

=

r

–3

!IO r

ñ

+

r ò òó ó r ñ

+

s P

The transformation a = X 2.9.1.1: ¨3g

g =  (¨ ). Solution: X

M¨

M Y



2

–

, ¨

¨

£ 2 ( ) + L

r

=

= AuX

1

– 2r

óò ò ó s

+j

Y M Y



2

òó óò s

r

,

=

r (ï

),

s

=

s

( ï ).

leads to an autonomous equation of the form +L

where  (¨ ) = X¯ (¨ ) ¨ .

2,

M

2.9.2. Equations of the Form — ( æ , â )â$äãÑã ä + ˜ ( æ , â ) âä ã + ™

(æ , â ) = 0

2.9.2-1. Argument of the arbitrary functions is  . 1.

ñóò ò ó

+ 3ññó ò + ñ

3

+ ! ( ï )ñ = 0. = ¨ OX ¨

The substitution 2.

3.

ñóò ò ó

+ 6ññó ò + 4ñ

ñ óò ò ó

+ 3 !„ññ

5.

6.

¨ 

 +  ( )¨ = 0. leads to a second-order linear equation: u

!„ó ò (ï

) = 0.

The substitution = ¨3
 ¨ leads to a third-order equation of the form 3.5.3.11: ¨*¨p


 + 3¨2
¨2

 + 4  ( )¨*¨2
+  
( )¨ 2 = 0.

óò

+

ñóò ò ó

+

! óò ñ

=¨ O X

 ¨

! 2ñ

3

+ [ ôñ + ! ( ï )]ñ„ó ò +

2

+ " ( ï )ñ = 0,



−1

M P

!„ó ò (ï )ñ

!

= ! ( ï ).

leads to a second-order linear equation: ¨ 

 + ( )¨ = 0.

= 0.

Integrating yields a Riccati equation: 
 +  ( ) + 12 

ñ óò ò ó

+ [2 ôñ + ! ( ï )]ñ

ñ óò ò ó

+ [3ñ + ! ( ï )]ñ

óò

+ ô! ( ï )ñ

2

2

=L .

= " ( ï ).

On setting ‡ = 
+  2 , we obtain ‡
 +  ( )‡ = ( ). Thus, the original equation is reduced to a first-order linear equation and a Riccati equation.

ñóò ò ó

óò

+ñ

3

+ [ñ + 3 ! ( ï )]ñ„ó ò – ñ

3

The substitution % ( )‡ = 0. 7.

M P

+ 4 ! ( ï )ñ +

3

The substitution 4.

−1



+ ! ( ï )ñ

2

+ ! ( ï )ñ

2

+ " ( ï )ñ + % ( ï ) = 0.

= ‡
 ‡ leads to a third-order linear equation: ‡


 +  ( )‡

 + ( )‡
 + + [ !„ó ò ( ï ) + 2 !

2

( ï )]ñ = 0.

The transformation =  ( )¨ ( H ),

H =ù

 ( )  , M

where

 ( ) = exp ¬ − ù

leads to an autonomous equation of the form 2.2.3.2: ¨p‘'

‘ + ¨*¨2‘
− ¨ 8.

ñóò ò ó

– (ù + 1)" (ï )ñ

÷ –1 ñ óò

= ! ( ï )ñ + "Ió ò ( ï )ñ

÷

– " 2 ( ï )ñ 1 1−

÷

2 –1

 , where ¨ = ¨ L + (1 − B ) X ( )¨  −1  S Solution: M Q of the second-order linear equation ¨u

 =  ( )¨ .

3

 ( )  ® , M = 0.

.

= ¨ ( ) is the general solution

© 2003 by Chapman & Hall/CRC

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468 9.

10.

11.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï ñóò ò ó

ô ï 2÷ +1 ñ

– ùñó ò + ! ( ï )ñ =

–3

.

 J 2 leads to Yermakov’s equation 2.9.1.2: The substitution ¨ =  − ´ ¨ 

 +  −2 [  ( ) − 1 B (B + 2)]¨ =  ¨ −3 . 4

ñóò ò ó

+ (2 !„ñ + " )ñó ò +

ñ óò ò ó

+ O 3 !„ñ +

!„óò ñ

2

+ "Ió ò ñ = 0,

!

Integrating yields a Riccati equation: 
 + 

"

ñ

P

ñ óò

+

! 2ñ

3

+

! óò ñ

2

2

= ! ( ï ),

+

=L .

+ (2 !" + % )ñ + "

= " ( ï ).

"

óò

"

+

Here,  =  ( ), = ( ), % = % ( ).

2

= 0.

ñ

‡
 , where ‡ = ‡ ( ) is the general solution of the linear equation:  ( )‡ ¨2
 
‡
 +  !‡ = 0, ‡

 − O + ¨ P  and ¨ = ¨ ( ) is the general solution of another linear equation: ¨p

 + % ( )¨ = 0. =

Solution:

12.

ñóò ò ó

+ [3 ! ( ï )ñ + 2" ( ï ) + % ( ï )ñ –1 ]ñó ò + ! 2 ( ï )ñ 3 + [ !„ó ò ( ï ) + 2 ! ( ï )" ( ï )]ñ 2 + ["Ió ò ( ï ) + " 2 ( ï ) + 2 ! ( ï )" ( ï ) –  ( ï )]ñ + %ƒòó ( ï ) + 2" ( ï ) % ( ï ) +

%

2

( ï )ñ

–1

= 0.

The solution satisfies the Riccati equation 
 +  ( ) 2 + [ ( ) − H ( , L )] + % ( ) = 0, where H = H ( , L ) is the general solution of another Riccati equation: H 
+ H 2 =  ( ). 13.

ñóò ò ó

+ O 2 !„ñ + 2" –

ñ

!

–

2"

ñ

ñóò P

2

+

%„ñ

+

!„ó ò

+

!

2

+"

ñ

óò

+

2 !"

ñ

+

2

"

ñ

2 3

= 0.

Here,  =  ( ), = ( ), and % = % ( ). The solution is determined by the Abel equation of the second kind ! 
= (ln ¨ )
 2 −  ( ) − ( ), where ¨ = ¨ ( ) is the general solution of the linear equation: ¨I

 + % ( )¨ = 0. Abel equations of the second kind are discussed in Section 1.3. 14.

ñóò ò ó

–

ƒñóò

+ ! ( ï )ñ =

ô 2 ó ñ

 The substitution ¨ =  −  J

2

–3

.

leads to Yermakov’s equation 2.9.1.2:

¨ 

 + V  ( ) − 1 Œ 2 W ¨ =  ¨ 4 15.

16.

ñóò ò ó

+ "Ió ò ñóò +

!„ñ

=

ô –2$ ñ

The substitution ¨ =  — J

ñóò ò ó

+

ƒþ

tan(

ï )„ñ óò

2

–3

,

!

= ! ( ï ),

−3

.

= " ( ï ).

"

leads to Yermakov’s equation 2.9.1.2:

¨ 

 + [  − 1 ( 
)2 − 1 

 ]¨ =  ¨ 4 2

+ ! ( ï )ñ = ô [cos(

ï

)]2ú

ñ

–3

This is a special case of equation 2.9.2.15 with = −

¡

−3

.

. ln cos( Π ).

2.9.2-2. Argument of the arbitrary functions is . 17.

ñ óò ò ó

=

ôñ óò

+

ó

 2 D ! (ñ

).

  , we obtain an equation of the form 2.9.2.34. −1 J 2 1   =L 2A Solution: X L 1 + 2 X  ( ) S  . M M  Q

Multiplying both sides by 

−2

© 2003 by Chapman & Hall/CRC

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469

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

18.

ñ óò ò ó

= ! (ñ )ñ

óò

Solution: X 19.

20.

= V

ñóò ò ó

.

M  ( )+ L

^ ó ! (ñ

=L

1

+  , where  ( ) = X

2

 ( )

.

M

)+*  Wñóò .

 ¨
=  ( ). The substitution ¨ ( ) =  − 9 
 leads to a first-order separable equation: u  1  = L 2 +  9 , where  ( ) = Xv ( ) . Solution: X M  ( )+ L 1 ƒ M

ï ñóò ò ó

+ï

ùñóò

=

÷ ! (ñ

2 +1

).



Multiplying both sides by 

−2 −1

, we obtain an equation of the form 2.9.2.34.

1 b . Solution for B ≠ −1:

X Q

L

1

+2X  ( )

1

+2X  ( )

J

−1 2

S M

M

=A

 

+1

+L

2.

= A ln | | + L

2.

B +1

2 b . Solution for B = −1:

X 21.

22.

ï ñóò ò ó

= ! (ñ )ñó ò .

ï ñóò ò ó

= V ï

Solution: X 23. 24.

ï 2 ñ óò ò ó

+

)+

ü

M  ( )+ L

ï ñ óò

= ! (ñ ).

=L

1

S M

J

−1 2

M


 leads to a first-order linear equation: !¨
= −¨ + 1 +  ( ). 

– 1WUñó ò .

+

2

1   , where  ( ) = X¯ ( )

,

.

g leads to an equation of the form 2.9.1.1: 

=  ( ). gg

(ô

+ ! (ñ ) = 0.

ï

2

+ õ )ñó ò ò ó +

ô ï ñóò

£ M


Šª
Š +  ( ) = 0.

( ô

2



ó

+ õ )ñó ò ò ó + ô

2

ó

 ñóò



2

+

leads to an autonomous equation of the form 2.9.1.1:

+ ! (ñ ) = 0.

This is a special case of equation 2.9.2.34 with ( ) =   2 

26.

M

The substitution  = A 

The substitution ˆ = X

25.

Q

The substitution ¨ ( ) = 

! (ñ

L

sin ï

ñ óò ò ó

+

ñóò ò ó

–

1 2

cos ï

1 2

sin ï

ñ óò

= ! (ñ ).

ñóò

= ! (ñ ).



+ .

This is a special case of equation 2.9.2.34 with = sin  .

27.

cos ï

This is a special case of equation 2.9.2.34 with = cos  . 2.9.2-3. Other arguments of the arbitrary functions. 28.

ñ óò ò ó

= ! (ñ )ñ

óò

+ " ( ï ).

Integrating yields a first-order equation:


= X  ( ) M

+ X ( )  + L .

M

© 2003 by Chapman & Hall/CRC

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470 29.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó

+ [ ! ( ï ) + " (ñ )]ñ

óò

! ó ò ( ï )ñ

+

= 0.


+  ( ) + Xw ( )

Integrating yields a first-order equation: 30.

31.

32.

ï ñóò ò ó

+ (ù + 1)ñó ò =

ïƒ÷ –1 ! (ñ ïƒ÷

=L .

M

).

The transformation ˆ =   , ¨ =   leads to an autonomous equation of the form 2.9.1.1: B 2 ¨2Šª

Š =  (¨ ).

ï 2 ñ óò ò ó

+ (ù + þ

+ 1)ï

ï 2 ñ óò ò ó

= ! (ñ

ï )(ï ñ óò

– ñ ).

ï 3 ñ óò ò ó

= ! (ñ

ï )(ï ñ óò

– ñ ).

ñ óò

+ ùþxñ =

ï ÷ –2ú ! (ñ ï ÷

).

1 b . For B ≠ , the transformation ˆ =   − + , ¨ =   leads to an autonomous equation of ¡ the form 2.9.1.1: (B − )2 ¨ Šª

Š =  (¨ ). ¡ 2 b . For B = , the transformation ˆ = ln  , ¨ =   leads to an autonomous equation of ¡ the form 2.9.1.1: ¨3Šª

Š =  (¨ ).

This is a special case of equation 2.9.3.42 with ( ) = % ( ) = 0. 33.

This is a special case of equation 2.9.4.61 with ( H ) = H . 34.

"Iñ óò ò ó

+

1 2

" óò ñ óò

= ! (ñ ),

= " (ï ).

"

Integrating yields a first-order separable equation: ( )( 
)2 = 2 X  ( ) M Solution for ( ) ≥ 0:

X 35.

ñ óò ò ó

–

r óò r ñ óò

+

Q

L

r óò óò O r s s

The transformation ˆ = X 2.9.1.1: ¨3Šª

Š =  (¨ ).

36.

37.

ó ! (ñ D ó

ñóò ò ó

= – ôñó ò + D

ñóò ò ó

+ ( + b )ñó ò + b “ñ =

óò ò ó s

–

j

 ( )

+2X

1

Y s

2

M

P

ñ

S M

=

r

s  , ¨ =

!IO

ñ s

2

r

,

P

 £ M ( ) .

A X =

r (ï

),

s

=

s

( ï ).

leads to an autonomous equation of the form

j

).

 ( –2

)

ó ! (ñ   ó

ï ñ óò ò ó

– ùñ

ï ñóò ò ó

= ! ( ïƒ÷

óò

– ô (ô

ï

+ ù )ñ =

).

 ˆ =  Á , ¨

2 b . For “ = @ , the substitution ¨ ¨ ªŠ

Š =  (¨ ). 2.9.1.1: 3

39.

3

=L

M

1.

  The transformation ˆ =   , ¨ =   leads to an equation of the form 2.9.1.1: ¨uŠª

Š =  1 b . For “ ≠ @ , the transformation of the form 2.9.1.1: (“ − @ )2 ¨2Šª

Š =  (¨ ).

38.

2

J

−1 2

+L

( − )

= 



= 



−2

 (¨ ).

leads to an autonomous equation

leads to an autonomous equation of the form

ï 2÷ +1  3D ó ! (ñ  D ó

).

 This is a special case of equation 2.9.2.35 with Y =   and j =  −  .

Df] )ñóò . The transformation H =      , ¨ H (  ¨ + B )¨2‘
= [  ( H ) + 1]¨ .

=  


leads to a first-order separable equation:

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2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

40.

41.

42.

ï 2 ñ óò ò ó

+ ï ñó ò = ! ( ïƒ÷ The transformation H H (  ¨ + B )¨2‘
=  ( H ).

Df]

).

=      , ¨

44. 45. 46. 47. 48.

leads to a first-order separable equation:

ó

– ô ï 2 ñó ò – ù ( ô ï + ù + 1)ñ = ï 3÷ +2  2 D ! (ñ ïƒ÷ ).  This is a special case of equation 2.9.2.35 with Y =   and j =  −  .

ï 2 ñ óò ò ó

ñóò ò ó

–

r óò r ñóò

r óò – ôpO r

+ô

ñ P

=

ó ó  3D r 2 !  ñD 

  , ¨ The transformation ˆ = XvY  2  M ¨2Šª

Š =  (¨ ). 43.

=  


,

r

=

r (ï

).

 =   leads to an equation of the form 2.9.1.1:

ï 2 ñ óò ò ó

+ ï ñó ò = ! (ñ + ô ln ï + õ ln2 ï ). g The substitution  =  leads to an equation of the form 2.9.1.5: 
g
g =  ( + a + 'a 2 ).

– (ù + 1) tan ï ñ„ó ò – ùñ = cos÷ –2 ï ! (ñ cos÷ ï ). This is a special case of equation 2.9.2.35 with Y = cos − 

ñóò ò ó

 and j = cos−   .

−1

+ (þ – ù ) tan ï ñ„ó ò – ù [(þ + 1) tan2 ï + 1]ñ = cos2ú +÷ï ! (ñ cos÷ ï ). This is a special case of equation 2.9.2.35 with Y = cos + −   and j = cos−   .

ñóò ò ó

ó

ó

+ ô tan ï ñó ò + õ ( ô tan ï – õ )ñ = cos2 D ï  3  ! (ñ  ). This is a special case of equation 2.9.2.35 with Y = cos   and j = 

ñóò ò ó

−





.

+ ô ï 2 tan ï ñó ò + ù ( ô ï tan ï – ù – 1)ñ = ï 3÷ +2 cos2 D ï ! (ñ ïƒ÷ ). This is a special case of equation 2.9.2.35 with Y = cos   and j =  −  .

ï 2 ñ óò ò ó

– ô ï 2 cot ï ñó ò – ù ( ô ï cot ï + ù + 1)ñ = ï 3÷ +2 sin2 D ï ! (ñ ïƒ÷ ). This is a special case of equation 2.9.2.35 with Y = sin   and j =  −  .

ï 2 ñ óò ò ó

2.9.3. Equations of the Form

› — ( æ , â )âäãÑã ä + š 

=0

˜  ( æ , â )(âäã )

= 0 (œ

= 2, 3, 4)

2.9.3-1. Argument of the arbitrary functions is  . 1. 2. 3. 4.

= ! ( ï )(ñó ò + ôñ )2 + ô The substitution ¨ = 
+ 

ñóò ò ó

2

ñ

. ¨ 
=  ¨ +  ( )¨ leads to a Bernoulli equation: u

2

.

= ! ( ï ) + " ( ï )( ï ñ„ó ò – ñ ) + % ( ï )(ï ñ„ó ò – ñ )2 . ¨ 
=   ( ) +  ( )¨ +  % ( )¨ The substitution ¨ ( ) =  
− leads to a Riccati equation: 3

ñóò ò ó

ï ñóò ò ó

+ ( ô + 1)ñó ò = ! ( ï )( ï The substitution ¨ =  
+ 

ñóò

– (ñó ò )2 = ! ( ï ).  1 b . The substitution =  ž  ¨

ññóò ò ó

2 b . The substitutions see 2.7.1.41.

+ ôñ )2 . ¨ 
=  ( )¨ leads to a first-order separable equation: u

−



.

.

   ( ). 2  . For  ( ) = ,  9 +: ,

leads to a similar equation: ¨*¨u

 − (¨2
)2 = 

 = A  J 2 leads to the equation ‡

 = 2  ( ) 

2

2

−2 −2



© 2003 by Chapman & Hall/CRC

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472 5.

SECOND-ORDER DIFFERENTIAL EQUATIONS 2ññó ò ò ó – (ñó ò )2 + ! ( ï )ñ 2 + ô = 0. 1 b . Differentiating with respect to  , we obtain a third-order linear equation: 2 


 + 2  ( ) 
+  
( ) = 0.

2 b . The substitution

=H

2

leads to Yermakov’s equation 2.9.1.2: H 

 + 14  ( ) H = − 14 !H

−3

.

3 b . If ‡ and  are two solutions of the second-order linear equation 4

 +  ( ) = 0, which satisfy the condition (‡" 
− ‡
  )2 =  , then = ‡" is a solution of the original equation. 6. 7. 8.

+ (ñó ò )2 + ! ( ï )ññó ò + " ( ï ) = 0. The substitution ‡ = 2 leads to a linear equation: ‡

 +  ( )‡
 + 2 ( ) = 0.

ññóò ò ó

– (ñ ó ò )2 + ! ( ï )ññ ó ò + " ( ï )ñ 2 = 0. The substitution ‡ = 
 leads to a first-order linear equation: ‡
 +  ( )‡ + ( ) = 0.

ññ óò ò ó

– ù (ñó ò )2 + ! ( ï )ñ 2 + ôñ 4÷ –2 = 0. 1 b . For B = 1, this is an equation of the form 2.9.3.7.

ññóò ò ó

2 b . For B ≠ 1, the substitution ¨ (1 − B )  ( )¨ +  (1 − B )¨ −3 = 0. 9.

=

1−

¨ 

 +  leads to Yermakov’s equation 2.9.1.2: u

– ù (ñó ò )2 + ! ( ï )ñ 2 + " ( ï )ñ ÷ +1 = 0. ¨ 

 + (1 − B )  ( )¨ + The substitution ¨ = 1−  leads to a nonhomogeneous linear equation: u (1 − B ) ( ) = 0.

ññóò ò ó

10.

+ ô (ñó ò )2 + ! ( ï )ññó ò + " ( ï )ñ 2 = 0. ¨ 

 +  ( )2 ¨ 
+ (  + 1) ( )¨ = 0. The substitution ¨ =  +1 leads to a linear equation: 3

11.

ññ óò ò ó

– 2(ñ ó ò )2 – ( !„ñ + 2" )ñ ó ò + ! ó ò ñ Integrating yields a Riccati equation:

12.

ññóò ò ó

– (ñó ò )2 + ( !„ñ 2 + " )ñó ò + !„óò ñ 3 – "Ió ò ñ = 0, 
Integrating yields a Riccati equation:  +  2 + L

13.

ñ óò ò ó

14. 15.

ññóò ò ó

+ " óò 
+ L

ñ

= 0, 2 +

= ! ( ï ), + = 0.

!

= ! ( ï ), − = 0.

!

= " ( ï ).

" "

= " ( ï ).

+ 2( ! + " )ñ + " + 2 % ]ñ ó ò + ! 2 ñ 4 + 2 !"Iñ 3 + (2 ! ó ò + " 2 + 2 !% )ñ 2 + (" ó ò + 2"/% )ñ + % òó + % 2 –  = 0. Here,  =  ( ), = ( ), % = % ( ),  =  ( ). The solution satisfies the Riccati equation 
 +  ( ) 2 + ( ) + % ( ) − H ( , L ) = 0, where H = H ( , L ) is the general solution of another Riccati equation: H 
+ H 2 =  ( ). + (ñ

óò

2

)2 + [2 !„ñ

2

= ! ( ï )(ñ ó ò )2 . The substitution ¨ ( ) = 

ññ óò ò ó

¨ 
= ¨ + [  ( ) − 1]¨ 2. 
 leads to a Bernoulli equation 1.1.5:  u

+ ! ( ï )(ñó ò )2 + " ( ï )ññó ò + % ( ï )ñ 2 = 0. The substitution ‡ = 
 leads to a Riccati equation: ‡
 + (1 +  )‡

ññóò ò ó

ô ï )ñ ó ò ò ó

= ! ( ï )( ï

ñ óò

2

+ !‡ + % = 0.

16.

(ñ +

– ñ )2 .

17.

– ô (ñó ò )2 + ! ( ï ) Df] + " ( ï ) = 0. ¨ 

 − ´ ( )¨ =  ( ). The substitution ¨ =  −   leads to a nonhomogeneous linear equation: u

2 The substitution = −  + "H leads to the equation "HH 

 + 2 HH 
−  3  ( )( H 
) = 0. On


u ¨
¨ ¨ ¨ setting = H  H , we obtain a Bernoulli equation:   + 2 + [ −  3  ( )] 2 = 0.

ñóò ò ó

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473

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

18.

19.

20.

21.

22.

ñóò ò ó

– ô (ñó ò )2 +

+ ! ( ï ) = 0.

õ 4fD ]

¨ 

 −  ( )¨ =  ¨ The substitution ¨ =  −   leads to Yermakov’s equation 2.9.1.2: u

ñóò ò ó

= ! ( ï )(ñó ò sinh ï – ñ cosh ï )2 + ñ .

ñ óò ò ó

= ! ( ï )(ñ

óò

cosh ï – ñ sinh ï )2 + ñ .

ñ óò ò ó

= ! ( ï )(ñ

óò

sin ï – ñ cos ï )2 – ñ .

ñóò ò ó

= ! ( ï )(ñó ò cos ï + ñ sin ï )2 – ñ .

The substitution ¨ sinh 3 ( )¨ 2 .

= 
sinh  − cosh 

The substitution ¨ cosh 3 ( )¨ 2 .

= 
sin  −

The substitution ¨ cos 3 ( )¨ 2 .

cos 

= 
cos  +

sin 

.

¨ 
= leads to a first-order separable equation: p

= 
cosh  − sinh 

The substitution ¨ sin 3 ( )¨ 2 .

−3

¨ 
= leads to a first-order separable equation: p

leads to a first-order separable equation: ¨p
=

¨ 
= leads to a first-order separable equation: p

2.9.3-2. Argument of the arbitrary functions is . 23.

24.

ñóò ò ó

+ ô (ñó ò )2 –

1 2

27.

ó

 ! (ñ

 9  M  ( )+ L ù

26.

=

).

 The substitution ¨ ( ) =  − (
 )2 leads to a first-order linear equation: ¨u
+ 2  ¨ = 2  ( ).  ó ñóò ò ó +  (ñóò )2 = V  _ ! (ñ ) + ·WUñóò . Solution:

25.

ñóò

1

=L

ñóò ò ó

+ ! (ñ )(ñó ò )2 + " (ñ ) = 0.

ñóò ò ó

+ ! (ñ )(ñó ò ) –

ñóò ò ó

=

2

+

ý

1 :   ,

where

 ( )=ù

 9   ( ) M

.

The substitution ¨ ( ) = (
 )2 leads to a first-order linear equation: ¨u
+ 2  ( )¨ + 2 ( ) = 0. 2

1 2

ñ óò =  ó " (ñ



).

 The substitution ¨ ( ) =  − (
 )2 leads to a first-order linear equation: ¨u
+ 2  ( )¨ = 2 ( ). 

ï ! (ñ )(ñ óò

)3 .

Taking to be the independent variable, we obtain a linear equation with respect to  =  ( ): 

= −  ( ) .



28.

ñóò ò ó

+ [ï

! (ñ

) + " (ñ )](ñ„ó ò )3 + % (ñ )(ñó ò )2 = 0.

Taking to be the independent variable, we obtain a linear equation with respect to  =  ( ): 

− % ( )
−  ( ) − ( ) = 0. 29.





ñ óò ò ó

+ ! (ñ )(ñ

óò

)4 + " (ñ )(ñ

óò

)2 + % (ñ ) = 0.

¨
+ 2  ( )¨ The substitution ¨ ( ) = (
 )2 leads to a Riccati equation: 3  (see Section 1.2).

2

+ 2 ( ) ¨ + 2 % ( ) = 0

© 2003 by Chapman & Hall/CRC

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474 30.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï (ñ óò

ï ñóò ò ó

+ô Solution:

 

M  ( )+ L ù

31.

32.

ï ñóò ò ó

+

ï ñóò ò ó

+ ô ï ( ñ óò

1 2

)2 – ! (ñ )ñó ò = 0.

ñóò

=

2

+ ln | |,

where

   ( )

 ( )=ù

+

M

1   .



)2 + " (ñ ).

The substitution ¨ ( ) =  (
 )2 leads to a first-order linear equation: ¨u
= 2  ( )¨ + 2 ( ).



) = V ï ! (ñ ) + ü – 1WUñó ò . 2

 

M  ( )+ L

ù

ï 3 ñ óò ò ó

=L

1

2

+

1   ,

where

,

   ( )

 ( )=ù

M

.

+ [ ï 4 ! (ñ ) + ô ](ñó ò )3 = 0. Taking to be the independent variable, we obtain an equation of the form 2.9.1.2 for  =  ( ): 

−  ( ) −  −3 = 0.



34.

1

ï ! (ñ )(ñ óò

Solution:

33.

=L

= ï –1 [ ! (ñ ) + " (ñ )( ï The substitution ¨ ( ) = 

ñ óò ò ó

ó



2

.

35.

+ D ! (ñ )(ñó ò )3 + ôñó ò = 0.  The substitution ˆ =  −  leads to an equation of the form 2.9.4.36 with ( H ) = !H 3 : 
Šª
Š −  ( )(
Š )3 = 0.

36.

ñ óò ò ó

37.

ñóò ò ó

– ñ ) + % (ñ )(ï ñ ó ò – ñ )2 ]ñ ó ò . 
− leads to a Riccati equation: ¨
=  ( ) + ( )¨ + % ( )¨

ñ óò

+ ! (ñ )(ñ ó ò )4 + " (ñ )(ñ ó ò )2 + % (ñ ) = 0. The substitution ¨ ( ) = (
 )2 leads to a Riccati equation: ¨3
+2  ( )¨



ï ñóò ò ó

2

+2 ( )¨ +2 % ( ) = 0.

+ ï 2ú +1 ! (ñ )(ñó ò )4 + þxñó ò = 0. This is a special case of equation 2.9.4.18 with B = 4 and Y =  − + .

2.9.3-3. Other arguments of arbitrary functions. 38.

= ! (ñ )(ñó ò )2 + " ( ï )ñó ò . Dividing by 
, we obtain an exact differential equation. Its solution follows from the equation:
ln |  | = Xv ( ) + Xw ( )  + L .

ñóò ò ó

M

M

Solving the latter for 
, we arrive at a separable equation. In addition, solution, with L 1 being an arbitrary constant. 39.

ñóò ò ó

=

! (ï ÷ ñ ú

ï ñ

)

(ï

ñóò

The transformation H

=   + , ¨

ó

2

= ! ( D ñ ÷ )(ñó ò )2 .  The transformation H =   ‘ ¨ 2 ¨
¨ H (B +  ) = [  ( H ) − 1] 2 .

ññóò ò ó

1

is a singular

– ñ )ñó ò .

H ( ¨ + B )¨2‘
= [  ( H ) − 1](¨ ¡

40.

=L

=

 


leads to a first-order separable equation:

− ¨ ).

 , ¨

= 
 

leads to a first-order separable equation:

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475

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

41.

ï 2 ñ óò ò ó

ï )(ï ñ óò

= ! (ñ

– ñ )(ñó ò )2 .

This is a special case of equation 2.9.4.20 with , = 2. 42.

ñóò ò ó

ï

–2

=

(ï

! O –ñ ) I

ñóò

Q

ï

ñ

P

The transformation H =   , ¨ [ Hž ( H ) − 1]¨ +  ( H ). 43.

ñóò ò ó

ï –5 ) 2 r (ñ ï –3 ) 2 )(ñóò

=

ñ ï P ñ óò

+ "3O

+I % O

ï

ñ

P

(ñó ò )2 S .


 leads to a Riccati equation: H ¨3‘
= H 2 % ( H )¨

=

2

+

)4 .

This is a special case of equation 2.9.4.21 with B = −3 2,

¡

= 1, and  (H ) = HY ( H ).

2.9.4. Equations of the Form — ( æ , â , â$ä ã )âäãÑã ä + ˜ ( æ , â , âä ã ) = 0 2.9.4-1. Arguments of the arbitrary functions depend on  or . 1. 2. 3.

4. 5. 6.

7.

8.

9.

ñóò ò ó

= ! ( ï )(ñó ò + ôñ )  +

ñóò ò ó

+ ! ( ï )ñó ò + " ( ï )(ñó ò )  = 0.

ñóò ò ó

=

ñóò ò ó

= ! ( ï )( ï

ô 2ñ

.

¨ 
=  ¨ +  ( )¨  . leads to a Bernoulli equation: u

The substitution ¨ = 
+ 

The substitution ‡ ( ) = 
leads to a Bernoulli equation: ‡
 +  ( )‡ + ( )‡  = 0.

ù (ù

– 1) ï

–2

The substitution ¨   +  − ´  ( )¨  .

ñ

+ ! ( ï )( ï

=   
 − B/ 

– ñ ) + " ( ï )(ï

ñóò

The substitution ¨ ( ) = 

ï ñóò ò ó

The substitution ¨ =  
+ 

ô

+ ! ( ï ) (ñ

=

ñóò ò ó

= ! (ï )

ü

óò

ñ„óò


−

+ ( ô + 1)ñó ò = ! ( ï )( ï

ñóò ò ó

– ùñ )  .

ñóò

ñóò

¨ 
= leads to a first-order separable equation: u

−1

– ñ ) .

¨ 
=   ( )¨ +  ( )¨  . leads to a Bernoulli equation: u

+

ôñ )

.

¨ 
=  ( )¨  . leads to a first-order separable equation: u

)2 – 2 ôñ .

Setting ‡ = 
, we rewrite the equation as follows: (‡
 −  )2 [  ( )]−2 = ‡ 2 −2  . Differentiating both sides with respect to  and dividing by (‡
 −  ), we obtain a second-order linear equation: "‡

 =  
‡
 +  3 ‡ −  
. There is also the solution: = 12  ( + L )2 .

ü

ñ óò (ï ñ óò

– ñ ).

The Legendre transformation  = ¨ug
, 1 ¨ g

g = . ¨
 ( g )£ a ¨

! 1 (ï )ñ óÈò ñóò ò ó

+

! 2 (ï )ñ ñóò ò ó

+

! 3 (ï )(ñ óò

¨ g
− ¨ =a2

)2 +

leads to an equation of the form 2.9.4.40:

! 4 (ï )ñ ñóò

+

! 5 ( ï )ñ

2

= 0.

The substitution ¨ ( ) = 
 leads to the Abel equation (  1 ¨ +  2 )¨2
+  1 ¨  4 ¨ +  5 = 0, where   =   ( ).

ñóò ò ó

= ! ( ï )(ñó ò sinh ï – ñ cosh ï )  + ñ .

The substitution ¨ sinh 3 ( )¨  .

= 
sinh  − cosh 

3

+ (  2 +  3 )¨

2

+

¨ 
= leads to a first-order separable equation: p

© 2003 by Chapman & Hall/CRC

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476 10.

11.

12.

13.

14.

15.

16.

17.

18.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó

= ! ( ï )(ñ

óò

cosh ï – ñ sinh ï )  + ñ .

ñ óò ò ó

= ! ( ï )(ñ

óò

sin ï – ñ cos ï )  – ñ .

The substitution ¨ cosh 3 ( )¨  .

= 
cosh  − sinh 

The substitution ¨ sin 3 ( )¨  .

ñóò ò ó

= 
sin  −

= ! ( ï )(ñó ò cos ï + ñ sin ï )  – ñ .

The substitution ¨ cos 3 ( )¨  .

ñóò ò ó

cos 

= 
cos  +

sin 

¨ 
= leads to a first-order separable equation: p

¨ 
= leads to a first-order separable equation: p

¨ 
= leads to a first-order separable equation: p

= ! (ñ )(ñó ò )2 + " (ñ )(ñó ò )  .

The substitution ¨ ( ) = 
leads to a Bernoulli equation: ¨
=  ( )¨ + ( )¨ 

ï ñ óò ò ó

+

ï ÷¿ú

ï

–1

–2

ú

+1



! (ñ )(ñ óò )÷

+ þxñ

óò

−1

.

= 0.

This is a special case of equation 2.9.4.18 with Y =  − + .

V ! (ñ )( ï ñó ò – ñ ) + " (ñ )(ï ñ„ó ò – ñ ) Wñó ò . The substitution ¨ ( ) =  
− leads to a Bernoulli equation: ¨u
=  ( )¨ + ( )¨  . 

ñóò ò ó

=

ñ óò ò ó

= ! (ñ )( ï

ñóò ò ó

+

D (÷

ñóò ò ó

+

r

ñ óò

–ñ )

) (ñ ó ò

1 2

)2 .

¨ g
− ¨ , The transformation  = a 3 1 . 2.9.4.40: ¨ g

g =  (−¨ g
) £ a ¨ –2)

ó ! (ñ )(ñ óò )÷

= −¨2g
, where ¨ = ¨ (a ), leads to an equation of the form

+ ôñó ò = 0.

 This is a special case of equation 2.9.4.18 with Y =  −  . 2–

÷ ! (ñ )(ñ óò )÷

–

r óò r ñóò

= 0,

r

=

r (ï

).

The substitution ˆ = X Y ( )  leads to an equation of the form 2.9.4.36: "
Šª
Š +  ( )(
Š )  = 0.

M

19.

!„ñóò ò ó

+

1 2

!„óò ñóò

=

!" (ñ )(ñ óò

)2 +

! ÷ % (ñ )(ñ óò )2÷

,

!

= ! ( ï ).

 £ M  ( ) leads to an autonomous equation of the form 2.9.4.13: 
Šª
Š = ( )(
Š )2 + % ( )(
Š )2  .

The substitution ˆ = X

2.9.4-2. Arguments of the arbitrary functions depend on  and . 20.

ñ óò ò ó

=

ï –2 ! (ñ  ï )(ï ñ óò

– ñ )(ñ

óò )

.

The transformation H =   , ¨ =  
 leads to a Bernoulli equation: H ¨u‘
= −¨ + H   ( H )¨  . There are particular solutions: = L3 and = L 1 (for , > 0).

© 2003 by Chapman & Hall/CRC

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2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

21.

ñóò ò ó

! (ï ÷ ñ ú

ï ñ

=

)

(ñó ò )

÷ ú ÷ ú

2 + +

.

The transformation H =   + , ¨ =  
  yields: H ( ¨ + B )¨ ‘
= H

¡

 +

−

+

2 ++  ( H )¨  + + + ¨ − ¨ 2 .

1 +1

 divide both sides of this equation by ¨  ++ and introduce the new dependent variable We +  = ¨  + + − ¨ −  + + . As a result, we obtain a first-order linear equation: 2 +



22.

ñóò ò ó

ù

–

=



‘
= − + H −  +1+  ( H ). (B + ) H ¡

üIù – Iü þ ï –1 ñ óò üIþ

+ ï

ñ  ! (ƒï ÷ ñ ú

–1 –

)(ñó ò ) 

+1

.

Passing on to the new variables H =   + and ¨ =  
  , we arrive at a first-order equation:

 The substitution 23.

ñ óò ò ó

þ

=

+

üIþ üIù

+

B (1 − , ) ¨ H ( ¨ + B )¨ ‘
= − ¨ 2 +  ( H )¨  +1 . ¡ ,   ¡ B , − 1 ¨ 1−  − ¨ −  leads to a linear equation: H ‘
= = ¡ +  ( H ). 1−, , ¡

üIù ñ

–1

(ñ

óò

 ñ – –1 ! (ï ÷ ñ ú

)2 + ï

)(ñ

óò ) +2 .

Passing on to the new variables H =   + , ¨ =  
  , we arrive at a first-order equation:

 The substitution 24.

ñóò ò ó

! (ï ÷ ñ ú

ï ñ

=

)

(1 + , ) ¨ H ( ¨ + B )¨ ‘
= ¨ + ¡ ¡ ,!B = ¡ ¨ , (ï

−

 + B ¨ , +1

–ñ )

ñóò

÷ ú ÷

2 +

ñóò ò ó

! (ï ÷ ñ ú

ï ñ

=

)

(ñó ò )

+  ( H )¨ 

+2

.





, +1 leads to a linear equation: H ‘
= − −  ( H ). B



− −1

.

This is a special case of equation 2.9.4.25 with , = 25.

2

÷ ú –÷' ï ÷ +ú ( ñ óò

2B +

B

¡ .



2 +

–ñ ) .

The transformation H =   + , ¨ =  
  leads to a first-order equation:

 −1 2  + + − ´  ++ H ( ¨ + B )¨ ‘
= H  + +  ( H )¨ (¨ − 1)  + ¨ − ¨ 2 . ¡  + 2 ++  on to the new variable  = ¨  ++ − ¨ Multiplying both sides by ¨ −  + + and passing  −1 we arrive at a Bernoulli equation: (B + ) H ‘
= − + H  + +  ( H )  . ¡ 26.

ñóò ò ó

=

ñóò ò ó

=

ï

–2

(ï

ñóò

! O –ñ ) I

ñóò

–ñ ) I ! O

ñ ï P ñ óò

Q

+ "3O

ï

–3

(ï

2

ï

ñ

P

+ï

–3

ñ

P

(ñó ò )  S .


 leads to a Bernoulli equation: H´‡ ‘
= [ H! ( H ) − 1]‡ +

The transformation H =   , ‡ =  H  ( H )‡  . 27.

ï

  + ,

− +

(ï

ñóò



–ñ ) 3 " O

ï

ñ

.

P The transformation  = −1 a , = −¨u a leads to an autonomous equation of the form 2.9.4.13: ¨2g

g =  (¨ )(¨2g
)2 + (¨ )(¨2g
)  .

© 2003 by Chapman & Hall/CRC

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478

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

28.

=

! (ï ÷ ñ ú

ï ñ

)

(ñó ò )

÷ ú –÷' ï ÷ +ú ( ñ óò



2 +

–ñ ) +

" (ï ÷ ñ ú

ï ñ

)

ñóò (ï ñ óò

– ñ ).

 The transformation H =   + , ¨ =  
   , followed by the substitution leads to a Bernoulli equation: (B +

ñóò ò ó

29.

=

! (ï ÷ ñ ú

ï ñ

)

÷ ú (ñó ò ÷ ú 2 + ) +

" (ï ÷ ñ ú

+

The transformation H =   + , ¨



ï ÷ ú ñóò (ï ñ óò – ñ ) + % ( ï ñ ) (ñóò ) ÷ ñ

 , followed  by the  substitution

)

ï ñ

=  


33.

34.

 +ú  (H

(ï

ñóò

–ñ ) . 2

+ ) = ¨  ++

−

1–ü (ñó ò )2 + ï –2 ! ( ïƒ÷ Df] )(ñó ò )  . ù 2–ü Passing on to the new variables H =      and ¨ =  
 , we have

ñóò ò ó

=

ô

¨

2−

 + B

¨

1−

 ,

 2−, ¨ H ( ¨ +  )¨ ‘
= −¨ 2 − +  ( H )¨  . ¡ 1−, ¡ ¨ Multiplying both sides by ¨ −  and introducing the new variable  = ¡ 2 − , H´ ‘
= ( , − 2) +  (H ). we obtain a first-order linear equation: ¡ ¡

2−

 + 

¨

1−

 ,

‘ 1 − ,  +  ( H ). we obtain a first-order linear equation: H´
=

32.

ú

  + ,

− +

− +

 1−, ¨ 2 ¨ H (  ¨ + B )¨ ‘
= + +  ( H )¨  . B 2− , Multiplying both sides by ¨ −  and introducing the new variable  =

31.

+

= ¨  ++ − ¨

1 1  + , leads to a Riccati equation: (B + )H ‘
= H  + + % ( H ) 2 + [ ( H ) − 1] + H −  + +  ( H ). ¡

¨

30.

‘
= [ ( H ) − 1] + H   +−1+  ( H )  .

)H

¡





2−,

1−,

B

2–ü ñóò + ñ 1– ! (D ó ñ ú )(ñóò ) . þ 1–ü  Passing on to the new variables H =   + and ¨ = 
  , we have

ñóò ò ó

ô

=–

ñóò ò ó

=–

ñóò ò ó

=–

þ

ñ óò ñ

ô ñóò

ln O

ô (ñ óò

)2 ln( ï

P

+ ! ( D

ó ñ ú )ñ óò

1−,

.

 The transformation H =   + , ¨ = 
  leads to a first-order equation:  H ( ¨ +  )¨ ‘
= − ¨ ln ¨ − ¨ 2 +  ( H )¨ . ¡ ¡ Dividing both sides by ¨ and passing on to the new variable  = ¨ +  ln ¨ , we obtain a ¡ ‘ first-order linear equation: H´
= − +  ( H ). ¡ ¡ ñóò ò ó = ï –2 ! (ïƒ÷ Df] ) exp O – ô ï ñóò P . ù The transformation H =      , ¨ =  
 leads to the first-order equation H (  ¨ + B )¨ ‘
= ¨ +  ( H ) exp O −  ¨ , which  can  be reduced, with the aid of the substitution = ¨ exp O  ¨ , B P B P to a linear equation: B±H ‘
= +  ( H ).

ù

ñóò

) + ! ( ïƒ÷

Df] )(ñ óò

)2 .

The transformation H =      , ¨ =  
 leads to the equation

 H (  ¨ + B )¨ ‘
= ¨ − ¨ B

2

ln ¨ + ¨

2

 ( H ).

Dividing both sides by ¨ 2 and passing on to the new variable  =  ln ¨ − B ¨ a first-order linear equation: B±H´ ‘
= − + B… ( H ).

−1

, we obtain

© 2003 by Chapman & Hall/CRC

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2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

2.9.4-3. Arguments of the arbitrary functions depend on  , , and
 . 35.

= ! ( ï )" (ñó ò ). The substitution ‡ ( ) = 
leads to a first-order separable equation: ‡
 =  ( ) (‡ ).

36.

ñ óò ò ó

37.

38.

ñóò ò ó

= ! (ñ )" (ñ ó ò ). The substitution ‡ ( ) = 
leads to a first-order separable equation: ‡"‡
=  ( ) (‡ ).  In addition, there may exist solutions of the form = #* + L , where # are roots of the equation ( # ) = 0, L is an arbitrary number, or = $ , where $ are roots of the equation  ( $ ) = 0. = ! ( ô ï + õÿñ + ö )" (ñ„ó ò ). For  = 0, we have an equation of the form 2.9.4.35. For  ≠ 0, the substitution ‡ ( ) =  + (  + ( )  leads to an equation of the form 2.9.4.36: ‡

 =  ( '‡ ) rO˜‡
 − .

ñóò ò ó

hP

ñ óò ò ó = ï –÷ –1 ! (ï ÷ ñ óò

).

¨ 
=  (¨ ) + B ¨ . The substitution ¨ ( ) =   
 leads to a first-order separable equation:  u

39.

= ñ –2÷ –1 ! (ñ ÷ ñó ò ). The substitution ¨ ( ) = 

40.

ñóò ò ó

ñóò ò ó

=

! (ñ ó ò 7 ï

ñ

)


leads to a first-order separable equation: !¨*¨
=  (¨ ) + B ¨ 2 . 

.

Setting ‡ = 
and passing on to the new variables a = X

‡ and ¨ = 2 £  , we have M  (‡ )

= (¨2g
)2 . Differentiating the latter with respect to  , we obtain a second-order linear equation:

¨2g

g = (a )¨ . Here, the function (a ) is defined parametrically: = 1 ‡ , a = X 4 41.

ñóò ò ó

=

! (ñ ó ò

7

ô ï ñ

) +

õ

−2

=

+ '

1 ating both sides with respect to  and passing on to the new variables a = X 2 we obtain an equation of the form 2.9.1.2: H g

g = ‡ (a ) H − ˜H −3 .

ñ óò ò ó

! (ñ ó ò

=

ü

ôñ

+

)

õï

‡

 (‡ )

.

.

Setting ‡ = 
, we rewrite the equation as follows: V £ 2‡
  (‡ )W

42.

M

2

−1

‡

. Differenti-

M , H =£  ,  (‡ )

.

Setting ‡ = 
, we rewrite the equation as follows: V ‡
  (‡ )W

−2

=

both sides with respect to  and passing on to the new variable a = X

+ ' 2 . Differentiating

M

‡

, we obtain a

 (‡ ) second-order linear equation for  =  (a ) integrable by quadrature: 2 g

g = 2 ' + ‡ (a ). Here, ‡ the function ‡ = ‡ (a ) is defined implicitly: a = X M .  (‡ ) 43.

ñóò ò ó

=

! (ñ ó ò

)

. + õÿñ 2 ü to be the independent variable, we obtain an equation of the form 2.9.4.42 for Taking −1 J 2  =  ( ): 

= −(  +  2 )  (1 
)(
)3 .

ô ï







© 2003 by Chapman & Hall/CRC

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480 44.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

= (ô

ñóò ò ó

=

ï

2

+

õï ñ

+ ö^ñ

! (ñ ó ò

)

2

+

 ï

+ ñ + )

) ! (ñ óò

–1 2

).

. +

The transformation  = #*a + $I‡ + L , = c\a . ‡ + Æ , where ‡ = ‡ (a ), reduces this equation by selecting appropriate constants # , $ , L , c , , and Æ , to an equation of the form 2.9.4.41, 2.9.4.42, or 2.9.4.43. 45.

ü

ô ï ñ

+ õ ï 3) 2 + ö ï

.

Setting ‡ = 
, we rewrite the equation as follows: V £ 2‡
  (‡ )W

+ £  +(. 1 ‡ Differentiating both sides with respect to  and passing to the new variables a = X M 2  (‡ ) and H = £  , we obtain a second-order linear equation: 2 H g

g = 2 ‡ (a ) H +  . Here, the function 1 ‡ ‡ = ‡ (a ) is defined implicitly: a = X M . 2  (‡ ) 46.

ñ óò ò ó

=

! (ñ ó ò

48.

49.

50.

51.

= 

)

. + õÿñ 3 ) 2 + ö^ñ ü to be the independent variable, we obtain an equation of the form 2.9.4.45 for Taking −1 J 2  =  ( ): 

= −(  +  3 J 2 + ( )  (1 
)(
)3 .

ô ï ñ 

47.

−2

ñ óò ò ó

=

! (ñ ó ò





)

. + ö ï 3) 2 + ý ï ü The substitution  ¨ =  + ' leads to an equation of the form 2.9.4.45 for ¨  (¨2
−   ) ¨ 

 = £  ¨ + (˜ 3 J 2 +  .

ñóò ò ó

=

ô ï ñ

+õ ï

2

! (ñ ó ò

= ¨ ( ):

M

)

. + õÿñ 2 + ö^ñ 3 ) 2 + ýIñ ü to be the independent variable, we obtain an equation of the form 2.9.4.47 for Taking −1 J 2  =  ( ): 

= −(  +  2 + ( 3 J 2 + )  (1 
)(
)3 .

ñ óò ò ó = ï

ô ï ñ 

–2

(ï ñ óò

óò

– ñ ) ! (ñ

M

).

The Legendre transformation  = ¨ug
, = a ¨2g
− ¨ of the form 2.9.3.14: ¨*¨3g

g = [  (a )]−1 (¨2g
)2 . = Vï

ñ óò ò ó

! (ñ ó ò

) + ñ" (ñ

óò

) + % (ñ

óò

)W

–1





(
 = a , 

 = 1 ¨2g

g ) leads to an equation

.

The Legendre transformation  = ¨ug
, = a ¨2g
− ¨ (
 = a , 

 = 1 ¨2g

g ) leads to a second-order linear equation: ¨3g

g = [  (a ) + aÝ (a )]¨3g
− (a )¨ + % (a ).

(ï

ñóò

+ ôñó ò – ñ + õ )ñó ò ò ó = ! (ñó ò ).

The contact transformation

n o
=  +  , n ol

o = 1  

 ,

− n = 0. where n = n (m ), leads to a linear equation:  (m ) n ol o m

= 
,

n =  
+  
− +  ,

Inverse transformation:

 = n o
− ,

= mn o
− n +  ,


= m ,





 = 1 n l o o .

© 2003 by Chapman & Hall/CRC

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2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

52.

53. 54.

+ " (ñ )ñ ó ò + % ( ï ) = 0. Integrating yields a first-order equation:

! (ñ ó ò )ñ ó ò ò ó

Xv (‡ ) ‡ + Xw ( ) M M


where ‡ =  .

+ Xv% ( )  = L ,

M

= ô 2 ñ + ! ( ï )" (ñó ò + ôñ ). ¨ 
=  ¨ +  ( ) (¨ ). The substitution ¨ = 
+  leads to a first-order equation: u

ñóò ò ó

+ ô ï )ñó ò ò ó + (ñó ò 2 + 2 ôñ )(ñó ò ò ó + ô ) = 0. The contact transformation

! (ñ óò


=  +  , m

n o
= 
,

n = 12 ( 
)2 +  ,



+ 2n where n = n (m ), leads to a linear equation:  (m ) n ol o Inverse transformation: 1

 = 55.

ñ óò ò ó

=

ï –1 I ! Oñ óò

–

ï

ñ P



 m −n o
 ,

(ï

– ñ )( ï

ñóò

2

ñóò ò ó

ô ï ñóò

+

= 
+ 



,

– ôñ ) =

ï 2I ! Oiñóò

n = 

+1


−   ,

+

ô ïñ P

n o
= 

.

+1

,

n ol

o =



leads to a linear equation:  (m )n l o o = (  + 1) n . Inverse transformation:  = (n o
)  57.

58. 59.


= n o
.

¨ 
= −¨ +  (¨ ). leads to a first-order separable equation:  u 

The contact transformation (  ≠ −1)

m

1 V 2 n − ( n o
)2 W , 2

= 0.

.


The substitution ¨ =  − 56.

=



 n ol

o =

,   + 

1 +1

,

=

 1

(mn o − n )( n o )−  +1 ,  +1

= ï –3 ! ( ï ñó ò – ñ )(ñó ò )–1 . The Legendre transformation  = ¨ g
, ¨2g

g = a [  (¨ )]−1 (¨2g
)3 .

ñóò ò ó

= ! ( ï )" ( ï ñó ò – ñ ). The substitution ¨ =  
−

ñóò ò ó

= ï –÷ –3 ñ ÷ ! ( ï ñ ó ò – ñ ). The transformation  = 1 a , ¨2g

g = ¨   (−¨2g
).


=

2

mn o
+ !n . (  + 1) n o


leads to an equation of the form 2.9.3.27:

¨ 
=   ( ) (¨ ). leads to a first-order separable equation: u

ñ óò ò ó

60.

= ï –1 ! (ñ )" ( ï ñó ò – ñ )ñó ò . The substitution ¨ ( ) =  
−

61.

ñ óò ò ó = ï –3 ! (ñ  ï )" (ï ñ óò

ñóò ò ó

= a ¨ g
− ¨



(  + 1)  +2 

 +  
− 

= ¨u a leads to an autonomous equation of the form 2.9.4.36:

leads to a first-order separable equation: ¨
=  ( ) (¨ ).



– ñ ). The transformation  = 1 a , = ¨u a leads to an autonomous equation of the form 2.9.4.36: ¨2g

g =  (¨ ) (−¨2g
).

© 2003 by Chapman & Hall/CRC

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482 62.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï 3 ñ óò ñóò ò ó

+ 2 ï 2 (ñó ò )2 = ! ( ï The contact transformation

m

ñóò

+ ñ ).


+ ,

=

n =

2

n o
=  ,


,

n ol

o =



where n = n (m ), leads to a linear equation:  (m ) n l o o =n . Inverse transformation:  = n o
, 63.

ñóò ò ó

=

ñ !IO ï ñ óò ï 2 ñ

=m

64.

n , n o


66.

n



 =

, ( n o
)2

1 2n − . n o
n ol

o ( n o
)3

¨ 
=  (¨ )+ ¨ − ¨ 2 . 
 leads to a first-order separable equation:  u

ï ñ óò ï ï ƒ ÷ ú –2 + ñ ! ( ñ )3  " O ñ

ñ óò ò ó = ñ (ñóò ) – ï –1 ñ óò 2

The transformation H =   + , ¨ H ( ¨ + B )¨ ‘
=  ( H ) (¨ ). 65.


=

.

P

The substitution ¨ ( ) =  –1

−

1

 

 + 2 


=  
 

P

.

leads to a first-order separable equation:

¡

= ù (ù – 1) ï –2 ñ + ! ( ï )" ( ïƒ÷ ñó ò – ù ïƒ÷ –1 ñ ). ¨ 
= B   −1 leads to a first-order separable equation: u The substitution ¨ =   
 − / ¨    ( ) ( ).

ñóò ò ó

+ ô ï ñó ò – ôñ = ï D +2 !  ï D +1 ñó ò – ï The contact transformation (  ≠ −1):

ï 2 ñ óò ò ó

m

= 
+ 



,

n = 

+1


−   ,

D¦ñ 

.

n o
= 

+1

,

n ol

o =

(  + 1)  +2 

 +  
− 

 2

leads to an autonomous equation of the form 2.9.1.1:  ( n ) n olo =  + 1. Inverse transformation:

 = (n o
)  67.

68.

=

,

 1 (mn o
− n )( n o
)−  +1 ,  +1

= – ï –1 ñó ò + ï –2 ! ( ïƒ÷ Df] )" ( ï ñó ò ). The transformation H =      , ¨ =  
 H (  ¨ + B )¨2‘
=  ( H ) (¨ ).

ñóò ò ó


=

mn o
+ !n . (  + 1) n o


leads to a first-order separable equation:

ó

= ñ –1 (ñ ó ò )2 + ñ! (  D ñ ú )" (ñ ó ò ¦ñ ).  The transformation H =   + , ¨ = 
 ‘ ¨ 2 ¨
¨ H ( +  ) =  ( H ) ( ).

ñ óò ò ó ¡

69.

1 +1

ó

(ñó ò + ñ )(ñó ò ò ó + ñó ò ) =  –2 ! (  The contact transformation

ó ñóò

leads to a first-order separable equation:

).

  −2 n ol

o = −

,  + 


= −n . where n = n (m ), leads to a linear equation:  (m ) n ol o m

=


 ,

n = 
+ ,

 n o
=  − ,

Inverse transformation:

70.



= mn o
.  = − ln n o
, = n − mn o , ñóò ò ó – ñóò =  2ó !   ó ñóò –  ó ñ   ñóò ò ó – ñ  .   1 b . The substitution ¨ = 
− leads to a first-order equation: ¨u
=  2  (  ¨ )(¨2
+ ¨ ).

© 2003 by Chapman & Hall/CRC

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483

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

2 b . The contact transformation




=  (  − ), m

n = ( 
)2 −



leads to a linear equation: n l o o

71.

72.

73.

74.

2

,

 n o
= 2 − 
,

n ol

o = 2 

−2



 − 


 − 

= 2  (m ).

= ! ( ï )" (ñó ò sinh ï – ñ cosh ï ) + ñ . ¨ 
= The substitution ¨ = 
sinh  − cosh  leads to a first-order separable equation: p ¨ sinh 3 ( ) ( ).

ñóò ò ó

= ! ( ï )" (ñó ò cosh ï – ñ sinh ï ) + ñ . ¨ 
= The substitution ¨ = 
cosh  − sinh  leads to a first-order separable equation: p ¨ cosh 3 ( ) ( ).

ñóò ò ó

= ! ( ï )" (ñó ò sin ï – ñ cos ï ) – ñ . The substitution ¨ = 
sin  − cos  sin 3 ( ) (¨ ).

ñóò ò ó

¨ 
= leads to a first-order separable equation: p

= ! ( ï )" (ñ ó ò cos ï + ñ sin ï ) – ñ . The substitution ¨ = 
cos  + sin  cos 3 ( ) (¨ ).

ñ óò ò ó

leads to a first-order separable equation: ¨ 
=

7

75.

+ 12 "Ió ò ñóò = ! (ñ ) %  ñóò "  , " = " (ï ). The substitution ¨ ( ) = 
£ leads to a first-order separable equation: ¨*¨u
=  ( ) % (¨ ).

76.

!  ôñóò – õ ï  ñ óò ò ó

"Iñóò ò ó

+  2 ôñó ò – 3 õÿñ   2 ôñó ò ñóò ò ó – õ  = 0, The contact transformation 2

=  ( 
)2 − ' ,

m

n = 2  ( 
)3 − 3  ,



+ 3n leads to a linear equation:  (m )n ol o Inverse transformation:  = 77. 78.



9

1 ( n o
)2 − m ,



=

= !  ñó ò 2 + ôñ  . The substitution ¨ ( ) = (
 )2 + 

ñóò ò ó

= ! (ñ )"  ñó ò 2 + 2 ôñ  – ô . The substitution ¨ ( ) = (
 )2 +2 

n o
= 3 
,

≠ 0.

n ol

o =

3






 − 

2

= 0.

2 1 n , ( n o
)3 − 81  3

ñóò ò ó



õ

3


=

1
n o , 3



 =

3 ˜n ol

o

. 2 !n o
n ol

o − 9

leads to a first-order separable equation: ¨u
= 2  (¨ ) +  .



leads to a first-order separable equation: ¨u
= 2  ( ) (¨ ).



ñ óò ò ó = !  ñóò – 2ô ï ñ óò

79.

+ 2 ôñ  . ¨ = + 1   2 leads to an autonomous equation of the form 2.9.4.78: The substitution ¨2

 =  (¨2
2 + 2  ¨ ) −  . 2

80.

ñóò ò ó

2

ó

– ñó ò =  2 !  ñó ò 2 – ñ 2   The contact transformation

m

 =  ( 
− ),

ñóò ò ó

–ñ  .

n = ( 
)2 −

2

,

 n o
= 2 − 
,

n ol

o = 2 



leads to an autonomous equation of the form 2.9.1.1: n l o o = 2  ( n ).

−2

 

 − 
 

 −

© 2003 by Chapman & Hall/CRC

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484 81.

82.

83.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ï ñ óò ò ó

ñ óò

=u ! 

ï ñ óò 2 + ôñ

. ¨ 
The substitution ( ) =  (  )2 +  +

1 2

leads to a first-order separable equation: ¨u
= 2  (¨ ) +  .



ñóò ò ó + 12 ñ óò =  –ó !   ó ñ óò 2 + ôñ

.  
2 ¨ The substitution ( ) =  (  ) +  leads to a first-order separable equation: ¨u
= 2  (¨ )+  .  !  ô (ñóò ) – õ ï  ñóò ò ó = V ôƒü (ñóò ) +1 – õ ( ü + 1)ñ±W V ôƒü (ñóò ) –1 ñóò ò ó – õW . The contact transformation (  ≠ 0, , ≠ −1)

m =  ( 
)  − ' ,

n = , ( 
) 

+1

n o
= ( , + 1) 
,

−  ( , + 1) ,

n ol

o =

( , + 1) 

 , ( 
)  −1 

 − 



= ( , + 1) n . leads to a linear equation:  (m )n ol o Inverse transformation:  (n o
)  m − ,   ( , + 1) 

 = 84.

85.

86.

r ñ óò ò ó

+

1 2

r òó ñ óò

=u ! 

r ñ óò 2 + ôñ 

r

,

The substitution ¨ ( ) = Y ( )(
 )2 +  2  (¨ ) +  .

r ñóò ò ó

+

1 2

r òóñóò

= ! ( ñ )" 

r ñóò 2 + 2ôñ 

The substitution ¨ ( ) = Y ( )(
 )2 + 2  2  ( ) (¨ ).

ñ óò ò ó

=

ï ÷ ñ ú (ñ óò ) 2÷ ÷ ++ú ú

+3 +2

H (

, ( n o
)   ( , + 1) 

=

),



r (ï

=

+1 +2

−

n

 ( , + 1)

,


=

n o
.

, +1

).

leads to a first-order separated equation: ¨u
=



–ô ,

r

=

r (ï

).

leads to a first-order separable equation: ¨u
=



= (ï

ñ óò

– ñ )(ñ

÷ +1 óò )– ÷ +ú +2 .

The Legendre transformation  = ¨ug
, = a ¨2g
− ¨ leads to an equation of the form 2.9.4.25: a ¨ B +1 ¨ g

g = 1 ¨
2  +1−   ¨
¨ ¨a Q  (a  ¨ ) S ( g )  +1 (a g − )  , where  = − B + + 2 ,  = −¡ .

¡

2.9.5. Equations Not Solved for Second Derivative 1.

! (ï )(ñ óò ò ó

– ô )2 = (ñó ò )2 – 2 ôñ + õ .

Differentiating with respect to  , we obtain ( 

 −  )(2 




 +  


 − 2 
−  
) = 0.

Equating the second factor to zero and making the transformation ˆ = X

(1)

 ¨ 
£ M  , =  , we

arrive at a second-order constant coefficient linear equation of the form 2.1.9.1:

¨ Šª

Š − ¨ = 1   
, 2

(2)

whose right-hand side is to be expressed in terms of ˆ . Substituting the solution of equation (2) into the original one, we obtain a relation connecting integration constants. 1  Equating the first factor in (1) to zero, we find the singular solution: =  ( + L )2 + . 2 2 © 2003 by Chapman & Hall/CRC

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485

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

2.

! (ï )(ñ óò ò ó

– ôñ )2 = (ñó ò )2 – ôñ 2 + õ . Differentiating with respect to  , we obtain ( 

 − 

)[2  ( 


 − 


) +  
( 

 −  ) − 2 
] = 0.

Equating the second factor to zero, we arrive at a third-order linear equation:


) +  
( 

 −  ) − 2 
= 0.

2  ( 


 − 

Equating the first factor to zero, one can find the singular solution.

ï

3.

= ! (ñ ó ò ò ó ). ¨ 
). The substitution ¨ ( ) = 
leads to an equation of the form 1.8.1.7:  =  (p

4.

ñ = ! (ñóò ò ó ). The substitution ¨ ( ) = 12 (
 )2 leads to an equation of the form 1.8.1.8:

5.

ñ =ô ï

=  (¨p
).



+õ ï

+ ö + ! (ñó ò ò ó ). ¨ 

 +2  ). The substitution ¨ = −  2 − ' − ( leads to an equation of the form 2.9.5.4: ¨ =  (p 2

ï ñóò

6.

= ñ + ô (ñó ò )2 + õÿñó ò + ö + ! (ñó ò ò ó ). The Legendre transformation  = ¨ug
, = a ¨2g
− ¨ (
 = a , 

 = 1 ¨2g

g ) leads to an equation of the form 2.9.5.5: ¨ = a 2 + 'a + ( +  (1 ¨2g

g ).

7.

! (ñ óò ò ó

8.

)+ Solution:

ï ñóò ò ó =

= 1 2

ñóò

L 1

. 2

+   ( L

1)

+L

2.

! (ñ óò ò ó

+ ñ ) = (ñó ò ò ó )2 + (ñó ò )2 . Differentiating with respect to  , we obtain

[ 
( 

 + ) − 2 

 ]( 


 + 
) = 0.

From the equation 


 + 
 = 0, it follows that: = # sin( + L

1)

+L

2,

where

#

2

=  (L

2 ).

Equating the expression in square brackets to zero, we arrive at the singular solution in parametric form: [2 −  

 (‡ )] ‡
 =ù M , = ‡ − 12   (‡ ). 2 V
Å 4  ( ‡ ) −   ( ‡ )W 9. 10.

ñóò

= ñ! (ñó ò ò ó¦ñ ). The transformation a =

ññóò ò ó

= (ñó ò )2 + 1 b . Solution:

2

, ¨ = (
 )2 leads to an equation of the form 1.8.1.11: ¨ = aª 2 (¨2g
).

!  ñóò ò ó¦ñ 

. =L

2 b . Solution:

exp( L

=L

where the constants L 3 b . Solutions:

1

1

2

 )+

sin( L

3

 ( L 22 ) exp(− L 2  ). 4 L 1 L 22

 )+L

2

cos( L

3

 ),

L 2 , and L 3 are related by the constraint ( L = A" £ −  (0) + L . 1,

2 1

+L

2 2)

L

2 3

+  (− L

2 3)

= 0.

© 2003 by Chapman & Hall/CRC

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486 11. 12. 13.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñ

=

ï ! (ï 3 ñ óò ò ó

).

The transformation  = 1 a ,

ï ñóò

– ñ = ! ( ïƒ÷

ï ñóò

– ñ = ! ( 

ï ñóò

– ñ = ! (ln ï

ï ñóò

– ñ = ! (sin ï

ñ„óò ò ó

ï ñ óò

– ñ = ! (r

),

¨ g

g ). = ¨u a leads to an equation of the form 2.9.5.4: ¨ =  (I

ñóò ò ó ). This is a special case of equation 2.9.5.16 with Y =   . ó ñóò ò ó

).

ñóò ò ó

).



This is a special case of equation 2.9.5.16 with Y =   . 14.

This is a special case of equation 2.9.5.16 with Y = ln  . 15.

).

This is a special case of equation 2.9.5.16 with Y = sin  . 16.

ñ óò ò ó

r

=

r (ï

).

  , ¨ =  
 − Y M Z

The transformation ˆ = X

¨ =  (2 ¨ Š
).

leads to an equation of the form 1.8.1.8:

2.9.6. Equations of General Form 2.9.6-1. Equations containing arbitrary functions of two variables. 1. 2.

3.

ñóò ò ó

=

H (ï , ñ óò

).

ñ óò ò ó

=

H (ñ , ñ ó ò

).

ñóò ò ó

=

H (ô ï

õÿñ , ñ óò

¨ 
=  ( , ¨ ). The substitution ¨ ( ) = 
leads to a first-order equation: u Autonomous equation. The substitution ¨ ( ) = "
 leads to a first-order equation: ¨*¨u
=   ( , ¨ ). +

).

The substitution  ¨ =  +  4.

ñóò ò ó

=

ñ 1 H O ï , ñ óò ï w

P

 leads to an equation of the form 2.9.6.2: ¨ 

 = O ¨ , ¨ 
−

hP

.

.

Homogeneous equation. This is a special case of equation 2.9.6.6 with , = 1. 5.

ô ï  ï

+ õÿñ + ö , ñó ò . P + õÿñ + ö + ñ + 1 b . For ý − ˜ƒ = 0, the substitution  ¨ =  +  the form 2.9.6.2.

ñóò ò ó

=

1

ô ï

HwO

+ ( leads to an autonomous equation of

2 b . For ý − ˜ƒ ≠ 0, the transformation

H =  −  0, where 

0

and

0

¨ = − , 0

are the constants determined by the linear algebraic system of equations



0

+

0

+ ( = 0,

ƒ…

0

+ý

0

+ = = 0,

© 2003 by Chapman & Hall/CRC

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487

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

leads to a homogeneous equation of the form 2.9.6.4:

¨ ¨ '‘

‘ = 1 ` O , ¨ ‘
, H H P

6.

7.

ñóò ò ó

=

ï –2 H (ï – ñ , ï 1– ñ óò

where

` (ˆ , ‡ ) =

 + 'ˆ 1  O ,‡ .  + 'ˆ ƒ +± ý ˆ P

).

Generalized homogeneous equation. The transformation a = ln  , ¨ =  −  ¨ g
+ , ( , − 1)¨ =  (¨ , 3 ¨ g
+ , ¨ ). equation of the form 2.9.6.2: ¨ug

g + (2 , − 1)3

ñ óò ò ó

=

ñ ï ÷ ú ï 2H O ñ

ï

,

ñ óò ñ

P

.

Generalized homogeneous equation. The transformation H =   + , ¨ ‘ first-order equation: H ( ¨ + B )¨
=  ( H , ¨ ) + ¨ − ¨ 2 . 8. 9.

10.

ñóò ò ó

=

ô 2ñ

ñ óò ò ó

= (ô

ñóò ò ó

=w H O

+

ï

+ ôñ ).

+ ô )ñ +

H (ï , ñ óò

–

ñ ï

12. 13. 14. 15. 16. 17.

18.

–ô

ñóò ò ó

=

ñóò ò ó

=

H (ï , ï ñ óò

¨ 
= −¨ + " ( , ¨ ). leads to a first-order equation:  u 

– ñ ).

¨ 
= " ( , ¨ ). leads to a first-order equation: u

ï –2 H (ñ , ï ñ óò

– ñ ). The substitution ¨ ( ) =  
−

leads to a first-order equation: ( + ¨ )¨u
=  ( , ¨ ).

+ ( ô + 1)ñ ó ò = H ( ï , ï ñ ó ò

+ ôñ ).

The substitution ¨ =  
+ 

ï 2 ñ óò ò ó

= 2ñ +

ï 2 ñ óò ò ó

= ô ( ô + 1)ñ +

H ( ï , ï ñ óò

óò

H (ï , ï „ñ óò

H (ï , ñ óò


+ 

= 2 ôññ

The substitution ¨ = 
− 

2

ñóò ò ó

=

).

ó

ó

ñóò ò ó

=

ñH ( D ó ñ ú

,



+ ôñ ). ¨ 
= (  + 1)¨ +  ( , ¨ ). leads to a first-order equation:  u

– ôñ 2 ).

ó

 –D H ( D ñ ,  D ñóò

The substitution ¨ =  ¨2

 − 2  ¨2
+  2 ¨ = 

¨ 
=  ( , ¨ ). leads to a first-order equation: u

leads to a first-order equation:  ¨ 
= 2¨ +  ( , ¨ ).

ñ óò ò ó

+



+ ñ ).

The substitution ¨ =  
+ The substitution ¨ = 

).

¨ 
= −  ¨ +  ( , ¨ ). leads to a first-order equation: u

The substitution ¨ ( ) =  
−

ï ñ óò ò ó

ï ñ

.

P

The substitution ¨ ( ) = 
− 11.

¨ 
=  ¨ +  ( , ¨ ). leads to a first-order equation: u

The substitution ¨ = 
− 

ï , ñ óò

=  
  leads to a

¡

H (ï , ñ óò

The substitution ¨ = 
+  2 2

leads to an

¨ 
=  ( , ¨ ). leads to a first-order equation: u

leads to a second-order autonomous equation of the form 2.9.6.2: ¨ 
−  ¨ ). (¨ , 2

ñóò ¦ñ

).

Equation invariant under “translation–dilatation” transformation. The transformation  H =   + , ¨ = 
  leads to a first-order equation: H ( ¨ +  )¨u‘
=  ( H , ¨ ) − ¨ 2. See also ¡ Paragraph 0.3.2-7.

© 2003 by Chapman & Hall/CRC

Page 487

488 19.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó

=

ï –2 H (ƒï ÷  Df] , ï ñ óò

).

Equation invariant under “dilatation–translation” transformation. The transformation H =      , ¨ =  
 leads to a first-order equation: H (  ¨ + B )¨u‘
=  ( H , ¨ ) + ¨ . See also Paragraph 0.3.2-7. 20.

21.

22.

23.

24.

25.

ñ óò ò ó

=

 2fD ] H (ï  Df] ,  –fD ] ñ óò

ñ óò ò ó

=

ô ] ñ óò

ñóò ò ó

= (

ñ óò ò ó

=

H (ï , ñ óò

sinh ï – ñ cosh ï ) + ñ .

ñóò ò ó

=

H (ï , ñ óò

cosh ï – ñ sinh ï ) + ñ .

ñ óò ò ó

=

ï –2 H (ô ñ

ñ óò ò ó

=

ñH (ô ï

+ õ ln ñ ,

ñóò ò ó

=

H (ï , ñ óò

sin ï – ñ cos ï ) – ñ .

ñóò ò ó

=

H (ï , ñ óò

cos ï + ñ sin ï ) – ñ .

ñóò ò ó

= (r

ñ óò ò ó

=

+

H (ï , ñ óò

– ô

27.

28.

29.

30.

]

).

The substitution ¨ = 
−    leads to a first-order equation: ¨ 
=  ( , ¨ ). 2

ó

+

ó )ñ



+

H  ï , ñ óò

 The substitution ¨ = 
− 

–

 leads to a first-order equation: ¨ 
= −  ¨ +  ( , ¨ ).

+ õ ln ï ,

ï ñ óò

).

=  
 leads to a first-order equation: (  ¨ +  )¨u‘
=

+  ln  , ¨

ñ óò ¦ñ

).

‘ = 
 leads to a first-order equation: (  ¨ +  )¨
=

The transformation H =  +  ln , ¨  (H , ¨ ) − ¨ 2.

¨ 
=  ( , ¨ ) sin  . The substitution ¨ = 
sin  − cos  leads to a first-order equation: u ¨ 
=  ( , ¨ ) cos  . The substitution ¨ = 
cos  + sin  leads to a first-order equation: u 2

+

r òó )ñ

+

H (ï , ñ óò


The substitution ¨ =  − Y

r ò òó ó r ñ

+w H O

ï , ñ óò

–

–

ñ óò ò ó

=

r ñ óò ]

+

r ó

+

r ñ

r óò r ñ P

H (ï , ñ óò

r

),

=

r (ï

).


leads to a first-order equation: ¨  = − Y ¨ +  ( , ¨ ).

r

,

=

r (ï

).

Y
 ¨
leads to a first-order equation: ¨  = − +  ( , ¨ ).

Y

32.

.

¨ 
=  ( , ¨ ) cosh  . The substitution ¨ = 
cosh  − sinh  leads to a first-order equation: u



Y  The substitution ¨ =  − 31.

óñ

¨ 
=  ( , ¨ ) sinh  . The substitution ¨ = 
sinh  − cosh  leads to a first-order equation: u

The transformation H =   (H , ¨ ) + ¨ . 26.

).

The transformation H =     , ¨ =  −   
 leads to a first-order equation: ( !H ¨ + 1)¨u‘
=  (H , ¨ ) −  ¨ 2 .

– r ),

r

=

r (ï , ñ

Y

).

The substitution ¨ = 
− Y ( , ) leads to a first-order equation: ¨ 
=  ( , ¨ ).

! 2 ñ óò ò ó

+

! !„óò ñóò

=

ž (ñ , „! ñóò

The substitution ¨ ( ) = 

),

!

= ! ( ï ).


leads to a first-order equation: ¨*¨u
= ` ( , ¨ ). 

© 2003 by Chapman & Hall/CRC

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489

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

33.

ñ óò ò ó

=

H (ï , ñ

). Let  ≠ Y ( ) + j ( ), i.e., the equation is nonlinear. Then its order can be reduced by one if the right-hand side of the equation has the following form:

˜ J «E¦±` (‡ ) + X V 1 / 


 (‡ + ) +  1 J 2 

 « 2 

−3 2

 ( , ) = 

−1

W t  § , M

(1)

where

˜ = X  −3 J 2 « −1  , ‡ =  −1 J 2 « −1 − ˜ ;  , M P M ` = ` (‡ ),  =  ( ), and = ( ) are arbitrary functions, and , is˜ an arbitrary constant. The integral in (1) can always be expressed in terms of « and . The following cases are −1

« = exp O, X 

possible:

1 b . For  


 ≠ 0,

 ( , ) = 

J «ç` (‡ ) + 1  4

−3 2

2 b . For  = 

 ( , ) = 

2

−2

V 2 / 

 − (  
)2 W

−2

+ 12 

 2  
−  
+ 2 ,  + , 2 

+ ' + ( ,  
≠ −2 , ,  
≠ 23 , ,

J «ç` (‡ )+ 1  2

−3 2

−2

 2  
−  
+2 ,  +  , 2 + 14 ¢  

J «

−3 2

˜

,

J «

−3 2

˜

.

¢ = 4!( −  2.

where

3 b . For  = ý − 2 ,! , −2

 ( , ) =  2 3

4 b . For  =

ú

[` ( +


) +   + 2 , ],

ú

where

= − Xv

−1

M

 .

,! + ý ,

 ( , ) = `  

−’  +

−2

−2

  
+ 23 ,  + 89 , 2 ’ ,

where

In all these cases, the transformation

a = Xv

−1

 , M

‡ =

J «

−1 2

−1

−

’ = Xv

−3

M

 .

˜

leads to the autonomous equation ‡ g

g + 2 ,!‡ g
+ , 2 ‡ = ` (‡ ), which is reducible, with the aid of the substitution H (‡ ) = ‡ g
, to an Abel equation: HH 
+2 ,H + , 2 ‡ = ` (‡ ) (see Subsection 1.3.1). If , = 0, the solution of the original equation for case 1 b is as follows:

X

‡ £ 2 i (M ‡ ) + L

1

=A X

M 



+L

2,

where

i (‡ ) = X ` (‡ ) ‡ . M

If , = 0, the solution of the original equation for case 2 b is given by: 2X 34.

ñóò ò ó

ü

M

‡

8 i (‡ ) − ¢C‡

ó

2

+L

1

=A X



2

M



+ ' + (

+L

2,

where

i (‡ ) = X ` (‡ ) ‡ . M

+ ôñó ò + õÿñ =  H  ñó ò ¦ñ , ñóò ò ó¦ñ  .  ¨ leads to an autonomous equation: 1 b . The substitution = ž

¨ 

 + (2 Œ +  )¨ 
+ ( Π

2





+ Œ +  )¨ =   ¨  + Œ ¨ , ¨   + 2 Œ ¨  + Œ 2 ¨  .

2 b . Particular solution: = , ´ , where , is a root of the algebraic (transcendental) equation , ( Œ 2 + Œ +  ) =  ( , Œ , , Œ 2 ).

© 2003 by Chapman & Hall/CRC

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490 35.

SECOND-ORDER DIFFERENTIAL EQUATIONS

ô 1 (ñ ó ò ò ó

)2 + ô

2

ñ óò ñ óò ò ó

ñóò ò ó

=

ñó'ò H

1

 ñóò ¦ñ ,

)2 + ô

5

ññ óò

+

ô 6ñ

2

ñóò ò ó¦ñ

 + ñH

2

 ñóò ¦ñ ,

ó

ñóò ò ó¦ñ

 +  H

 = ž  ¨ leads to an autonomous equation.

The substitution 37.

óò

+ ô 4 (ñ

=

 = ž  J 2 ¨ leads to an autonomous equation.

The substitution 36.

ô 3 ññ óò ò ó

+

H  ï ñ óò ò ó , ï 2 ñ óò ò ó

–ï

ó

  HÊ ñ óò ¦ñ , ñ óò ò ó ¦ñ 

ñóò ò ó¦ñ

.

3

 ñóò ¦ñ ,

3

are related by the constraint

.

+ ñ  = 0.

ñóò

Solution: =L

1

 ln  + L 2  + L 3 ,

where L 2 is an arbitrary constant and the constants L  ( L 1 , L 3 ) = 0. 38.

H (ñ óò ò ó¦ñ , ñ ñóò ò ó

and L

– ñó ò 2 ) = 0.

1 b . Solution:

=L

where the constants L

L 2 , and L

1,

2 b . Solution:

1

H (ñ 3 ñ óò ò ó , ñ ñ óò ò ó

L 2 , and L

1,

exp( L

1

3

3

 )+L

2

exp(− L

3

 ),

are related by the constraint  ( L

3

=L

where the constants L 39.

1

cos( L

3

 )+L

2

sin( L

3

2 3,

4L

1

L 2 L 32 ) = 0.

 ),

are related by the constraint   − L

L

2 1+

−L

2 2,

2 3 , −(

L 22 ) L

2 3

 = 0.

óò 2 ) = 0.

+ñ

Solutions: =A where the constants L 40.

Hv”Èñ

+

ñ óò 2 ï , ñ óò ò ó

+

L 2 , and L

1,

ñ óò

2ñ

–ñ

óò ò ó

óò 3

3

ü

L 1

2

+ 2L

2

 + L 3,

are related by the constraint  ( L

1

L

3

L 1 ) = 0.

= 0.

•

Solutions can be found from the relation ( − L 1 )2 = 2 L 2 ( − # ) + L 22 , where  ( L The question of whether there are other solutions calls for further investigation. 41.

HwOiñ

ñ óò ò ó ñ óò

+

ôñ óò , ñ D

ñ óò ò ó ñ óò

+1

1,

# ) = 0.

= 0.

P

A solution of this equation is any function that solves the first-order separable equation:


= L where the constants L 42.

HwO

ñ óò ò ó ñ óò

+ ñó ò ,

]

1

ñ óò ò ó ñ óò

and L

P

2

1

−

 +L , 2

are related by the constraint  (L

2, −

L 1 ) = 0.

= 0.

A solution of this equation is any function that solves the first-order separable equation:


= L 1  − + L 2 , where the constants L

1

and L

2

are related by the constraint  ( L

2, −

L 1 ) = 0.

© 2003 by Chapman & Hall/CRC

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491

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

2.9.6-2. Equations containing arbitrary functions of three variables. 43.

H (ñ óò ò ó , ï ñ óò ò ó

ï 2 ñ óò ò ó

– ñó ò ,

– 2ï

+ 2ñ ) = 0.

ñóò

Solution:

=L

where the constants L

^_ 44.

L 2 , and L

1,

+L

2

 + L 3,

are related by the constraint  (2 L

3

H  ñóò ò ó , ï ñ óò ò ó

=L

H  ï 3 ñ óò ò ó , ï ñ óò ò ó

1,

L 2 , and L

3

ï 2 ñ óò ò ó

+ 2ñó ò ,

2



1

+L

2

 + L 3,

are related by the constraint  (2 L

+ï

ñóò =L

where the constants L

HÊ ï D +2 ñ óò ò ó , ï ñ óò ò ó

L 2 , and L

1,

3

=L 1,

H  ñ 3 ñ óò ò ó , ñ ñóò ò ó

L 2 , and L

3

+ï

Solution:

48.

49.

50.

1,

L 2 , and L

3

=L

– ñ , ñó ò ò ó – ñ , ñó ò ò ó – ñó ò  = 0.

H  ï , ñ óò

+

ôñ , ñ óò ò ó

–ô

2

Hv” ï

–

ñ óò 2 + 1 óò ñ óò ò ó ñ , ñ

+

L 2, 4 L 1 L

3−

L 22 ) = 0.



3

,

1

ô ï ñ óò

3, 2

L 1 , − L 2 ) = 0.

– ôñ  = 0.

 − + L 2  + L 3.

1



2

1, (

 +1) L 2, − L 3  = 0.

 = 0.

+ 2L

2

 + L 3, 1

L

3

−L

2 2,

L 1 , − L 2 ) = 0.

leads to a first-order equation:  ( , ¨ , ¨u
+ ¨ , ¨2
) = 0.

ñ , ñ óò ò ó

The substitution ¨ = 
+ 

L

are related by the constraint  ( L

H  ï , ñ óò

The substitution ¨ = 
−

+

ñóò 2 – ññóò 2

where the constants L

+

2

are related by the constraint    (  +1) L

ï ññóò ò ó

+ ñó ò 2 ,

 +L

are related by the constraint  (2 L

Solution: The constants L

1

óò , ï 2 ñ óò ò ó

+ ( ô + 1)ñ

1, −

– ñ  = 0.

Solution:

47.

L 2 , 2 L 3 ) = 0.

– ñó ò , 2ññó ò ò ó – (ñó ò )2  = 0.

where the constants L

46.

1, −

Reference: E. L. Ince (1964).

Solution:

45.

2



1

+ ôñó ò  = 0.

leads to a first-order equation:  ( , ¨ , ¨u
−  ¨ , ¨2
) = 0.

ñ óò 2 + 1 (ñ óò 2 + 1)3 ) 2 • ñ óò ò ó , ñ óò ò ó

= 0.

It is known that all functions of the form ( − L 1 )2 + ( − L 2 )2 = # 2 , where # = # ( L 1 , L 2 ) is determined from the algebraic (transcendental) equation  ( L 1 , L 2 , # ) = 0, are solutions of the original equation. 51.

ó

H   ñóò ò ó , ñ óò ò ó

+ ñó ò ,

ñ

–ï

ñóò

– ( ï + 1)ñó ò ò ó  = 0.

Solution: where the constants L

=L 1,

L 2 , and L

3

1



−



+L

2

 + L 3,

are related by the constraint  ( L

1,

L 2 , L 3 ) = 0.

© 2003 by Chapman & Hall/CRC

Page 491

492 52.

SECOND-ORDER DIFFERENTIAL EQUATIONS

H O ñ óò ò ó ¦ñ , ñ ñ óò ò ó

óò

– (ñ

Solution:

=L

where the constants L 53.

H  ñóò ò ó

–ñ ,

ñóò ò ó

L 2 , and L

1,

1

HÊ ï ñ óò ò ó , ï 2 ñ óò ò ó

1,

–ï

ñ óò

L 2 , and L

where the constants L –ñ

Solution:

ñ óò

+ñ ,

1,

H  ñóò ò ó

+ñ ,

ñóò ò ó

L 2 , and L

ñ óò ò ó ï ñ óò ò ó , ñ óò ñ óò

where the constants L 56.

1,

2

ñ óò ò ó ¦ñ

exp(− L

3

 P

L 2 , and L

2

cosh  + L

= 0.

 ),

3

3

3, −

ln ï  = 0.

óò

)2

=L

1

1,

HÊ ñ , ï ñ óò 2, ï ñ óò ò ó

+

1 2

HÊ ñ , r ñ óò 2, r ñ óò ò ó

+

1 2

L 2 , and L

3

1

2L

2 3,

1)

= 0.

L 1, L

2 1

−L

2 2)

= 0.

1,

L 3, L

1

+L

2)

= 0.

= 0.

•

exp( L

2

 ) + L 3,

are related by the constraint  ( L

=L

L 2L

 ln  + L 2  + L 3 ,

1

ñ óò ò ó

1

3,

are related by the constraint  ( L (ñ

4L

2 3,

2, −

L 2 L 3 , − ln L 1 ) = 0.

3, −

L 1, L

sin ï – ñó ò cos ï , (ñó ò ò ó )2 + (ñó ò )2  = 0.

where the constants L

58.

óò ò ó ï

–ñ

– ln

Solution:

57.

 )+L

ü

are related by the constraint  (− L

3

=L

ñ óò ò ó ,  ñ óò ñ óò

3

sinh  + L

1

Solution:

Hv”

exp( L

 exp  – ï

are related by the constraint  ( L

3

=L

where the constants L

55.

ñ¦ñ óò ò ó ü

sinh ï – ñó ò cosh ï , (ñó ò ò ó )2 – (ñó ò )2  = 0.

Solution:

54.

óò

)2 , Ññ + ñ

sin  + L

2

cos  + L

3,

are related by the constraint  ( L

2 1

+L

2 2)

= 0.

ñ óò  = 0. ¨ The substitution ( ) =  (
 )2 leads to a first-order equation:   , ¨ , 12 ¨2
 = 0.  r òó ñ óò 

r

= 0,

=

r (ï

).

The substitution ¨ ( ) = Y ( )(
 )2 leads to a first-order equation:   , ¨ , 12 ¨2
 = 0. 

2.9.7. Some Transformations 1.

ñóò ò ó

+

ñóò ò ó

=

1 ñ ï –3 w H O ï , ï

P

= 0.

The transformation ˆ = 1  , ¨ =   leads to the equation ¨pŠª

Š +  (ˆ , ¨ ) = 0. 2.

3.

ù (ù

+ 1) ï

–2

ñ

+ï

3

The transformation ˆ =  

÷ H ( ï 2÷

2 +1

öï

+1

,

, ¨ = 

ïƒ÷ ñ

).

leads to the equation (2B + 1)2 ¨2Šª

Š =  (ˆ , ¨ ).

+ý ñ , ï = 0. +õ ô +õ P (˜ + leads to the equation ¨3Šª

Š + ¢ The transformation ˆ = M , ¨ =  +    +  where ¢ =  − ˜( .

ñóò ò ó

+ (ô

ï

+ õ )–3 HwO

ô ï

−2

 (ˆ , ¨ ) = 0,

M

© 2003 by Chapman & Hall/CRC

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493

2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS

4.

+ ô ï ñó ò + õÿñ + H ( ï , ñ ) = 0. Á The transformation  = ˆ , = ˆ  ¨ , where the parameters @ and “ are found from the simultaneous algebraic equations

ï 2 ñ óò ò ó

2“ + 1 + (  − 1) @ = 0,

2

“

+ (  − 1)“…@ + @

2

= 0,

leads to an equation of the form

¨ ªŠ

Š + @ 2 ˆ −  5.

= ù (ù + 1) ï –2 ñ + ï 3÷   2 The transformation ˆ = 

ñóò ò ó

ó

ñóò ò ó

H ( ô ï 2÷ +1

ó

+1

ó

7.

ñóò ò ó

=

ñ

+

ó

+ ý )3

Hv”

ñ óò ò ó

ñ

ï )v H ”

+ sinh–3 (

2

=



ô 2 ó ö 2 ó

+

+ý

ï

coth(

ñ óò ò ó

2

=

ñ

+ cosh–3 (

ï )v H ”

ï

tanh(

ï 2 ñ óò ò ó

1 4

+

ñ

+

ó

  ñ ö  2 ó + ý • 

,

7 ï Hv”ô

ln ï + õ ,

12.

ñ

.

cosh( Π )

ñ

.

¨ ªŠ

Š = Œ leads to the equation 3

ï

cosh(

7 ï •

)•

−2

 (ˆ , ¨ ).

.

leads to the equation ¨ ªŠ

Š = Œ

−2

 (ˆ , ¨ ).

= 0.

¨ ªŠ

Š +  leads to the equation 3

−2

 (ˆ , ¨ ) = 0.

ô ï ï

–ô ñ , . ï +1 | 2 – 1| •  −  ü ¨ ¨ ªŠ

Š =  (ˆ , ¨ ) + ¨ . leads to the equation 43 , = The transformation ˆ = ln 2  +1 | − 1|

|ï

2

ñ óò ò ó

) ñóò ò ó

3 2

– 1|

+

ñ

2

=v H ” ln

+ sin–3 (

ï )H

” cot(

ï

ü

),

The transformation ˆ = cot( Œ  ), ¨ = 13.

)•

sinh( Π )

The transformation ˆ =  ln  +  , ¨ = £  11.

ï

sinh(

),

The transformation ˆ = tanh( Œ  ), ¨ = 10.

õ

ñ

),

The transformation ˆ = coth( Œ  ), ¨ = 9.

leads to the equation ¨3Šª

Š = (2 Œ )−2  (ˆ , ¨ ).

  2 +  ¨  , = 2  leads to the equation  (  2 + (  + M M ¨ Šª

Š = (2 ¢eŒ )−2  (ˆ , ¨ ), where ¢ =  − ˜( . M

The transformation ˆ =

8.

). leads to the equation  2 (2B + 1)2 ¨2Šª

Š =  (ˆ , ¨ ).

+ , ¨ = 

= 2 ñ +  3  H ( ô 2  + õ ,  ñ ).   The transformation ˆ =   2  +  , ¨ = 

 3 ( ö  2 ó

ïƒ÷ ñ

+õ ,

6.

2

 (ˆ Á , ˆ  ¨ ) = 0.

−2

ñ óò ò ó

+

2

ñ

+ cos–3 (

ï )H

” tan(

ï

),

The transformation ˆ = tan( Œ  ), ¨ =

ñ ï

sin(

)•

sin( Π )

ñ cos(

ï

cos( Π )

= 0. leads to the equation ¨3Šª

Š + Œ

)•

−2

 (ˆ , ¨ ) = 0.

= 0.

¨ ªŠ

Š + Œ leads to the equation 3

−2

 (ˆ , ¨ ) = 0.

© 2003 by Chapman & Hall/CRC

Page 493

494 14.

SECOND-ORDER DIFFERENTIAL EQUATIONS sin( ï + ô ) ñ , = 0. sin( ï + õ ) sin( ï + õ ) • sin( Œ  +  ) ¨ , = leads to the equation The transformation ˆ = sin( Œ  +  ) sin( Œ  +  )

ñóò ò ó

2

+

ñ

+ sin–3 (

ï

+ õ ) Hv”

¨ Šª

Š + [ Œ sin(  −  )]−2  (ˆ , ¨ ) = 0. 15.

(ï

2

) ñóò ò ó

+v H ” arctan ï + õ ,

3 2

+ 1)

ñ

7 ï

The transformation ˆ = arctan  +  , ¨ = £  16.

(ï

2

) ñ óò ò ó

3 2

+ 1)

+

H ”

arccot ï + õ ,

7 ï

ñ 2

2

ñóò ò ó

+

H (ï , ñ

) = 0.

The transformation  = Y ( H ),

+1

2

leads to the equation ¨ ªŠ

Š + ¨ +  (ˆ , ¨ ) = 0. = 0.

+1•

The transformation ˆ = arccot  +  , ¨ = £  17.

= 0.

+1•

2

+1

leads to the equation ¨3Šª

Š + ¨ +  (ˆ , ¨ ) = 0.

= ¨ £ !Y ‘
leads to the equation

'
‘
‘'
‘ 3 Y ‘'

‘ 2 ¨ ¨ '‘

‘ + ¬ 1 Y − O ® + ‘ 2 Y
4 Y ‘
P

−2

3J 2 ( !Y ‘
)   Y , ¨ £ !Y ‘
 = 0.

‘ The sign of the parameter  must coincide with that of the derivative Y
. 18.

+ ! ( ï , ñ )(ñó ò )3 + " ( ï , ñ )(ñó ò )2 = 0. Taking to be the independent variable, we obtain the following equation with respect to  =  ( ): 

− ( , )
−  ( , ) = 0.

ñóò ò ó



19.

H ( ï , ñ , „ñ óò , ñ óò ò ó



) = 0. Applying the Legendre transformation  = ¨ug
, = a ¨2g
− ¨ , where ¨ = ¨ (a ), and using the relations 
= a and 

 = 1 ¨2g

g , we arrive at the equation 1 O ¨ g
, a ¨ g
− ¨ , a , ¨

= 0. gg P Given a solution of the original equation, the corresponding solution of the transformed equation is written in parametric form as:

a = 
,

¨ =  
− ,

where

= ( ).

© 2003 by Chapman & Hall/CRC

Page 494

Chapter 3

Third-Order Differential Equations 3.1. Linear Equations 3.1.1. Preliminary Remarks 1 b . A homogeneous linear equation of the third order has the general form

 3 ( ) 


 +  2 ( ) 

 +  1 ( ) 
+  0 ( ) = 0. Let

0

=

0(

(1)

 ) be a nontrivial particular solution of this equation. The substitution =

0(

 ) X H ( )  M

leads to a second-order linear equation:



3 0

H

+ (3 


+

2 0)

3 0

H
+ (3 



+ 2

3 0


+

1 0)

2 0

H = 0,

(2)

where the prime denotes differentiation with respect to  . 2 b . Let 1 = 1 ( ) and 2 = 2 ( ) be two nontrivial linearly independent particular solutions of equation (1). Then the general solution of this equation can be written in the form: =L

+L

1 1

2 2

+L

O

3

where

j = exp O − X 



X

2

2 3

1



M P



j M

 −

1

X


−
 1 2

1 2

2

−2

j

 , M P

(3)

.

For specific equations described below in 3.1.2–3.1.9, often only particular solutions will be given, and the general solution can be obtained by formula (3). 3 b . A nonhomogeneous linear equation of the third-order has the form

 3 ( ) 


 +  2 ( ) 

 +  1 ( ) 
+  0 ( ) = ( ).

(4)

Let 1 = 1 ( ) and 2 = 2 ( ) be two linearly independent particular solutions of the corresponding homogeneous equation (1). Then the general solution of equation (4) is defined by formula (3) with:

j =¢

1  Ž O 1+ X L 3

−2 −

4 b . The substitution



= H exp O −

derivative is absent: where Y  =   

H


+  − Y 2
− 13 Y 3

3

¢  Ž

1 X 3 2 2

+Y



 , M P 

2 3

1

where

 =X

 

2 3

M

 , ¢ =


−
. 1 2

1 2

 reduces equation (1) to a form from which the second M P

 H
+  − 13 Y 2

− 13 Y 1 Y

2

+

2 27

Y

3 2

+Y

0

 H = 0,

( , = 0, 1, 2). 495

© 2003 by Chapman & Hall/CRC

Page 495

496

THIRD-ORDER DIFFERENTIAL EQUATIONS

3.1.2. Equations Containing Power Functions 3.1.2-1. Equations of the form  3 ( )


 +  0 ( ) = ( ). 1.

ñóò ò ó ò ó

+

Ė

= 0.

Solution:

a

 + L 3 L 1 exp(− ,! ) + L L

=

+L

1

2

where , = Π1 J 3 . 2.

ñóò ò ó ò ó

+

Ė

ô ï

=

2

=¨ +

Solution:

+õ 1

¨2


 + Œ ¨ = 0.

3.

ô ï ñ

ñ óò ò ó ò ó

=

ñ óò ò ó ò ó

+ (ô

ñóò ò ó ò ó

+

ñ óò ò ó ò ó

+ (3 ô

ñóò ò ó ò ó

=

Œ

ï

2 2

exp 

1 2

,!  cos 

2

3 2

,!  + L

3

exp 

1 2

,!  sin 

2

if Π= 0, 3 2

,! 

if Œ ≠ 0,

+ö .

( 

2

+ ' + ( ), where ¨ is the general solution of the equation 3.1.2.1:

+õ .

This is a special case of equation 5.1.2.3 with B = 3. 4.

ï

+ õ )ñ = 0.

For  = 0, this is an equation of the form 3.1.2.1. For  ≠ 0, the substitution ˆ =  +  leads to an equation of the form 3.1.2.3: 
Šª
Šª
Š + ˆ = 0. 5.

ô ï 3ñ

õï

=

.

The substitution ˆ =  6.

7.

2

ï

–ô

3

2

ï 3 )ñ

leads to an equation of the form 3.1.2.126: 2ˆ "
Šª
Šª
Š +3
Šª
Š + 14 ˆ

= 14  .

= 0.

Integrating, we obtain a second-order nonhomogeneous linear equation: 

 +  
+ (  2  2 −  ) = L exp  12  2  (see 2.1.2.31 for the solution of the corresponding homogeneous equation).

ô ï _ ñ

.

This is a special case of equation 5.1.2.4 with B = 3.

1 b . For ý = −9, −7, −6, −9 2, −3, −3 2, 1, and 3, see equations 3.1.2.17, 3.1.2.14, 3.1.2.11, 3.1.2.19, 3.1.2.10, 3.1.2.18, 3.1.2.3, and 3.1.2.5, respectively. 2 b . The transformation  = a

8.

9.

ñóò ò ó ò ó

+ [ô

ï ÷

– 3ô

3 3

2

ù ï 2÷

–1

, = ‡"a

leads to an equation of similar form: ‡


 = − a −:

‡ . : : ¨ 3 b . For ý ≠ −3, the transformation ˆ =  ( +3) J 3 , =  J 3 leads to an equation of the form 3.1.2.69: ˆ 3 ¨2Šª

Šª
Š + (1 − @ 2 )ˆ ¨2Š
+ ( @ 2 − 1 − @ 3 ˆ 3 )¨ = 0, where @ = : 3+3 . −1

+

ôù (ù

−2

– 1) ïƒ÷

–2

−6

]ñ = 0.

   +1   +1 X H ( )  leads v . The substitution = exp O − Particular solution: 0 = exp O − B +1 P B +1 P M


 B/  −1 ) H = 0. to a second-order linear equation of the form 2.1.2.47: H  −3   H  +(3  2  2  −3 

ï ñóò ò ó ò ó

– ô 2(ô

ï

+ 3)ñ = 0.





Particular solution: 0 =    . The substitution =    X H ( )  leads to a second-order M equation of the form 2.1.2.108: "H 

 + 3(  + 1)H 
+ 3  (  + 2)H = 0.

© 2003 by Chapman & Hall/CRC

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497

3.1. LINEAR EQUATIONS

10.

11.

ï 3 ñ óò ò ó ò ó

= ô (ô

– 1)ñ .

2

=  (L 1   1 + L This is a special case of equation 3.1.2.175. Solution: 2 where B 1 and B 2 are roots of the quadratic equation B + B +  2 − 1 = 0.

ï 6 ñ óò ò ó ò ó

=

ôñ

+

õï

2

( ï – ô )3 ( ï – õ )3 ñó ò ò ó ò

13.

(ô

−2

ï

2

+

õï

óò ò ó ò ó

+ ö )3 ñ

üIñ

=

ï 7 ñ óò ò ó ò ó

=

ôñ

+

õï

3

ï 7 ñ óò ò ó ò ó

ï

+ (ô

ï 9 ñ óò ò ó ò ó

+ (ô



= a

+ õ )ñ = 0.

ï )ñ

– 3ô

3

= a

2 2

ï 9 ñ óò ò ó ò ó

=

ôñ



=

ï 9 ) 2 ñ óò ò ó ò ó

=

ôñ

.

ôñ

.

−1

2

+ ' + (

leads to a constant coefficient

−1

,

= ¨ a

−2

leads to a linear equation of the form 3.1.2.3:

−1

,

= ¨ a

−2

leads to a linear equation of the form 3.1.2.4:

,

= ¨ a

−2

leads to a linear equation of the form 3.1.2.6:

=¨ a

−2

−1

.

The transformation  = a

ï 3 ) 2 ñ óò ò ó ò ó

2

= 0.

The transformation  = a ¨2g

g
g + (3  2 a −  3 a 3 )¨ = 0.

18.

  ),

.

.

The transformation  ¨2g

g
g − ( 'a +  )¨ = 0.

17.

3

– ö^ñ = 0, ô ≠õ.  − ¨ , = leads to a constant coefficient linear equation: The transformation a = ln û  −  ûû (  −  )2 û ¨ 2 ¨


2 ¨

2 ¨
(  −  )3 ( g g g − 3 g g + 2 gû ) − ( û = 0.

The transformation  ¨2g

g
g + a ¨ +  = 0.

16.

+L

leads to a constant coefficient linear equation of the

, ¨ = M + '   + (  linear equation: ¨3Šª

Šª
Š + (4 !( −  2 )¨2Š
= , ¨ .

15.

2

ó

The transformation ˆ = X

14.

 

.

The transformation  = a −1 , = ¨ a form 3.1.2.2: ¨3g

g
g +  ¨ +  = 0.

12.

2

,

¨ g

g
g + a 3 ¨ = 0. leads to an equation of the form 3.1.2.5: u

This is a special case of equation 5.1.2.8 with B = 1. 19.

This is a special case of equation 5.1.2.9 with B = 1. 3.1.2-2. Equations of the form  3 ( )


 +  1 ( )
 +  0 ( ) = ( ). 20.

21.

ñ óò ò ó ò ó

+

ôƒõÿñ óò

ñóò ò ó ò ó

+

ô ï ñóò

+ô

2

ï

(3 –

õ

–ô

ï 2 )ñ

= 0.

Integrating, we obtain a second-order nonhomogeneous linear equation:

 +  
 + (  2  2 +  −  ) = L exp  12  2  (see 2.1.2.31 for the solution of the corresponding homogeneous equation).

Solution:

+ ôùñ = 0, ù = 1, 2, 3, uuu = ‡ (  −1), where ‡ is the solution of the second-order linear equation ‡

 + ‡ = L .

© 2003 by Chapman & Hall/CRC

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498 22.

23.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ó

+

ñóò ò ó ò ó

+

ô ï ñ óò

– 2 ôñ = 0.

ô ï ñóò

+ õ (ô

The substitution ¨ ¨2

 +  ¨ = 0.


− 2 leads to a second-order linear equation of the form 2.1.2.2:

=

ï

õ 2 )ñ

+

24.

25.

26.

= 0.



Particular solution: 0 =   . The substitution ¨ = 
+  equation of the form 2.1.2.12: ¨u

 −  ¨2
+ (  +  2 )¨ = 0. −

ñóò ò ó ò ó

+

ô ï ñóò

ï

+ ( ôƒõ

+

ô

+

õ 3 )ñ

leads to a second-order linear

= 0.

Integrating yields a second-order linear equation: "

 −  
 + (  +  2 ) = L  2.1.2.108 for the solution of the corresponding homogeneous equation with L = 0).

ñóò ò ó ò ó

ï

+ (ô

−



(see

+ õ )ñó ò + ôñ = 0.

Integrating yields a second-order nonhomogeneous linear equation: 

 + (  +  ) = L 2.1.2.2 for the solution of the corresponding homogeneous equation).

ñ óò ò ó ò ó



ï

+ (ô

+ õ )ñ

óò

– ôñ = 0.

(see




 −  +  (  +  )2 leads to a second-order linear equation of the form 2.1.2.67: žˆ H Šª

Š + 3 H Š
+  −2 ˆ 2 H = 0.

Particular solution:

27.

ñóò ò ó ò ó

+ (ô

ñ óò ò ó ò ó

+ (2 ô

ñ óò ò ó ò ó

+ (ô

ï

0

=  +  . The transformation ˆ =  +  , H =

+ õ )ñó ò + 3 ôñ = 0.

 ˆ =  +  leads to a linear equation of the form 3.1.2.21 with B = 3: The substitution  
Šª
Šª
Š + ˆ 
Š + 3  = 0. 28.

ï

+ õ )ñ

óò

+

ôñ

= 0.

 ˆ =  + 12  leads to a linear equation of the form 3.1.2.48 with B = 1: The substitution  
Šª
Šª
Š + 2 ˆ 
Š +  = 0. 29.

30.

31.

32.

ï

–

õ 2 )ñ ó ò

+

ôƒõ ï ñ

= 0.

The substitution ¨ = 
+  ¨2

 −  ¨2
+  ¨ = 0.

ñóò ò ó ò ó

ï

+ (ô

–

õ 2 )ñ óò

leads to a second-order linear equation of the form 2.1.2.108:

ï

+ ô (õ

+ 1)ñ = 0.

Integrating yields a second-order linear equation: "

 −  
 +  for the solution of the corresponding homogeneous equation with L

ñóò ò ó ò ó

+ (ô

ï

ñóò ò ó ò ó

+ (ô

ï

+ õ )ñó ò + ö ( ô

ï

+

õ

= L  − = 0).

(see 2.1.2.108

+ ö 2 )ñ = 0.

The substitution ¨ = 
+ ( leads to a second-order linear equation of the form 2.1.2.12: ¨2

 − ( ¨2
+ (  +  + ( 2 )¨ = 0. + õ )ñó ò + ö

Particular solution:

0

ï (ö 2 ï

2

+

ô ï

+

õ

– 3 ö )ñ = 0.

= exp  − 21 (˜ 2  . The substitution

= exp  − 12 (˜ 2  X H ( )  leads to

a second-order linear equation of the form 2.1.2.31: H 

 − 3 (˜"H 
+ (3 ( 2  33.



ñóò ò ó ò ó

+

ô ï 2 ñ óò

+ô

ï ñ

2

M

+  +  − 3 ( ) H = 0.

= 0.

This is a special case of equation 3.1.2.47 with B = 1.

© 2003 by Chapman & Hall/CRC

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499

3.1. LINEAR EQUATIONS

34.

35.

36.

37.

38.

39.

40.

41.

ô ï 2 ñ óò

ñóò ò ó ò ó

+

ñóò ò ó ò ó

–ô

ñóò ò ó ò ó

+

ñóò ò ó ò ó

+ ( ô – 1) õ

ñóò ò ó ò ó

+ (ô

ï

ñóò ò ó ò ó

+ (ô

ï

ï ñ

– 2ô

= 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.7: ¨2

 +  2 ¨ = 0. 2

ï 2 ñ óò

+ô

ï ñ

2

= 0.

Integrating yields a second-order linear equation: "

 +  
 −  = L exp  12  2  (see 2.1.2.108 for the solution of the corresponding homogeneous equation with L = 0).

ô ï 2 ñ óò

ï

+ õ (ô

2

+

õ 2 )ñ

= 0.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.2.31: ¨ 

 −  ¨ 
+ (  2 +  2 )¨ = 0. 2

ï 2 ñ óò

+

õ 2 ï (ƒô õ ï

2

+ 2 ô + 1)ñ = 0.

Integrating, we obtain a second-order nonhomogeneous linear equation:

 − ' 
 + (  2  2 +  ) = L exp  − 12 ' 2  (see 2.1.2.31 for the solution of the corresponding homogeneous equation). 2

+ õ )ñó ò + 2 ô

ï ñ

= 0.

Integrating yields a second-order nonhomogeneous linear equation:

 + (  (see 2.1.2.4 for the solution of the corresponding homogeneous equation). 2

–

õ 2 )ñ óò

+

ô ï

õ ï )ñ

(2 –

= 0.

Integrating yields a second-order linear equation: "

 +  
 +  2 = L  the solution of the corresponding homogeneous equation with L = 0).

ñóò ò ó ò ó

+ (ô

ñóò ò ó ò ó

– (3 õ

ï

2

+ õ )ñó ò + ö ( ô

ï

2

õ

+

+

ö 2 )ñ



+ ) =L

(see 2.1.2.13 for

= 0.

The substitution ¨ = 
+ ( leads to a second-order linear equation of the form 2.1.2.13: ¨ 

 − ( ¨ 
+ (  2 +  + ( 2 )¨ = 0. 2

ï

2

+ ô + 3 õ )ñó ò + 2 õ

ï (õ 2 ï

1 b . Particular solutions with  > 0: 2 b . Particular solutions with  < 0:

ñóò ò ó ò ó

ï

+ (ô

2

+

õï

ü ï [(ô

+ ö )ñó ò +

2 1

– ô )ñ = 0.

= exp 

1 2

'

2

+ £   ,

 ' 2  cos   £ |  |  ,

1 1 = exp 2

3 b . Particular solutions with  = 0: 42.



2

1

+

= exp 

ü 2)ï

2

+

1 2

õï

' 2  ,

2

2

= exp  2 = exp

=  exp 

1 2

1 2 1 2

'

2

− £   .

 ' 2  sin   £ |  |  .

' 2  .

+ ö – 3 ü ]ñ = 0.

= exp  − 21 ,! 2  v X H ( )  leads M


to a second-order equation of the form 2.1.2.31: H  − 3 ,!"H  + [(  + 3 , 2 ) 2 + ' + ( − 3 , ] H = 0. Particular solution:

43.

ñóò ò ó ò ó

+ (ô

ñóò ò ó ò ó

+

ï

4

+

0

õ ï )ñ óò

= exp  − 12 ,! 2  . The substitution

– 2( ô

ï

3

+ õ )ñ = 0.

This is a special case of equation 3.1.2.49 with B = 2. 44.

45.

ô ƒï ÷ ñ óò

– 2ô

ïƒ÷ –1 ñ

= 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.7: ¨2

 +   ¨ = 0.

+ ô ï ÷ ñ ó ò + ôù ï ÷ –1 ñ = 0. Integrating yields a second-order nonhomogeneous linear equation:

 +   2.1.2.7 for the solution of the corresponding homogeneous equation).

ñ óò ò ó ò ó

=L

(see

© 2003 by Chapman & Hall/CRC

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500 46.

47.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+

ñóò ò ó ò ó

+

ô ïƒú

+1

ø

49.

50.

52.

ñ

ô ï 2÷ ñ óò

−1

ôù ï 2÷ –1 ñ

+

=L

ø



1

= 0.

= 0.

ø

Á 2 (‡ ) + L 2 

Á (‡ ) n Á (‡ ) + L 3 "n Á 2 (‡ ), where @ =

Á (‡ ) and n Á (‡ ) are the Bessel functions.

ñ óò ò ó ò ó

+ 2ô

ï ÷ ñ óò

+ ôù

ñóò ò ó ò ó

+ (ô

ï 2÷

+

õ ïƒ÷

–1

ñ óò ò ó ò ó

+ (ô

ï 2÷

+

õï ÷

–1

= L 1¨

ï ÷ –1 ñ

£ Ñ  +1 1 , ‡ = ; 2(B + 1) 2(B + 1)

= 0.

¨ 2 ¨ ¨ Solution: 2 + L 3 2 . Here, 1 and 2 are a fundamental set of solutions of a second-order linear equation of the form 2.1.2.7: 2¨p

 +   ¨ = 0. 2 1

+ L 2¨ 1¨

ï 2÷

)ñó ò – 2( ô

–1

+

õ ïƒ÷

–2

)ñ = 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.10: ¨2

 + (  2  + '  −1 )¨ = 0.

óò

)ñ

– ö [( ô + ö 2 ) ï

3

÷

+ ( õ + 3 ö^ù ) ï

÷

2 –1

+ ù (ù – 1) ï

÷

–2

]ñ = 0.

(˜  +1 (˜  +1 XvH ( )  leads to . The substitution = exp O B +1 P B +1 P M


 2 2  a second-order equation of the form 2.1.2.47: H  +3 (˜ H  +[(  +3 ( ) +(  +3 (˜B )  −1] H = 0. Particular solution:

51.

+ 3) ïƒú

+ ô (þ

The substitution  = a , = ¨ a −2 leads to an equation of the form 3.1.2.45 with B = − − 5: ¡ ¨2g

g
g + a − + −5 ¨2g
−  ( + 5)a − + −6 ¨ = 0. ¡

Solution:

48.

ñóò

ï ñóò ò ó ò ó

+

ôñóò

ï ñ óò ò ó ò ó

+

ô ï ñ óò

0

+ õ (õ

2

ï

= exp O

+ ô )ñ = 0.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.2.108:  ¨2

 − ' ¨2
+ (  2  +  )¨ = 0. – [õ (ô +

õ 2 )ï 

+ ô + 3 õ 2 ]ñ = 0.





 X H (

)  leads to a second-order M


 H  + 3( ' + 1)H  + [(  + 3  2 ) + 6  ] H = 0. linear equation of the form 2.1.2.108: " Particular solution:

53.

54.

55.

ï ñóò ò ó ò ó

+ (õ – ô

2

0

ï )ñ óò

= 

. The substitution

+ ôƒõÿñ = 0.

The substitution ¨ = 
+   ¨2

 −  ¨2
+  ¨ = 0.

ï ñóò ò ó ò ó

+ (ô

ï

2

ï ñóò ò ó ò ó

+ (ô

ï

3

= 

leads to a second-order linear equation of the form 2.1.2.108:

+

õ ï )ñ óò

– 2( ô

ï

+

õ ï )ñ óò

– 2( ô

ï

+ õ )ñ = 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.2: ¨ 

 + (  +  )¨ = 0. 2

+ õ )ñ = 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.4: ¨2

 + (  2 +  )¨ = 0.

56.

(ô

ï

+ õ )ñ„ó ò ò ó ò

57.

(ô

ï

+ 2)ñ„ó ò ò ó ò

ó

+ ö^ñó ò + ü ( ôƒü

2

ï ñóò

3

The substitution ¨ = (  +  )¨ 

 − , (  + 

ó

–ô

3

Particular solutions:

ï

+

ñ

= 0.

õü

2

+ ö )ñ = 0.


+ , leads to a second-order linear equation of the form 2.1.2.108: )¨ 
+ ( , 2  + , 2 + ( )¨ = 0. + 2ô 1

2

= ,

2

 =  − .

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501

3.1. LINEAR EQUATIONS

58.

ï

( ô„ö

+

õ^ö

– ô )ñ„ó ò ò ó ò

ó

Particular solutions: (ô

60.

(ô ï

61.

(ô

ï

59.

+ õ )ñ

óò ò ó ò ó

1

ï

+ (ö

64.

65.

66.

67.

+ õ )ñ„ó ò ò ó ò ó + [( ö – ôƒü ) ï 2

= 0.

+ õ )ñ

óò ò ó ò ó

ï

– (ô

ï 2 ñ óò ò ó ò ó

3 3

ï 2 ñ óò ò ó ò ó

+ (ô

ï

ï 2 ñ óò ò ó ò ó

+ (ô

ï

2

– 3ô

1

ï 2ñ

– 6ñó ò + ô

ý

–õ

ü ]ñ óò

+

õ 3 )ñ ó ò

ï

+ ü (ö

+ ý )ñ = 0.


+ , leads to a second-order linear equation of the form 2.1.2.108: )¨2
+ ( (˜ + )¨ = 0. M

The substitution ¨ = (  +  )¨2

 − , (  + 

ï

+

M

2

2





=

,

ï

+ 2õ

= ¨

ï

ï (ô 2 ï

+ ôƒõ

2

– 3ô –

õ 2 )ñ

= 0.

= exp  − 21  2  .

2

= 0.

¨ 


 + 6 2 2 ¨ 

 + leads to an equation of the form 3.1.2.173:  3 2

2

+

õï

–þ

+

õï

+ ö )ñó ò – ü [( ô +

2

–þ

)ñ

óò

+ (þ

ï

– 1)(ô

+ õ )ñ = 0.

The substitution ¨ =  
+ ( − 1) leads to a second-order linear equation of the form ¡ 2.1.2.108:  ¨ 

 − ( + 1)¨ 
+ (  +  )¨ = 0. ¡ 2

The substitution ¨ = 
− ,  2 ¨2

 + ,! 2 ¨2
+ [(  + , 2 )

ï 2 ñ óò ò ó ò ó

+ (ô

ï ÷

ï 2 ñ óò ò ó ò ó

+ (ô

ïƒ÷

ï 2 ñ óò ò ó ò ó

– 3ô

ï ÷

–

õ

2

+1

–

õ

– õ )ñ

2

ü 2 )ï

+ ô ( õ – 1) ï

óò

2

+

õï

+ ö ]ñ = 0.

leads to a second-order linear equation of the form 2.1.2.135: + ' + ( ]¨ = 0.

÷ –1 ñ

= 0.

The substitution ¨ =  
+ (  − 1) leads to a second-order equation of the form 2.1.2.67:  ¨ 

 − (  + 1)¨ 
+   −1 ¨ = 0. 2

– õ )ñó ò + ô ( õ – 1) ïƒ÷

ñ

= 0.

The substitution ¨ =  
+ (  − 1) leads to a second-order linear equation of the form 2.1.2.67:  ¨3

 − (  + 1)¨2
+   ¨ = 0. +1

(ù + ô

ï ÷

+1

)ñ

óò

ï ÷ (ù

+ô

ï (ô ï

+ õ )ñ„ó ò ò ó ò

ó

+ ï (ö

ï

+ ý )ñó ò – 2( ö

–ù

2

+ 2ô

2

ï 2÷

   +1 , 1 = exp O B +1 P   +1 . 1 =  , 2 = 

2 b . Particular solutions with B = −1:

69.

ñ

+ s[( ô s2 + ö ) ï + õ s2 + ý ]ñ = 0.

óò

1 b . Particular solutions with B ≠ −1:

68.

3

v . 2 = 

The substitution ¨ = 
+ ° leads to a second-order linear equation of the form 2.1.2.108: ¨ 
+ [( !° 2 + ( ) + ˜° 2 + ]¨ = 0. (  +  )¨2

 − ° (  +  )2

The substitution (  3 − 12)¨ + 2  = 0. 63.

+ õ )ñó ò + ô„ö

=  +  ,

+ ý )ñ

Particular solutions: 62.

ï

– ö 3(ô

ï

+2

)ñ = 0.

2

=  exp O

  +1 . B +1 P

+ ý )ñ = 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.108: (  +  )¨ 

 + ( (˜ + )¨ = 0. M

ï 3 ñ óò ò ó ò ó

+ (1 – ô 2 ) ï

ñóò

+ (õ

ï

3

+ô

2

– 1)ñ = 0.

For  = A 1, we have a constant coefficient equation of the form 3.1.2.1. For  = 0, we obtain the Euler equation 3.1.2.175.

© 2003 by Chapman & Hall/CRC

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502

THIRD-ORDER DIFFERENTIAL EQUATIONS 1 b . If  ≠ 0 and  is a positive integer greater than 1, then the solution is: = where Œ 1 , Œ 2 , and Œ degree ≤ 3(  − 1).

3

1−

 ½

3



=1

.

L  exp(− Œ   )  ( ),

are roots of the cubic equation Œ

3

.

=  and  ( ) are polynomials of

2 b . Denote the solution of the original equation for arbitrary (including complex)  by  . Then the following recurrence relation holds:



+3

=   + (2  + 3)

−1



 − (  + 1)(2 + 3)(

−2


 −

−3

 ),

(1)

where the prime denotes differentiation with respect to  .  Since the functions 1 1 =  −  , corresponding to three values of Œ determined by the 3 equation Œ =  , form a fundamental set of solutions, formula (1) makes it possible to find  all  for any integer values of B not divisible by 3. In particular, 2 = ( −1 + Œ )  −  , where 3 Œ =.

70.

71.

72.

73.

74.

75.

76.

77.

+ (4 ï 3 + ô ï )ñ ó ò – ôñ = 0. = L 1  ø Á 2 ( ) + L 2  ø Á ( ) n Á ( ) + L Solution: Bessel functions; 4 @ 2 = 1 −  .

ï 3 ñ óò ò ó ò ó

3

"n Á 2 ( ), where ø Á ( ) and n Á ( ) are the

+ ï [ ô ï 2 + 3 õ (1 – õ )]ñ ó ò + 2 õ ( ô ï 2 + õ 2 – 1)ñ = 0. 1 b . Particular solutions with  > 0: 1 = ' sin  £   , 2 = ' cos  £   .

ï 3 ñ óò ò ó ò ó

2 b . Particular solutions with  < 0:

1

3 b . Particular solutions with  = 0:

1

= ' exp  − £ −   ,

= ' ,

2

= ' +1 .

2

= ' exp   £ −   .

+ ï ( ô ï 2 + õ ï + ö )ñó ò + ( ü – 1)(ô ï 2 + õ ï + ö + ü 2 + ü )ñ = 0. The substitution ¨ =  
+ ( , − 1) leads to a second-order linear equation of the form 2.1.2.131:  2 ¨2

 − ( , + 1) ¨2
+ (  2 + ' + ( + , 2 + , )¨ = 0.

ï 3 ñ óò ò ó ò ó

+ ô ïƒ÷ ñó ò + ( õ – 1)(ô ïw÷ –1 + õ 2 + õ )ñ = 0. The substitution ¨ =  
+ (  − 1) leads to a second-order linear equation of the form 2.1.2.132:  2 ¨2

 − (  + 1) ¨2
+ (   −1 +  2 +  )¨ = 0.

ï 3 ñ óò ò ó ò ó

+ ï ( ô ïƒ÷ + õ )ñó ò – 2( ô ïƒ÷ + õ )ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.118:  2 ¨2

 + (   +  )¨ = 0.

ï 3 ñ óò ò ó ò ó

+ ï ( ô ïƒ÷ + õ – ö )ñó ò + ( ö – 1)(ô ïw÷ + õ + ö 2 )ñ = 0. The substitution ¨ =  
+ ( ( − 1) leads to a second-order linear equation of the form 2.1.2.132:  2 ¨2

 − ( ( + 1) ¨2
+ (   +  + ( 2 )¨ = 0.

ï 3 ñ óò ò ó ò ó

+ ( ô ï 2÷ + 1 – ù 2 ) ï ñó ò + [ õ ï 3÷ + ô (ù – 1) ï 2÷ + ù 2 – 1]ñ = 0. The transformation ˆ =   B , H =   −1 leads to a constant coefficient linear equation: H Šª

Šª
Š + !H Š
+ ˜H = 0.

ï 3 ñ óò ò ó ò ó ï 2(ô ï

+ õ )ñó ò ò ó ò ó + ( ö ï – õÿþ 2 – õÿþ )ñó ò + (þ – 1)(ö + ôþ 2 + ôþ )ñ = 0. The substitution ¨ =  
+ ( − 1) leads to a second-order linear equation of the form ¡ 2.1.2.172:  (  +  )¨3

 − ( + 1)( +  )¨3
+ ( ( +  2 +  )¨ = 0.

¡

¡

¡

© 2003 by Chapman & Hall/CRC

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503

3.1. LINEAR EQUATIONS

78.

79.

ï (ô ï

2

õï

+

+ ö )ñó ò ò ó ò

ñóò

– 2ñ = 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.179: (  2 + ' + ( )¨2

 + ¨ = 0.

ï 5 ñ óò ò ó ò ó

= ô (ï

ï 6 ñ óò ò ó ò ó

+

– 2ñ ).

ñóò a

 2VL  2VL

=

Solution: 80.

+ï

ó

ô ï 2 ñ óò

1

+L

2

exp  £    + L

£ −   + L 1 + L 2 cos 

+ (õ – 2ô

ï )ñ

3

exp  − £    W

£ −   W 3 sin 

if  > 0, if  < 0.

= 0.

The transformation  = a , = ¨ a −2 leads to a constant coefficient linear equation of the form 3.1.2.82 with  2 = 0: ¨2g

g
g +  ¨2g
−  ¨ = 0. −1

3.1.2-3. Equations of the form  3 ( )


 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( ). 81.

ñóò ò ó ò ó

+ 3 ôñó ò ò ó + 3 ô

 =  − (L

Solution: 82.

ñ óò ò ó ò ó

2

ñóò

+

1+L

ô 3ñ

= 0.

2 2  + L 3  ).

ô 2 ñ óò ò ó + ô 1 ñ óò + ô 0 ñ = 0. A third-order constant coefficient linear equation. . Denote ( Œ ) = Œ 3 +  2 Œ 2 +  1 Œ +  0 . . 1 b . Let the characteristic polynomial ( Œ ) be factorizable: . ( Œ ) = ( Œ − Œ )( Œ − Œ )( Œ − Œ ), where Œ , Œ , and Œ are real numbers. 1 2 3 1 2 3   L 1   + L 2    + L 3    if all the roots Œ  are different, +

1

=

Solution:

2 b . Let

.

=L

84.

ôñóò ò ó

1

+ (õ

ï

ñóò ò ó ò ó

+

ñóò ò ó ò ó

+ 3 ôñó ò ò ó + 2( õ

2

1+L 1+L

( Œ ) = ( Œ − Œ 1 )( Œ

Solution: 83.

(L (L



1



2

3

1 + L 3  2  )   2 + L 3  2 )  1

if Œ if Œ

3

+ 2  1 Π+  0 ), where 

+

−



1



+ ö )ñó ò + ( ôƒõ

(L

ï

2

cos “/ + L

3

2 1

1 1

=Œ =Œ

86.

87.

2

≠ Œ 3, = Œ 3.

<  0.

sin “/ ), where “ = Å  0 −  21 .

+ ô„ö + õ )ñ = 0.

 Integrating yields a second-order linear equation: "

 + ( ' + ( ) = L  −  (see 2.1.2.2 for the solution of the corresponding homogeneous equation with L = 0).

ï

ï

+ ô 2 )ñó ò + õ (2 ô

+ 1)ñ = 0.

This is a special case of equation 3.1.2.113 with B = 0 and 85.

2

ñóò ò ó ò ó

+

ôñóò ò ó

+ (õ

ï

2

+

öï

+ ý )ñó ò + ô ( õ

The substitution ¨ = 
+  ¨2

 + ( ' 2 + (˜ + )¨ = 0. M

ñ óò ò ó ò ó

+

ôñ óò ò ó

+

õ ï ÷ ñ óò

+

ñóò ò ó ò ó

+ 3ô

ï ñóò ò ó

+ 3ô

ï ñóò

2 2

2

+

öï

= 1.

+ ý )ñ = 0.

leads to a second-order linear equation of the form 2.1.2.6:

ôƒõ ï ÷ ñ

The substitution ¨ = 
+  ¨2

 + '  ¨ = 0.

ï

¡

= 0.

leads to a second-order linear equation of the form 2.1.2.7: + (ô

3

ï

3

+ õ )ñ = 0.

The substitution = ¨ exp  − 12  2  leads to a constant coefficient linear equation of the form 3.1.2.82 with  2 = 0: ¨2


 − 3  ¨2
+  ¨ = 0.

© 2003 by Chapman & Hall/CRC

Page 503

504 88.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+

ô ï ñóò ò ó

ï

+ ( ôƒõ

Particular solutions: 89.

90.

ñ óò ò ó ò ó

ï ñ óò ò ó

+ 3ô

–

=

−

2 2

+

1

ï

+ (2 ô





ô

ï ñ óò ò ó

+ 3ô

2 1

+ L 2¨ 1¨

õ 2 )ñ óò

+

,

=

−

óò

+

2

+ õ )ñ

ôƒõÿñ

= 0.



 X

exp  2 ' − 12    2

ôƒõ ï ñ

ï

+ (2 ô

¨ 2 ¨ 2 + L 3 2 . Here,

ï

+ 2õ

2 2

ñóò ò ó ò ó

ï ñóò ò ó

+ 3ô

ï

+ 3( ô

óò

+ ô )ñ

ï

+ õ (2 ô

ï ñóò ò ó

+ 3ô

+ [2( ô

2

+ õ )ï

3 3

   ¨ 2

= exp  The substitution ¨2


 + [(  − 3  2 ) + ( ]¨ = 0.

ñóò ò ó ò ó

ï

+ ô )ñó ò + ( ô

2 2

− 12

92.

+ 1)ñ = 0.

2

2

+

õï

94.

ñ óò ò ó ò ó

ï

+ (ô

+ õ )ñ

óò ò ó

leads to a linear equation of the form 3.1.2.4:

+ ô ]ñó ò + 2 õ

ï (ô ï

+ 1)ñ = 0.

2

96.

ñ óò ò ó ò ó

+ ( ôƒõ

ï

+

ô

ñóò ò ó ò ó

ï

óò ò ó

+ õ )ñ

+

óò

+ ö )ñ

ôƒõ 2 ï ñ óò

= ,

1

+

õ^ö^ñ

= 0.

=

2

– ôƒõ −





ñ

2

−





.

ï

+ 1)ñ = 0.

Integrating yields a second-order linear equation: "

 +  
 + (˜ = L  for the solution of the corresponding homogeneous equation with L = 0).

ñóò ò ó ò ó

ï

+ (ô

+

õ

+ ö )ñó ò ò ó + ( ô„ö

ñóò ò ó ò ó

ï

+ (ô

ï

 1

ï

+ ,

õ^ö 2

+ s)ñ„ó ò + s( ô = 

 2

ï

2

−





(see 2.1.2.28

+ õ )ñ = 0.

, where Œ

+ 2 ô )ñó ò + ô [( ö – ôƒõ ) ï

+ õ )ñó ò ò ó + ( ö

(see 2.1.2.28 for

= 0.

+ õ )ñó ò ò ó + [( ôƒõ + ö ) ï + ô ]ñ„ó ò + ö ( õ

+ (ô

Particular solutions: 1 =  equation Œ 2 + (Œ + ° = 0. 97.

ô

+

= 2.

¡

Integrating yields a second-order linear equation: "

 +  
 + ( = L  the solution of the corresponding homogeneous equation with L = 0).

Particular solutions: 95.

ï

+ ( ôƒõ

= 1.

¡

+ ö )ñ = 0.

This is a special case of equation 3.1.2.113 with B = 1 and 93.

 .

= 0.

This is a special case of equation 3.1.2.113 with B = 1 and 91.

M

¨ Solution: 1 and 2 are linearly independent solutions ¨ p

of a second-order equation of the form 2.1.2.28:   +  ¨2
+ 14  ¨ = 0.

ñ óò ò ó ò ó

= L 1¨

ô

+

1

and Œ

2

are roots of the quadratic

+ õ ]ñ = 0.

 − 21  2  . The substitution = exp  − 12  2  X H ( )  leads to a M   ) H 
+[  2  2 +( ( −2  ) −  ]H = 0. second-order linear equation of the form 2.1.2.31: H 

 +(  −2  Particular solution:

98.

99.

ñóò ò ó ò ó

+ (ô

ï

ñóò ò ó ò ó

+ (ô

ï

0 = exp

+ õ )ñó ò ò ó + ( ö

ï

+ ý )ñó ò + [ ô„ö

ï

2

+ ( ô„ý +

õ^ö )ï

+

ö

+ õ^ý ]ñ = 0.

Integrating yields a second-order linear equation: "

 + ( (˜ + ) = L exp  − 12  2 − '  (see M 2.1.2.2 for the solution of the corresponding homogeneous equation with L = 0). + õ )ñó ò ò ó + ( 

ï

2

+

ï

+ )ñó ò – ü [ 

ï

2

+ ( ôƒü +  ) ï +

ü

2

+

õü

+ ]ñ = 0.

The substitution ¨ = 
− , leads to a second-order linear equation of the form 2.1.2.31: ¨2

 + (  +  + , )¨2
+ [ ƒ… 2 + ( , + ý ) + , 2 + , + = ]¨ = 0.

© 2003 by Chapman & Hall/CRC

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505

3.1. LINEAR EQUATIONS

100.

– ï 2 ñó ò ò ó + ( ô + õ – 1)ï ñ„ó ò – ôƒõÿñ = 0. The following three series, converging for any  , make up a fundamental set of solutions:

ñóò ò ó ò ó

1

2

3

101.

102.

103.

½

=1

 (  − 3)(  − 3)  ( − 3B + 3)(  − 3B + 3) 3   , (3B )!

=1

(  − 1)(  − 1)(  − 4)(  − 4)  (  − 3B + 2)(  − 3B + 2) 3   (3B + 1)!

=1+ ÷

½



= + ÷ =





2

2

½

(  − 2)(  − 2)(  − 5)(  − 5)  (  − 3B + 1)(  − 3B + 1) 3   (3B + 2)!

+ ÷



=1

105.

ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

–

õÿñóò

= 0.

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

ïƒ÷ –1 ñ óò

– 2ô

1

.

2

2

= cos   £ −   ,

ïƒ÷ –2 ñ

+ 2ô

= ,

= exp(− £  ),

= exp( £  ).

2

= sin   £ −   .

= 0.

2

= .

+ ô ï ÷ ñ ó ò ò ó + õ ï ÷ –1 ñ ó ò – 2( ô + õ ) ï ÷ –2 ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.45: ¨2

 +   ¨2
+ (  +  )  −1 ¨ = 0.

ñ óò ò ó ò ó ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

+

ôƒõ ƒï ÷ ñ óò

ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

– (ô

1

ïƒ÷

ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

= exp  –1

õ 2 (ô ïƒ÷

õ ï 2 )ñ óò

–

– õ )ñ = 0.

'  cos 

− 12

ïƒ÷

1

=



−

õ ïƒú ñóò 0

ôù ïƒ÷

+

–

= .



,

–1 2

õ ïƒú

–1

ñ

3 2

2

'  ,

−



 X

+1

= sin 

õ 2 )ñ óò

–

=

2

õ ï (ô ïƒ÷

+

£  , 1 1 = cos  2 

+ ( ôƒõ

+ ô ïƒ÷ ñó ò ò ó + Particular solution:

ñóò ò ó ò ó

+

2

Particular solutions: 108.

ïƒ÷ ñ

– ôƒõ

Particular solutions: 107.

+2

+ ô ïƒ÷ ñó ò ò ó – 2 ô ïƒ÷ –2 ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.2.45: ¨ 

 +   ¨ 
+   −1 ¨ = 0.

Particular solutions: 106.

,

ñóò ò ó ò ó

Particular solutions: 104.

+1

+

2

= exp  − 21 '  sin 

2

3 2

'  .

+ 3)ñ = 0. 1 2

 2£   .

ôƒõÿù ïƒ÷ –1 ñ

exp O 2 ' −



= 0.

B +1

 

+1

P3M

 .

= 0.

109.

+ ô ïƒ÷ ñó ò ò ó + õ ïƒú ñó ò + õ ïƒú –1 ( ô ïƒ÷ +1 + þ )ñ = 0. ¨ 
+   ¨ = 0. The substitution ¨ = 

 + ' +2 leads to a first-order linear equation: u

110.

ñóò ò ó ò ó

ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

Particular solutions: 111.

ïƒ÷

– õ (2 ô 1

=



+ 3 õ )ñó ò +



,

2

= 

õ 2 (ô ïƒ÷ 



+ 2 õ )ñ = 0.

.

+ ô ïƒ÷ ñó ò ò ó + ( ôƒõ ïƒ÷ – õ 2 + ö )ñó ò + ö ( ô ïƒ÷ – õ )ñ = 0.   Particular solutions: 1 =  1 , 2 =  2 , where Œ 1 and Œ equation Œ 2 + Œ + ( = 0.

ñóò ò ó ò ó

2

are roots of the quadratic

© 2003 by Chapman & Hall/CRC

Page 505

506 112.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó

+ (õ

ïƒú

Particular solution: 113.

114.

ïƒ÷ ñóò ò ó

ï ÷

+ 3ô

+ (2 ô

ñóò ò ó ò ó

= ( ïƒ÷ – ô )ñó ò ò ó + ( ô

ïƒú

+ 2õ

2 2

L 2¨ 1¨

ïƒ÷

ñóò ò ó ò ó

+ (ô

ïƒ÷



1

,

ïƒ÷

+ õ )ñó ò ò ó + ( ô„ö

ñóò ò ó ò ó + (ô ïƒ÷

¡

– õ )ñó ò ò ó + ö

Particular solution: 117.

ñóò ò ó ò ó

0

2

õ^ö

+

 1

119.

ñ óò ò ó ò ó

+ (ô

ï ÷

+ ö )ñ

ñ óò ò ó ò ó

+ (ô

ï ÷

+

óò ò ó

123.

ïƒ÷ +ú

)ñó ò + õ (2 ô

+þ

ïƒú

–1

)ñ = 0.

õ ƒï ÷ ñ

. , where Œ

2

+þ

)ñó ò + (þ

 2

= 

2



= 

+ö

and Œ

ïƒú )ñ

1

are roots of the quadratic

2

ïƒ÷

+ ö 2 )( ô

, where Œ

ïƒ÷

– õ ( ôƒõ

1

+

and Œ

õ

– ö )ñ = 0.

are roots of the quadratic

2

= 0.

.

ïƒ÷

õ ïƒú )ñ óò

+

ï ÷

ñóò ò ó ò ó

+ (ô

ïƒ÷

+

ñóò ò ó ò ó

+ ( ôƒõ

ïƒ÷

õ ï )ñ óò ò ó

+

ñóò ò ó ò ó

+ (ô

ïƒ÷

+

+ õ (ô 1

ïƒú ñ

+ ôƒõ

÷

+ ( ôù + õ ) ï

ô ïƒ÷

+ ( ôƒõ

v 1 = 

–1 1

ï ÷

+1

+ 2)ñ

= exp  − 12 ' 2  ,

−

ïƒ÷

+1

–1

]ñ

= 0.

óò

ï ÷

+ õ [ö

2

v X 2 = 

2

=

−

2





+ ö^ñó ò + ö ( ô

ôƒõ ï ÷ ñ

+

–1

+ (ù – 1) ï

÷

–2

]ñ = 0.

= 0.

= exp  − 12 ' 2  X exp 

+

õ

– ö 2 )ñó ò + ö ( ôƒõ exp  2 (˜ − 12 '  

−

,

= ,

óò

õ^ö ï

+

+ õ )ñó ò ò ó + ôƒõ

õ ïƒú )ñ óò ò ó

2

ïƒ÷ ñóò

– ôƒõ

2

ïƒ÷ –1 ñ

ïƒ÷

+

õ ïƒú )ñ

= cos   £ (  ,

2 b . Particular solutions with ( < 0:

1

= exp  − £ − (  ,

+ (ô

ïƒ÷

+

õ ïƒú )ñ óò ò ó

ï ñóò ò ó ò ó

+ 3ñó ò ò ó +

The substitution ¨ ¨2


 +  ¨ = 0.

1

ô ï ñ

=

+ ( ôƒõ

=

= 0.

−

v

,

ïƒ÷ +ú 2

=

M

+1

' 2   . M – ô„ö

ïƒ÷

+ õ )ñ = 0.

 .

= 0.

= 0.

1

ñóò ò ó ò ó

ïƒ÷

1 2

.

1 b . Particular solutions with ( > 0:

Particular solutions: 124.



+ [ ô„ö

õ ï )ñ ó ò ò ó

Particular solutions: 122.

–1

= 0.

 Integrating yields a second-order linear equation: "

 +   
 + '  −1 = L  − v (see 2.1.2.45 for the solution of the corresponding homogeneous equation with L = 0).

Particular solutions: 121.



−  . 0 = 

Particular solutions: 120.

ïƒú ñóò

=

+ ( ïƒ÷ + ô )ñó ò ò ó + ( ô

Particular solution: 118.

ïƒ÷

– õ )ñó ò +

Particular solutions: 1 =  , equation Π2 + (Π+ + ( 2 = 0. 116.

+ ôù

õ ïƒú )ñ

¨ 2 ¨ ¨ Solution: 2 + L 3 2 . Here, 1 and 2 form a fundamental set of solutions ¨ 2
u ¨

1   + 2 ' +2¨ = 0. of the second-order linear equation:  +  2 1+

Particular solutions: 1 =  equation Π2 + Π+  = 0. 115.

+

v . 0 = 

ñóò ò ó ò ó

= L 1¨

ïƒ÷

– ö 2 )ñó ò – ö ( ô„ö

+ −

õ^ö ïƒú

v X

+

= sin   £ (  .

2

2

= exp   £ − (  .

õÿþ ƒï ú –1 – ö 2 )ñ óò ï ÷ +ú – ô„ö ƒï ÷ + ö ( ôƒõ ƒ

exp O 2 (˜ −

' + ¡

+1

+ 1 P3M

+

õÿþ ïƒú

–1

)ñ = 0.

 .

leads to a constant coefficient linear equation of the form 3.1.2.1:

© 2003 by Chapman & Hall/CRC

Page 506

507

3.1. LINEAR EQUATIONS

125.

ï ñ óò ò ó ò ó

óò ò ó

– 3ùñ

=  3

Solution:

ï ñ

+ô +2

O

¨2


 +  ¨ = 0.

126. 2 ï

ñ óò ò ó ò ó

óò ò ó

+ 3ñ

+ô

½

= 0,



1 2

ï ñ

M M



¨ O

3P

=õ ,



= 0, 1, 2,

uuu

, where ¨ is the general solution of equation 3.1.2.1:

P

2

ô

 ‘

4

ù

≠ 0.

H  LlÁ X  ¡ £ , where Œ 1 , Œ 2 , and Œ 3 are roots of the cubic equation M 3 0 2H +  Á =1 2 Œ 3 +  = 0; Œ 4 = − F for  > 0 and Œ 4 = + F for  < 0. In addition, the constants L*Á are related by the constraint £  ( L 1 + L 2 + L 3 + L 4 ) +  = 0, and the integrals are taken along =

Solution:

straight lines.

127.

128.

129.

130.

ï ñóò ò ó ò ó

+ 3ñó ò ò ó +

ô ï 2ñ

=õ .

ï ñóò ò ó ò ó

+ 3ñó ò ò ó +

ô ï 4ñ

=

õï

ï ñ óò ò ó ò ó

+

ôñ óò ò ó

ôƒõÿñ óò

+

õ 3ï ñ

ï ñóò ò ó ò ó

+ ( ô + õ )ñó ò ò ó – ï

¨ 


 +  ¨ =  . leads to an equation of the form 3.1.2.3: I

The substitution ¨ =  The substitution ¨ =  +

.

¨ 


 +  3 ¨ = ' . leads to an equation of the form 3.1.2.5: I

– ôñ = 0,

ñóò

Solution:

½

= where = 1 = −1, ý ý 3 = −1. 131.

132.

1

==

134.

135.

2

+

ôñóò ò ó

+ ( õ – ö 2)ï

ï ñ óò ò ó ò ó

+

ôñ óò ò ó

+ [( ö –

3

Á

= 0, ý

ï ñóò ò ó ò ó

LlÁ X

=1 2

ñóò

ô

;

:

¡

¡

|a | 

> 0.

|a − 1|(  −2) J 2 

−1 2

= 1; for  > 0, =

3

= 1 and ý

õ ï )ñ

= 0.

+ ö (ô –

õ ï )ñ

– ö ( ô„ö +

õ

> 0,

−

3

g M

a,

= + F ; for  < 0, =

3

= −F

and

The substitution ¨ = 
− ( leads to a second-order linear equation of the form 2.1.2.108:  ¨2

 + ( (˜ +  )¨2
+ ( ' + ! ( )¨ = 0.

Particular solutions: Œ 2 + Œ + ( = 0. 133.

= 0.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.2.108:  ¨2

 + (  − ' )¨2
+  2  ¨ = 0.

ï ñóò ò ó ò ó

+

ï ñóò ò ó ò ó

+ (ô

ôñóò ò ó

+

1

õ 2 )ï

= 

õ ƒï ÷ ñ óò

+ ôƒõ ]ñ

 1

,

+ õ )ñó ò ò ó – ô

2

The substitution ¨ = 
+   ¨2

 +  ¨2
−  ¨ = 0.

ï ñóò ò ó ò ó

+ (ô

ï

+ õ )ñó ò ò ó +

= 

 2

, where Œ

+ õ ( ô + ù – 1)ïw÷

The substitution ¨ = 

 + ' 

ï

2

óò

õÿñ

ö ï ñóò

−1

–1

ñ

1

= 0. and Œ

2

are roots of the quadratic equation

= 0.

leads to a first-order linear equation:  ¨ 
+  ¨ = 0.

= 0. leads to a second-order linear equation of the form 2.1.2.108: – ö^ñ = 0.

The substitution ¨ =  
− leads to a second-order linear equation of the form 2.1.2.108:  ¨2

 + (  +  − 1)¨3
+ (˜ ¨ = 0.

© 2003 by Chapman & Hall/CRC

Page 507

508 136.

137.

THIRD-ORDER DIFFERENTIAL EQUATIONS + ( ô ï + 3)ñ ó ò ò ó + ( õ ï + 2 ô )ñ ó ò + ( ö ï + õ )ñ = 0. The substitution ¨ =  leads to a constant coefficient linear equation of the form 3.1.2.82: ¨2


 +  ¨2

 +  ¨2
+ ( ¨ = 0.

ï ñ óò ò ó ò ó ï ñóò ò ó ò ó

+ (ô

Solution: 138. 139.

ï

ï

+ 3)ñó ò ò ó + ô ( õ −1

=

+ 2)ñó ò + õ [ õ ( ô – õ ) ï + ô ]ñ = 0.

2 2 V L 1  (  −  )  + L 2  −   J 2 cos  3 '  + L 3  −   J 2 sin  3 '  W . 2 2

+ [ ô ( õ + 1)ï + õ ]ñ„ó ò ò ó + ô 2 õ ï ñó ò – ô  Particular solutions: 1 =  , 2 =  −  .

ï ñóò ò ó ò ó

2

õÿñ

= 0.

ï ñóò ò ó ò ó

– ( ï + 2 ô )ñó ò ò ó – ( ï – 2 ô – 1)ñ„ó ò + ( ï – 1)ñ = 0.  Solution: = L 1  +   +1 [ L 2 ®  +1 ( )+ L 3 ;  +1 ( )], where ®  ( ) and Bessel functions.

;  (

) are the modified

140. 2 ï ñó ò ò ó ò ó – 4( ï + ô – 1)ñ„ó ò ò ó + (2 ï + 6 ô – 5)ñ„ó ò + (1 – 2 ô )ñ = 0.   = L 1  +    J 2 [ L 2 ®  ( 2) + L 3 ;  ( 2)], where ®  ( H ) and Solution: modified Bessel functions. 141. 2 ï ñó ò ò ó ò ó + 3(2 ô Solution:

ï

ï

+ ü )ñ„ó ò ò ó + 6( õ =

.

½

4

Á

+

ôƒü )ñ óò

 ‘ . LlÁÑX  ¡  [ ( H )]( 

J

−2) 2

0

=1

ï

+ (2 ö

+ 3õ

H , M

ü )ñ L

= −L

4

ü

= 0,

1

−L

2

;  (H

) are the

> 0. −L

3,

where ( H ) = H 3 + 3 !H 2 + 3 ˜H + ( ; Œ 1 , Œ 2 , and Œ 3 are roots of this polynomial, which are assumed to be different; Œ 4 = − F for  > 0 and Œ 4 = + F for  < 0. 142.

143. 144.

145.

146.

+ ( ô ï + õ )ñó ò ò ó + [( ô„ö + s – ö 2 ) ï + õ^ö ]ñó ò + s[( ô – ö ) ï + õ ]ñ = 0.   Particular solutions: 1 =  1 , 2 =   2 , where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + (Œ + ° = 0.

ï ñóò ò ó ò ó

+ ( ô ï 2 + õ + 2)ñó ò ò ó – ôƒõ ( õ + 1)ñ = 0. This is a special case of equation 3.1.2.145 with B = 2.

ï ñóò ò ó ò ó

+ ( ô ï 2 + õ )ñó ò ò ó + 4 ô ï ñó ò + 2 ôñ = 0. Integrating the equation twice, we arrive at a first-order linear equation: 
 + (  L 1 + L 2 .

ï ñóò ò ó ò ó

+ ( ô ïƒ÷ + õ + 2)ñó ò ò ó – ôƒõ ( õ + 1) ïƒ÷ –2 ñ = 0. The substitution ¨ =  
+  leads to a second-order linear equation of the form 2.1.2.45: ¨ 

 +   −1 ¨ 
−  (  + 1)  −2 ¨ = 0.

ï ñóò ò ó ò ó

+ (ô

ïƒ÷

+ 3)ñó ò ò ó + (2 ô

ï ñóò ò ó ò ó

+ (ô

ïƒ÷

+1

ï ñóò ò ó ò ó

+ (ô

ïƒ÷

1

=

−1

ïƒ÷

1

=

−1

+ 3)ñó ò ò ó + ( ôƒõ

Particular solutions:

1

=

−1

–1

+

õ ï )ñ óò

+ õ (ô

cos  £   ,

+ 3)ñó ò ò ó + ô ( õ

Particular solutions: 148.

+  − 2) =

ï ñóò ò ó ò ó

Particular solutions: 147.

2

ï

2

,

+ 2ô 2

=

−1

ïƒ÷

–1



–

  .

−1 −

2

+ 1)ñ = 0.

sin  £   .

ñóò + 2 õ (ôƒõ ïƒ÷ 3

+ 2) ïƒ÷

exp  − 21 '  cos 

ïƒ÷

=

ïƒ÷

'  ,

õ 2 ï )ñ óò

+1 2

+

ô ïƒ÷

=

+ õ (ô

–

−1

ïƒ÷

õ 2 ï )ñ

= 0.

exp  − 21 '  sin 

–1

2

3 2

'  .

– õ )ñ = 0.

© 2003 by Chapman & Hall/CRC

Page 508

509

3.1. LINEAR EQUATIONS

ï

+ õ )ñ„ó ò ò ó ò

149. ( ô

ó

+ [ õ ( ô + 1)ï +

Particular solutions:

1

= ,

õ

2 2

+ 1]ñó ò ò ó + =

−





õ 2 ï ñ óò

–

õ 2ñ

= 0.

.

150. ( ô ï + õ )ñ ó ò ò ó ò ó + ü ( ô ï + õ )ñ ó ò ò ó + ( ö ï + ý )ñ ó ò + ü ( ö ï + ý )ñ = 0. The substitution ¨ = 
+ , leads to a second-order linear equation of the form 2.1.2.108: (  +  )¨2

 + ( (˜ + )¨ = 0. 151.

ñ óò ò ó ò ó ( ô ï + õ )„

M

+ ( ö ï + ý ) ñ óò ò ó

+ [( ô + ö ) ï + õ + ý ]ñ„ó ò + ( + 2 )[( ö – ô ) ï + ý – õ ]ñ = 0. Particular solutions: 1 = exp( ° 1  ), 2 = exp( ° 2  ), where ° 1 and ° 2 are roots of the quadratic equation ° 2 + “±° + Œ + “ 2 = 0.

ï + õ )„ñ óò ò ó ò ó

152. ( ô

ï + ý )ñ óò ò ó – ü

+ (ö

 , 1 = 

Particular solutions:

[(3 ôƒü + 2ö ) ï + 3 õ

+ 2 ý ]ñ„ó ò + ü 2 [(2 ôƒü + ö ) ï + 2 õ

ü

 . 2 =  

ü + ý ]ñ

= 0.

153. ( ô ï + õ )ñ ó ò ò ó ò ó + ( ö ï + ý )ñ ó ò ò ó + s ï ( ô ï + õ )ñ ó ò + s[ ö ï 2 + ( ô + ý ) ï + õ ]ñ = 0. The substitution ¨ = 

 + ˜°  leads to a first-order linear equation: (  +  )¨u
+( (˜ + )¨ = 0. 154.

(1 – ï )ñ ó ò ò ó ò ó

+ ï (ô ï

– 2 ô + 1)ñ ó ò ò ó + (– ô ï 2

= ,

Particular solutions:

1

(ô ï

+1

2

2



+ 2 ô – 1)ñ ó ò + 2 ô ( ï

M

– 1)ñ = 0.

= .

155. ( ô ï + õ )ñ„ó ò ò ó ò ó + ( ö ï + ý )ñó ò ò ó + s ïƒ÷ ( ô ï + õ )ñó ò + s ïƒ÷ –1 [ ö ï 2 + ( ôù + ý ) ï + õÿù ]ñ = 0. The substitution ¨ = 

 + °˜  leads to a first-order linear equation: (  +  )¨u
+( (˜ + )¨ = 0. 156.

–1)ñ„ó ò ò ó ò ó + ï ( ƒ ô õ ïƒ÷

Particular solutions: 157.

158.

159.

161.

 . 2 = 

2

= ,

+(2 õ ïƒ÷ – ô 2 õ ïƒ÷

+2

+ ô ) ñ óò +2ôƒõ (ô ï 2

M

–1)ïw÷ ñ

= 0.

+ 3 ï ñ ó ò ò ó – 3ñ ó ò + ô ï 2 ñ + õ = 0. = (¨u  )
 , where the function ¨ = ¨ ( ) satisfies a constant coefficient linear Solution: equation of the form 3.1.2.2: ¨u


 +  ¨ =  .

ï 2 ñ óò ò ó ò ó

+ 6 ï ñó ò ò ó + 6ñó ò + ô ï 2 ñ = õ . The substitution ¨ =  2 leads to a constant coefficient linear equation of the form 3.1.2.2: ¨2


 +  ¨ =  .

ï 2 ñ óò ò ó ò ó ï 2 ñ óò ò ó ò ó

– 3(ù + þ Solution: =

160.

1

–2 õ ïƒ÷ – ô 2 ) ñ óò ò ó

)ï

»

ñ óò ò ó

−1



+ 3ù (3þ

( > − 3“ − 1)

óò

+ 1)ñ

+»

–ï

2

Á

−1

( > − 3 @ − 2)

ñ ½

3



=1

where the    are three roots of the cubic equation  

3

=0

ï 2 ñ óò ò ó ò ó

+ 6 ï ñ ó ò ò ó + 6ñ The substitution ¨ = 

ï 2 ñ óò ò ó ò ó

– 2(ù + 1)ï Solution: =

where

.

™

L

óò

2

ñ óò ò ó

+L L 1 (  1

=0

þ ,ù

= 0,

 L   Ÿ ¼ ,

= 1, 2, 3,

uuu

> = M ,  M

= 1.

+ ô ï 3ñ = õ . ¨ 


 +  ¨ =  . leads to a linear equation of the form 3.1.2.3: p – (ô 2 2

ï

2

– 6ù )ñ

óò

+ 2ô

+ L 3  2  +1  − 4B + 2) + L 2 



4

2 .

ï ñ

ù

= 0,

= 1, 2, 3,

uuu

if  = 0,  2  ( ) + L 3  Æ ( ) if  ≠ 0, −

( ) and Æ ( ) are some polynomials of degree ≤ 2B + 2.

© 2003 by Chapman & Hall/CRC

Page 509

510 162.

163.

164.

165.

166.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ï 2 ñ óò ò ó ò ó

+ 3 ï ñó ò ò ó + (4 ô 2 ï 2 D + 1 – 4 ô 2 õ 2 )ñó ò + 4 ô = L 1 ø 2 (  ) + L 2 ø (  ) n (  ) + L 3 n Solution:    Bessel functions.

168.

169.

170.



= 0.

(  ), where

2

ø  (H

) and n ( H ) are the



+ ô ï 2 ñó ò ò ó + ( õ ï + ö )ñó ò + ô ( õ ï + ö )ñ = 0. The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.2.111:  2 ¨2

 + ( ' + ( )¨ = 0.

ï 2 ñ óò ò ó ò ó

+ ô ï 2 ñ ó ò ò ó + ( õ ï ÷ + ö )ñ ó ò + ô ( õ ï ÷ + ö )ñ = 0. The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.2.118:  2 ¨ 

 + ( '  + ( )¨ = 0.

ï 2 ñ óò ò ó ò ó ï 2 ñ óò ò ó ò ó

– ( ï + ô ) ï ñó ò ò ó + ô (2 ï + 1)ñ„ó ò – ô ( ï + 1)ñ = 0.  = L 1  +  (  +1) J 2 V L 2 ø  +1  2 £   + L 3 n  +1  2 £   W , where ø  ( H ) and n  ( H ) Solution: are the Bessel functions.

ï 2 ñ óò ò ó ò ó

– (ï

2

– 2 ï )ñó ò ò ó – ( ï



= L 1 Solution: Bessel functions. 167.

ï D ñ

3 2 –1

ï 2 ñ óò ò ó ò ó

+£  VL

– 2 ï ( ï – 1)ñ„ó ò ò ó + ( ï

2 2

+ô

2

®  (

2

– 14 )ñó ò + ( ï

)+L

– 2ï +

  = L 1 + £   J 2 V L Solution: modified Bessel functions.

2

®  (

3

1 4

2

;  (  )W ,

– 2ï +

ô

2

– 14 )ñ = 0.

where ®  ( ) and

– ô 2 )ñó ò + ( ô

2

;  (

) are the modified

– 14 )ñ = 0.

2) + L

3 ;  (  2)W , where ®  ( H ) and ;  ( H ) are the

ï 2 ñ óò ò ó ò ó

– 3( ï – ô ) ï ñó ò ò ó + [2 ï 2 + 4( õ – ô ) ï + ô (2 ô – 1)]ñ„ó ò – 2 õ (2 ï – 2ô + 1)ñ = 0. = L 1 ¨ 12 + L 2 ¨ 1 ¨ 2 + L 3 ¨ 22 . Here, ¨ 1 and ¨ 2 are a fundamental set of solutions Solution: of a second-order linear equation of the form 2.1.2.108:  ¨p

 + (  −  )¨2
+  ¨ = 0.

ï 2 ñ óò ò ó ò ó

+ ï [( ô + ö ) ï + õ ]ñ„ó ò ò ó The substitution ¨ = 
+ ( with B = 1:  2 ¨2

 +  (  + 

+ [( ô„ö +  ) ï 2 + ( õ^ö +  ) ï + ]ñ„ó ò + ö (  ï 2 +  ï + )ñ = 0. leads to a second-order linear equation of the form 2.1.2.146 )¨2
+ ( ƒ… 2 + ý± + = )¨ = 0.

+ ( ô ï ÷ +1 + õ ï )ñ ó ò ò ó + [ ô ( õ – 2)ï Particular solutions: 1 =  + 1 , 2 =  + equation 2 + (  − 3) + ( −  + 2 = 0.

ï 2 ñ óò ò ó ò ó

¡

÷

+ ö ]ñ ó ò + ô ( ö – õ + 2)ï ÷ –1 ñ = 0. 2 , where 1 and 2 are roots of the quadratic

¡

¡

¡

171. 2 ï (ï – 1)ñ„ó ò ò ó ò ó + 3(2 ï – 1)ñ„ó ò ò ó + (2 ô ï + õ )ñó ò + ôñ = 0. Solution: = L 1 ¨ 12 + L 2 ¨ 1 ¨ 2 + L 3 ¨ 22 . Here, ¨ 1 and ¨ 2 form a fundamental set of solutions of the equation 2 ( − 1)¨3

 + (2 − 1)¨3
+  12  + 14  − 12  ¨ = 0, which is reduced, by means of the substitution  = cos2 ˆ , to the Mathieu equation 2.1.6.29: 2¨uŠª

Š = (  +  − 2 +  cos 2ˆ )¨ . 172. ( ô 2 ï 2 + ô 1 ï + ô 0 )ñó ò ò ó ò ó + ( õ 1 ï + õ 0 )ñó ò ò ó + ( ö 1 ï + ö 0 )ñó ò – þ ö 1 ñ = 0. is a positive integer. A solution of this equation is a polynomial of Here, ( 1 ≠ 0 and ¡ degree that can represented as follows:

¡

= where c

½+ 

O− =0

1 

( 1P

= M , ´ ®

M



,ž + ®´ −+

−1

[( 

2

+  1  +  0)c

3

+ (  1  +  0)c

2

 + ( 0c ].  + ,

Á Á =  +1 with @ ≠ −1. @ +1

© 2003 by Chapman & Hall/CRC

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511

3.1. LINEAR EQUATIONS

173.

174.

175.

176.

177. 178.

ï 3 ñ óò ò ó ò ó

+ 6ï

2

ñóò ò ó

+ (ô

ï

3

– 12)ñ + 2 õ = 0.

= (¨u  2 )
 , where ¨ = ¨ ( ) satisfies a constant coefficient linear equation of Solution: the form 3.1.2.2: ¨ 


 +  ¨ =  . + ô ï 2 ñ ó ò ò ó + õ ï ñ ó ò + ( ô – 2) õÿñ = 0. This is a special case of equation 3.1.2.175. Solution: = L 1  B 1 and B 2 are roots of the quadratic equation B 2 − B +  = 0.

ï 3 ñ óò ò ó ò ó

180.

181.

182.

183. 184.

 + L 2   1 + L 3   2 , where

+ ô ï 2 ñó ò ò ó + õ ï ñó ò + ö^ñ = 0. The Euler equation. The substitution a = ln | | leads to a constant coefficient linear equation of the form 3.1.2.82: g

g
g + (  − 3)
g
g + (2 −  +  ) g
+ ( = 0.

ï 3 ñ óò ò ó ò ó

+ 3 ô ï 2 ñó ò ò ó + 3 ô ( ô – 1)ï ñ„ó ò + [ õ ï 3 + ô ( ô – 1)(ô – 2)]ñ = 0. The substitution ¨ =   leads to a constant coefficient linear equation of the form 3.1.2.1: ¨2


 +  ¨ = 0.

ï 3 ñ óò ò ó ò ó

+ 3 ô ï 2 ñó ò ò ó + 3 ô ( ô – 1)ï ñ„ó ò + [ õ ïƒ÷ + ô ( ô – 1)(ô – 2)]ñ = 0. The substitution ¨ =   leads to an equation of the form 3.1.2.7: ¨u


 + ' 

ï 3 ñ óò ò ó ò ó ï 3 ñ óò ò ó ò ó

+ 3(1 – ô )ï

2

ñóò ò ó

+ ï [4 õ

= L 1   ø Á 2 ( ' v ) + L Solution: are the Bessel functions. 179.

2−

ö ï 2}

2 2

2

 

ø

2 2

+ ( ô ï 2 + õ )ñó ò ò ó + 2(2 ô – 9) ï ñ„ó ò + 2( ô – 6)ñ = 0. Integrating the equation twice, we arrive at a first-order linear equation:  L 1 + L 2 . + ï 2 ( ô ï + õ )ñó ò ò ó + ö ï Particular solutions: 1 =   1 , B 2 − B + ( = 0.

+ ô ïƒ÷ ñó ò ò ó + õ ï ñó ò + õ ( ô Particular solutions: 1 =  + 1 , equation 2 − +  = 0.

+ 2 ôƒõ ï + ö )ñ ó ò + ( ô 2 õ ï 2 + õ^ö – 2 ö )ñ = 0. −    2 , where B 1 and B 2 are roots of the quadratic 2=

2

ïƒ÷

ï 3 ñ óò ò ó ò ó

2

– 2)ñ = 0. =  + 2 , where

–2

¡

+ ï 2 ( ô ïƒ÷

ï 3 ñ óò ò ó ò ó

+

+ õ )ñó ò ò ó + ï ( ô ïƒ÷ + õ – 1)ñó ò + ( ô Particular solutions: 1 = cos(ln  ), 2 = sin(ln  ).

ï 2 (ô ïƒ÷


 +[(  −6) 2 +  ] =

+ ö ( ô ï + õ – 2)ñ = 0.  2 , where B 1 and B 2 are roots of the quadratic equation 2=

+ ï 2 (2 ô ï + õ )ñ ó ò ò ó + ï ( ô 2 ï  Particular solutions: 1 =  −    1 , equation B 2 − B + ( = 0.

¡

3

ñóò

ï 3 ñ óò ò ó ò ó

ï 3 ñ óò ò ó ò ó

¨ = 0.

ö + 3ô (ô – 1)]ñ„óò + [4 õ 2 ö 2 ( ö – ô ) ï 2 } + ô (4 b 2 ö 2 – ô 2 )]ñ = 0. Á ( ' v ) n Á ( ' v ) + L 3   n Á 2 ( ' v ), where ø Á (‡ ) and n Á (‡ ) + 1 – 4b

ï 3 ñ óò ò ó ò ó ï 3 ñ óò ò ó ò ó

−3

õ + ö + 1)ñóò ò ó ï + [  ï 2÷ + ( ô„ö +  ) ï ÷ +

+ +

1

¡

ïƒ÷

õ^ö ]ñ óò

and +

õ

¡

2

are roots of the quadratic

– 3)ñ = 0.

+ ( ö – 1)(

ï 2÷

+

ï ÷

+ )ñ = 0.

The substitution ¨ =  
+ ( ( − 1) leads to a second-order linear equation of the form ¨ 
+ ( ƒ… 2  + ý±  + = )¨ = 0. 2.1.2.146:  2 ¨2

 +  (   +  )2

185. ( ô ï + õ ) ï 3 ñ ó ò ò ó ò ó + ( ö Particular solutions: B 2 − B + ° = 0.

ï

+ ý )ï 2 ñ  1, 1=

+ s( ô ï + õ ) ï  2 , where B 2=

óò ò ó

+ s[( ö – 2 ô ) ï + ý – 2 õ ]ñ = 0. 1 and B 2 are roots of the quadratic equation

ñ óò

© 2003 by Chapman & Hall/CRC

Page 511

512 186. 187.

THIRD-ORDER DIFFERENTIAL EQUATIONS + 6 ï 5 ñó ò ò ó – ôñ + 2 õ ï = 0. The substitution  = a −1 leads to an equation of the form 3.1.2.62: a 2 
g
g
g −6
g + a

ï 6 ñ óò ò ó ò ó

ï 2 (ƒï ÷ + ô )ñ óò ò ó ò ó + ï ( õ ï +1 +2ù ïƒ÷ + ö ï )ñ óò ò ó

2

−2 'a = 0.

+[2 õ ü ï +1 + ù (ù –1) ïƒ÷ ]ñó ò + õ ü ( ü –1) ï ñ = 0. Integrating the equation twice, we arrive at a first-order linear equation: (  +  )
 +( '  + ( ) = L 1 + L 2 .

3.1.3. Equations Containing Exponential Functions 3.1.3-1. Equations with exponential functions. 1.

ñóò ò ó ò ó

ó (ô 2  2 ó

– ô

ó

+ 3 ô

+



 

2



)ñ = 0.



 



X H ( )  leads to P  M  
Œ

2 2 a second-order linear equation of the form 2.1.3.27: H  +3    H  +(3    +3 Π ) H = 0. Particular solution:

2.

ñóò ò ó ò ó

+

ô ó ñóò

0

= exp O

óñ

+ ô

=

Œ

P

õ  ó

= exp O

. The substitution

.





= ' “ −1   + L Integrating yields a second-order linear equation:  +    for the solution of the corresponding homogeneous equation with  = L = 0).





(see 2.1.3.1

3.

ñóò ò ó ò ó

+ ô ñó ò + õ ( ô + õ 2 )ñ = 0. The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.3.10: ¨2

 −  ¨2
+ (    +  2 )¨ = 0.

ó

ó

4.

ñóò ò ó ò ó

ó

ó

+ ô ñó ò + [ ô ( – õ )  – õ 3 ]ñ = 0.   Integrating yields a second-order linear equation: "

 +  
 + (   +  2 ) = L   2.1.3.10 for the solution of the corresponding homogeneous equation with L = 0).

(see

5.

ñóò ò ó ò ó

+ ( ô – õ 2 )ñó ò + ôƒõ ñ = 0. The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.3.10: ¨2

 −  ¨2
+    ¨ = 0.

ó

6.

ñóò ò ó ò ó

+ ( ô – õ 2 )ñó ò + ô ( – õ )  ñ = 0.   = L   (see 2.1.3.10 Integrating yields a second-order linear equation: "

 +  
 +   for the solution of the corresponding homogeneous equation with L = 0).

ó

7.

ñóò ò ó ò ó

+ ( ô + õ )ñó ò The substitution ¨ = ¨2

 − ( ¨2
+ (    + 

ó

8.

ñóò ò ó ò ó

+ ( ô 2  + õ )ñó ò – ö ( ô 2  + õ + ö 2 )ñ = 0. The substitution ¨ = 
− ( leads to a second-order linear equation of the form 2.1.3.27: ¨2

 + ( ¨2
+ (   2   +    + ( 2 )¨ = 0.

9.

ñóò ò ó ò ó

– 3 ô

ó

ó

ó

ó

ó (ô  ó

+ )ñó ò +

Particular solutions: 10.

ñ óò ò ó ò ó

– (3 ô

ó

+ ö ( ô + õ + ö 2 )ñ = 0. 
+ ( leads to a second-order linear equation of the form 2.1.3.10: + ( 2 )¨ = 0.

 

2 2

ó

ó

1

= exp O

+ 3 ô



ó

ó

ô ó (2ô 2  2 ó Œ



 

+ õ )ñ

óò



P

,

+ ô



2

2

–

=  exp O

)ñ = 0.

ó (2ô 2  2 ó 

 



P

.

– 2õ –

2

)ñ = 0.

   −  £  , 2 = exp O   Œ Œ P    £ 2 b . Particular solutions with  < 0: 1 = exp O   cos   −   , 2 = exp O  Œ P Œ 1 b . Particular solutions with  > 0:

1

= exp O



Œ



 

+ £ 

 P

P

.

sin   £ −   .

© 2003 by Chapman & Hall/CRC

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513

3.1. LINEAR EQUATIONS

11.

12.

13.

ñ óò ò ó ò ó

+

ôñ óò ò ó

+

õ  ó ñ óò

ó

+ ôƒõ

 ñ

The substitution ¨ = 
+  ¨2

 +    ¨ = 0.

ñóò ò ó ò ó

+

ôñóò ò ó

+ ( õ

ñóò ò ó ò ó

+

ôñóò ò ó

+ ( õ

ó

= 0.

leads to a second-order linear equation of the form 2.1.3.1:

ó

+ ö )ñó ò + [ õ ( ô + ) 

+ ô„ö ]ñ = 0.

  Integrating yields a second-order linear equation: "

 + (   + ( ) = L  −  (see 2.1.3.2 for the solution of the corresponding homogeneous equation with L = 0). 2



ó

ö ó )ñ óò

+

The substitution ¨ = 
+  ¨2

 + (   2   + (   )¨ = 0.

+ ô ( õ

2

ó



+

ö ó )ñ

= 0.

leads to a second-order linear equation of the form 2.1.3.27:

14.

ñóò ò ó ò ó

+

ô ó ñóò ò ó

– õ 2 ( ô

+ õ )ñ = 0. The substitution ¨ = 
−  leads to a second-order linear equation of the form 2.1.3.27: ¨2

 + (    +  )¨2
+  (    +  )¨ = 0.

15.

ñ óò ò ó ò ó

+

ô  ó ñ óò ò ó

– õÿñ

16.

17.

óò

2

= exp   £   .

2 b . Particular solutions with  < 0:

2

= sin  £ −   .

+

+ ôƒõ

ó ñóò

= cos  £ −   ,

ñóò ò ó ò ó

ô ó ñóò ò ó

1

+

õ 3ñ

= 0.

ñóò ò ó ò ó

+

ô ó ñóò ò ó

+ ôƒõ

ó ñóò

+

õ 2 (ô  ó

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.3.27: ¨2

 + (    −  )¨2
+  2 ¨ = 0.

ñóò ò ó ò ó

+

ô ó ñóò ò ó

1

ñ óò ò ó ò ó

+

ô  ó ñ óò ò ó

ñóò ò ó ò ó + ô ó ñ óò ò ó ñ óò ò ó ò ó

+

ô  ó ñ óò ò ó

0

23.

ñóò ò ó ò ó

+ ( ô

ñóò ò ó ò ó

+ ( ô

ó



=

ó

+ ( õ

=

+ ( ôƒõ

+ 3 õ )ñó ò +



,

2

– ö 2 )ñ

óò

v

ó

–

õ

1

=

+ õ )ñó ò ò ó – ôƒõ



3 2

'  ,

õ 2 (ô  ó .

– ö ( ô„ö



ó

+

+ ö )ñó ò + ö ( ô

2

 =  : 1 ,

−

= 



2

.

+ [ô ( õ + )

Particular solutions: 22.

ó

– õ (2 ô 1

– õ )ñ = 0.

= exp  − 12 '  cos 

Particular solutions: 1 equation ý 2 + 'ý + ( = 0. 21.

= 0.

= exp  − £   ,

Particular solution: 20.

 ñ

1

Particular solutions: 19.

ó

– ôƒõ

1 b . Particular solutions with  > 0:

Particular solutions: 18.

ó

 2



õ 2 ]ñ ó ò

–

,

2

ó

 ñ

=

−



 X

= exp  − 21 '  sin 

2

3 2

'  .

+ 2 õ )ñ = 0.

õ ó )ñ ó

+ ôƒõ

= 0.

– õ )ñ = 0.

 =  : 2 , where ý

2

ó 

2

and ý

1

ó

 ñ

exp O 2 ' −

= 0.

Œ



 



2

P3M

are roots of the quadratic

 .

= 0.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.3.27: ¨2

 +    ¨2
−    ¨ = 0.

ó

+ õ )ñó ò ò ó – ö 2 ( ô

ó

+

õ

+ ö )ñ = 0.

The substitution ¨ = 
− ( leads to a second-order linear equation of the form 2.1.3.27: ¨2

 + (    +  + ( )¨2
+ ( (    +  + ( )¨ = 0.

© 2003 by Chapman & Hall/CRC

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514

THIRD-ORDER DIFFERENTIAL EQUATIONS

ó

24.

+ ( ô  + õ )ñ ó ò ò ó The substitution ¨ = ¨2

 + (    +  − ( )¨2


25.

ñóò ò ó ò ó

26.

ñ óò ò ó ò ó

ó

ó

+ ö ( ô  + õ )ñ ó ò + ö 3 ñ = 0. 
+ ( leads to a second-order linear equation of the form 2.1.3.27: + ( 2 ¨ = 0.

ó

+ ( õD + 2 ô )ñó ò ò ó – ô ( õD  Particular solutions: 1 =   ,

ñ óò ò ó ò ó = (  ó

2

ó – ô )ñ ó ò ò ó + ( ô 

Particular solutions: 1 equation ý 2 + ý +  = 0.

+ ô )ñó ò – 2 ô 3 ñ = 0.  =  − +   .

ó

– õ )ñ ó ò + õ  ñ .   : 1 =  , 2 =  : 2 , where ý

and ý

1

are roots of the quadratic

27.

ñ óò ò ó ò ó

+ ( ô  + õ )ñ ó ò ò ó The substitution ¨ = ¨2

 + (    +  + ° )¨2


ó

28.

ñóò ò ó ò ó

ó

+ ( ö  + ý )ñ ó ò – s[( ô s + ö )   + õ s + ý + s2 ]ñ = 0. 
− ° leads to a second-order linear equation of the form 2.1.3.27:  + [( !° + ( )  + ˜° + + ° 2 ]¨ = 0.

+ ( ô + õ )ñó ò ò ó The substitution ¨ = ¨2

 + (    +  + ° )¨2


+ ( ö 2  + ý )ñó ò – s( ö 2  + ô s  + õ s + ý + s2 )ñ = 0. 
− ° leads to a second-order linear equation of the form 2.1.3.27:   + ( (  2  + !°  + ˜° + + ° 2 )¨ = 0.

29.

ñóò ò ó ò ó

ó

+ ( ô

ó

2

ó

M

ó

ó

+ õ )ñó ò ò ó + ( ö

ó

ó

,

ó

M

ó

+ ý )ñó ò – ü [ ü ( ô + ü )  2  + ( ô + 3 ü/ + õ ü + ö ) -



. The substitution Particular solution: 0 = exp O   ΠP to a second-order linear equation of the form 2.1.3.27. 30.

ñ óò ò ó ò ó

+ (2 ô

ó + õ )ñ ó ò ò ó



Particular solutions: 31.

ñóò ò ó ò ó

ó

+ ( ô

1

0

ó

ó (ô   ó





= exp O −  

– õ )ñó ò ò ó +

Particular solution:

ñóò ò ó ò ó

+ ô

Œ

ö  ó ñ óò

=





.



,

P

ó

õ  ó )ñ óò

+ ô





ó [ô ( õ

=  exp O −  

2

– õ ( ôƒõ

ó

óò

+ 2 õ + 3 )ñ

= exp O

+ ö

Œ

ó )ñ 

ó



P

Œ

ó ,

2

+

 



+ 2 )



+

õ

+ ý ]ñ = 0.

XZH ( )  leads M P

ó + õ

+

2

]ñ = 0.

.

= 0.

32.

+ (  + ô )ñó ò ò ó + ( ô +  Particular solution: 0 =  −  .

33.

ñóò ò ó ò ó

+ ( ô + õ  )ñó ò ò ó + ö^ñó ò + ö ( ô + õ  )ñ = 0. ¨ 
+ (    +     )¨ = 0. The substitution ¨ = 

 + ( leads to a first-order linear equation: u

34.

ñóò ò ó ò ó

ó

+ ( ô

ó

ó

+

õ ó )ñ óò ò ó

ó

ó

 

( + )

ó

 ñ

= 0.

ó

ó – ö 2 ]ñóò ó + ö [ ôƒõ (  + )

+ õ ( ö + b ) 

ó

ó

35.

+ ô ( õ  + 2 )ñó ò ò ó – [ ôƒõ (  + ) + ]ñó ò – 2 ô

  Particular solutions: 1 =   , 2 =  −  +  “ .

36.

( ô + õ )ñ ó ò ò ó ò ó – ô Particular solution:

37.

ó

ó ( õ^öD

óñ

0

– ô„ö

 Á −v   . X exp O 2 (˜ −  2 =  @ P M

−v  , 1 = 

Particular solutions:

ñóò ò ó ò ó

+ [ ôƒõ

+ ôƒõ

3

ó

 ñ

ó

+

õb ó ]ñ

= 0.

= 0.

= 0.  =  + .

+ ô + ö )ñó ò ò ó ò ó – ( õ^ö 3 D  Particular solutions: 1 =  v ,

ó

+ ô 3 + ö 3 )ñó ò + −  + . 2 = 

ô„ö (ô

2

– ö 2 )ñ = 0.

© 2003 by Chapman & Hall/CRC

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515

3.1. LINEAR EQUATIONS

38.

( ô



ó

óò ò ó ò ó

+ õ )ñ

+ ( ö

ó 

óò ò ó

+ ý )ñ

+ ü ( ô

ó 

óò

+ õ )ñ

+ ü ( ö

ó 

+ ý )ñ = 0.

£ ,  , 2 = sin   £ ,  . 1 = cos   £ − ,  , 2 = exp   £ − ,  . 1 = exp  − 

1 b . Particular solutions with , > 0: 2 b . Particular solutions with , < 0:

3.1.3-2. Equations with power and exponential functions. 39.

ñóò ò ó ò ó

+

ô ó ñóò

+

õ ï (ô  ó

Particular solution: 40.

ñóò ò ó ò ó

ï

+ (ô

+ õ ) 

Particular solution: 41.

ñóò ò ó ò ó

ï

+ (ô

+ õ ) 

42.

43.

ñóò ò ó ò ó + ô ó ñ óò ò ó

= exp 

0

ó ñóò

+

= 0.

óñ

– 2 ô

= 0. 2

= (  +  ) .

0

ô ï  ó ñóò ò ó

'  . 2

óñ

õ ƒï ÷ ñ óò

õ ïƒ÷

+

+ (õ

ï

2

– ô

ó )ñ óò

+

ô ï 2   ó ñóò ò ó

Particular solutions: 45.

ñóò ò ó ò ó

+ (ô

ï

+

ñóò ò ó ò ó

+ ( ôƒõ

õ ó )ñ óò ò ó

ï  ó

ñóò ò ó ò ó

+

ô ïƒ÷ ( õ  ó

48.

ñ óò ò ó ò ó

+ (ô

ï ÷

– 2 õ

50.

ï ñóò ò ó ò ó

+

ôñóò ò ó

ï ñóò ò ó ò ó

+

ô ï  ó ñóò

  , 2





=  ,

ó )ñ ó ò ò ó

+ ï ( õ

–

ó

= exp 

ô 2 õ ï  ó ñóò ïƒ÷  ó

2

=  −



õ  ó (2ô ï ÷

= exp O

1

2

Œ



 

óñ

 P

,

+ ö )ñó ò + [ õ (

2

õ ó ñ

+ )ñó ò – 2 ô

+ Π.

= 0.

3

ïƒ÷ ñ

ó

– õ  + 3 )ñ ó ò ó + õ [ ô ïƒ÷ ( õ 2

ï

= 0.

 2  X exp  12   2   . M

− 21 –ô

 . 2 =  −

= ,

1

Particular solutions: 49.

− 12

+ 2 )ñó ò ò ó – ( ôƒõ

Particular solutions:

= 0.

+ 2)ñó ò + ôƒõ

+ ô )ñó ò ò ó +

1

óñ

= .

ï  ó

= exp 

õ ó

+

+ 2 ô

2

2

2

+ ô (õ

1

Particular solutions: 47.

= ,

1

Particular solutions: 46.

ï  ó ñóò

– 2ô

+ 3)ñ = 0.

£   , 2 = sin  1  2 £   . 1 = cos   2 1 2£ −   , 2 = exp  − 12  2 £ −   . 1 = exp  − 2  1 2

2 b . Particular solutions with  < 0:

ñóò ò ó ò ó

+ ù )ñ = 0.

õ ï (ô ï 2   ó

+

1 b . Particular solutions with  > 0:

44.

ï  ó

(ô

–1

¨ 
+    ¨ = 0. leads to a first-order linear equation: u

The substitution ¨ = 

 + ' 

ñóò ò ó ò ó

– 3 õ )ñ = 0.

2

=  +  .

ó ñóò

+

õ 2ï

− 12

– ô

0

Particular solution:

+

=  exp O

+ ô ) 

ó

Œ



 



P

ó

= 0.

– ) + 2 õ

ó

–

2

]ñ = 0.

.

+ ô„ö ]ñ = 0.

 ¨ 
+  ¨ = 0. The substitution ¨ = 

 + (   + ( ) leads to a first-order linear equation:  u – 2 ô

The substitution ¨ =  ¨2

 +    ¨ = 0.

óñ

= 0.


− 2 leads to a second-order linear equation of the form 2.1.3.1:

© 2003 by Chapman & Hall/CRC

Page 515

516 51.

52.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ó

ó

ï ñ óò ò ó ò ó

+ (ô

ï ó

54.

55.

56. 57.

óò ò ó

+ 3)ñ

Particular solutions: 53.

ó

= (   – ô ï )ñ ó ò ò ó + ( ô  – õ ï )ñ ó ò + õ  ñ .   Particular solutions: 1 =  : 1 , 2 =  : 2 , where ý equation ý 2 + ý +  = 0.

ï ñ óò ò ó ò ó

1

ó

ï

+ ô (õ −1

=

ó

 ñ óò

+ 2) 

+ õ ( ôƒõ

2

exp  − 21 '  cos 

ó

3 2

and ý

1

ï ó

'  ,

2

+

ô  ó −1

=

are roots of the quadratic

2

–õ

2

ï )ñ

= 0.

exp  − 21 '  sin 

ó

2

3 2

'  .

+ ï ( ô ï  + õ )ñó ò ò ó + [ ô ( õ – 2)ï - + ö ]ñó ò + ô ( ö – õ + 2)- ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 3)B + ( −  + 2 = 0.

ï 2 ñ óò ò ó ò ó

ó

ó

+ õ ï 2  ñó ò ò ó + ô ï ñó ò + ô ( õ – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B equation B 2 − B +  = 0.

ï 3 ñ óò ò ó ò ó

ó

ï 3 ñ óò ò ó ò ó

and B

1

ó

2

are roots of the quadratic

ó

+ ï 2 ( ô + õ )ñó ò ò ó + ï ( ôƒõ + ö – õ )ñó ò + ö ( ô – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 1)B + ( = 0.

ó

( ô + õ ï )ñó ò ò ó ò ó – ô Particular solution:

ó

( ô

+

õ ï 2 )ñ óò ò ó ò ó

ó

58.

( ô ï  + õ )ñó ò ò ó ò ó + Particular solution:

59.

(ô

+ õ )ñó ò ò ó ò

0

– ô

Particular solution:

ï 2 ó

óñ

ó

0

õÿñ

+

Particular solution:

= 0.  =   + ' .

óñ

= 0.

= 



+ ' 2 .

= 0. − . 0 =  +  

õÿñ 0

= 0. = 

2

+ 

−



.

3.1.4. Equations Containing Hyperbolic Functions 3.1.4-1. Equations with hyperbolic sine. 1.

2.

+ ôñó ò ò ó + õ sinh2 ï ñó ò + ôƒõ sinh2 ï ñ = 0. leads to a second-order linear equation of the form 2.1.4.1: The substitution ¨ = 
+  ¨2

 +  sinh2  ¨ = 0.

ñóò ò ó ò ó ñóò ò ó ò ó

sinh ÷ (

ô

+

ï )ñ óò ò ó

+

õÿñóò

+ ôƒõ sinh ÷ (

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

ï )ñ

= 0.

= cos   £   ,

2

= exp  − £ −   ,

= sin   £   . 2

= exp   £ −   .

3.

+ ô sinh ÷ ( ï )ñó ò ò ó + õ ïƒú ñó ò + õ ïƒú –1 [ ô ï sinh ÷ ( ï ) + þ ]ñ = 0. + leads to a first-order linear equation: ¨u
+  sinh ( Œ  )¨ = 0. The substitution ¨ = 

 + ' *

4.

ñ óò ò ó ò ó

ñóò ò ó ò ó

+

ô

sinh ÷ (

ï )ñ ó ò ò ó

Particular solutions:

1

+ ôƒõ sinh ÷ (

= exp 

− 12

ï )ñ ó ò 2

'  cos 

+ 3 2

õ 2 [ô

'  ,

sinh÷ ( 2

ï

) – õ ]ñ = 0.

= exp  − 21 '  sin 

2

3 2

'  .

© 2003 by Chapman & Hall/CRC

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517

3.1. LINEAR EQUATIONS

5.

ñóò ò ó ò ó

+

sinh ÷ (

ô

ï )ñ óò ò ó

Particular solutions: 6.

7.

1

10.

ñóò ò ó ò ó

+

ô ï

sinh ÷

ï ñóò ò ó

ñ óò ò ó ò ó

+

ô ï

2

sinh ÷ (

12.

13.

14.

16.

1

18.

– 2ô

ï )ñ óò

= ,

2

õ ï (ô ï

+

2

sinh ÷

ï

ñ óò ò ó ò ó

+ ( ô sinh ÷

ï

+

õ ï )ñ ó ò ò ó

sinh ÷ (

1 2

= .

2

+ 2 ô sinh ÷ (

ï )ñ ó ò

2

1

ï

+ õ (ô

) + 2 õ ]ñ = 0.

+ 3)ñ = 0.

£  . 2 = sin  

2

ï

ï

– õ 2 )ñ ó ò + ö ( ô sinh ÷ ï – õ )ñ = 0. 2 = exp( Œ 2  ), where Œ 1 and Œ 2 are roots of the

ö

£  , 1 = cos   1 2

sinh ÷ (

õ 2 [ô

.

– ô sinh ÷

2

) + 3 õ ]ñó ò +

ï )ñ

= (sinh ÷ ï – ô )ñó ò ò ó + ( ô sinh ÷ ï – õ )ñó ò + õ sinh ÷ ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ quadratic equation Œ 2 + Œ +  = 0.

= 0.

sinh ÷

= exp  − 12 ' 2  ,

ï

+ 2)ñ

+ ôƒõ sinh ÷

óò

ï ñ

= exp  − 12 ' 2  X exp 

2

and Œ

1

1 2

ï ñ óò ò ó ò ó

are roots of the

2

= 0.

' 2   . M

+ ï ( ô sinh2 ï + õ )ñ ó ò – 2( ô sinh2 ï + õ )ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.4.1: ¨2

 + (  sinh2  +  )¨ = 0.

+ ( ô ï 2 sinh ÷ ï + õ ï )ñó ò ò ó + [ ô ( õ – 2)ï sinh ÷ Particular solutions: 1 =  + 1 , 2 =  + 2 , where ¡ equation 2 + (  − 3) + ( −  + 2 = 0.

ï 2 ñ óò ò ó ò ó

¡

¡

¡

¡

ï

+ ö ]ñó ò + ô ( ö – õ + 2) sinh ÷ ï ñ = 0. 1 and 2 are roots of the quadratic

¡

+ ï 2 ( ô sinh÷ ï + õ )ñó ò ò ó + ï ( ôƒõ sinh ÷ ï + ö – õ )ñó ò + ö ( ô sinh÷ ï – 2)ñ = 0. Particular solutions: 1 =  + 1 , 2 =  + 2 , where 1 and 2 are roots of the quadratic ¡ ¡ equation 2 + (  − 1) + ( = 0.

ï 3 ñ óò ò ó ò ó

sinh ÷ ï ñó ò ò ó ò ó

+ ôñó ò ò ó + ôƒõÿñó ò +

sinh ÷

ï ñóò ò ó ò ó

1

= exp 

+ ôñó ò ò ó + õ sinh ÷

– õ sinh÷

õ 2(ô

− 12

'  cos 

ï ñóò

+

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

sinh ÷

ï ñóò ò ó ò ó

+ ôñó ò ò ó – õ (2 ô + 3 õ sinh ÷

Particular solutions:

17.

ï



= 

2

ï 

ñóò ò ó ò ó

Particular solutions: 15.

,

+ (õ

ï )ñ ó ò ò ó

Particular solutions: 11.



=

+ ô sinh ÷ ï ñ ó ò ò ó + ( ôƒõ sinh÷ ï + Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0.

Particular solutions:

9.



ñ óò ò ó ò ó

Particular solutions: 8.

– õ [2 ô sinh ÷ (

1

=





,

2

= 

2

3 2

ôƒõÿñ

ï )ñ

= 0.

'  ,

2

= exp  − 21 '  sin 

= 0.

= cos   £   ,

2

= exp  − £ −   ,





ï )ñ óò

+

õ 2 (ô

2

3 2

'  .

= sin   £   . 2

= exp   £ −   .

+ 2 õ sinh ÷

ï )ñ

= 0.

.

sinh ÷ ï ñó ò ò ó ò ó + ñó ò ò ó + [( õ – ô 2 ) sinh÷ ï + ô ]ñó ò + õ (1 – ô sinh ÷ ï )ñ = 0.   Particular solutions: 1 =  1 , 2 =  2 , where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + Œ +  = 0. sinh ÷ (

ï )ñ óò ò ó ò ó

+ô

Particular solutions:

ï 2 ñ óò ò ó 1

ï ñóò

– 2ô

= ,

2

2

+ 2 ôñ = 0.

= .

© 2003 by Chapman & Hall/CRC

Page 517

518 19. 20.

THIRD-ORDER DIFFERENTIAL EQUATIONS sinh ÷ ï ñó ò ò ó ò ó + ( ô sinh ÷ ï + ô Particular solutions: 1 =  ,

sinh ÷

ï ñóò ò ó ò ó

sinh ÷

ï

+ (ô

Particular solutions: sinh ÷

ï

21.

ï ñ óò ò ó ò ó

1

ï

3

22.

sinh ÷

ï ñóò ò ó ò ó

1

Particular solutions:

ï

3

23.

sinh ÷

ï ñóò ò ó ò ó

1

25.

sinh ÷ ï ñó ò ò ó ò ó + ô Particular solutions:

ï

3

ï

3

sinh ÷

ï ñóò ò ó ò ó

ï

+ ï )ñ

−1

=

+ï

Particular solutions:

1

2

=

−1

+ (ô

ï ñóò

–ô

=

−2

,

= 0.

ï )ñ óò

+

ôñ

sinh÷

ï

=

−1

2

= .

+ 2)ñ

ï ñóò

+ ô (sinh÷

óò

+ 2(2 sinh ÷

ï

– ô )ñ = 0.

+ 6(2 sinh ÷

ï

– ô )ñ = 0.

= .

+ ï ( ô – sinh ÷ ï )ñó ò + ( ô – 3 sinh ÷ = cos(ln  ), 2 = sin(ln  ). + ô )ñó ò ò ó + ï [ ô – ( õ + 1) sinh ÷

ï

=

2 −

,

2

2

=

 2   . M

1 2

ï

+ ï )ñ = 0.

sin  £   .

3

2

= 0.

ï

2

2

,

ñ

2

= exp  − 21  2  X exp 

ï ñóò

– 6 ï sinh ÷

(sinh÷ 1

2

2

cos  £   ,

ï 2 ñ óò ò ó 1

óò ò ó

– 2 ï sinh ÷

ô ï 2 ñ óò ò ó

+

Particular solutions: 24.

= exp  − 12  2  ,

ô ï 2 ñ óò ò ó

+

+ 1)ñó ò ò ó + ô −  . 2 = 

+ 1)ñó ò ò ó + ô ( ï + 2 sinh ÷

ï

+ (3 sinh ÷

Particular solutions:

ï

ï )ñ

ï ]ñ óò

.

= 0. + õ (2 sinh ÷

ï

– ô )ñ = 0.

3.1.4-2. Equations with hyperbolic cosine. 26.

27.

28.

+ ôñó ò ò ó + õ cosh(2ï )ñ„ó ò + ôƒõ cosh(2ï )ñ = 0. leads to a second-order linear equation of the form 2.1.4.9: The substitution ¨ = 
+  ¨2

 +  cosh(2 )¨ = 0.

ñóò ò ó ò ó

+ ôñó ò ò ó + õ cosh2 ï ñó ò + ôƒõ cosh2 ï ñ = 0. The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.4.10: ¨2

 +  cosh2  ¨ = 0.

ñóò ò ó ò ó ñóò ò ó ò ó

cosh÷ (

ô

+

ï )ñ óò ò ó

õÿñóò

+

+ ôƒõ cosh÷ (

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

ï )ñ

= 0.

= cos   £   ,

2

= exp  − £ −   ,

= sin   £   . 2

= exp   £ −   .

29.

+ ô cosh÷ ( ï )ñó ò ò ó + õ ïƒú ñó ò + õ ïƒú –1 [ ô ï cosh÷ ( ï ) + þ ]ñ = 0. + leads to a first-order linear equation: ¨u
+  cosh ( Œ  )¨ = 0. The substitution ¨ = 

 + ' *

30.

ñóò ò ó ò ó

ñóò ò ó ò ó

cosh÷ (

ô

+

ï )ñ óò ò ó

Particular solutions: 31.

ñ óò ò ó ò ó

+

ô

cosh÷ (

ï )ñ ó ò ò ó

Particular solutions: 32.

1

1

+ ôƒõ cosh÷ (

= exp 

− 12

'  cos 

– õ [2 ô cosh÷ ( =





,

2

= 

+ ô cosh÷ ï ñó ò ò ó + ( ôƒõ cosh÷ ï + Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0.

ñóò ò ó ò ó

cosh÷ (

ï )ñ óò + õ 2 [ô 2 3



ö



ï

2

'  ,

) + 3 õ ]ñ

2

óò

= exp 

+

õ 2 [ô

ï

) – õ ]ñ = 0.

− 21

'  sin 

cosh÷ (

ï

2

3 2

'  .

) + 2 õ ]ñ = 0.

. – õ 2 )ñó ò + ö ( ô cosh÷ ï – õ )ñ = 0. 2 = exp( Œ 2  ), where Œ 1 and Œ 2 are roots of the

© 2003 by Chapman & Hall/CRC

Page 518

519

3.1. LINEAR EQUATIONS

33.

ñóò ò ó ò ó

+

ô ï

cosh÷

ï ñóò ò ó

Particular solutions: 34.

ñóò ò ó ò ó

+

ô ï

2

cosh÷ (

36.

38.

39.

40.

41.

42.

44.

46.

= ,

2

cosh÷

£  . 2 = sin  

2

cosh÷ (

ï

2

1 2

2

= .

+ 3)ñ = 0.

2

+ 2 ô cosh÷ (

ï )ñ óò

ï ï )ñ

= (cosh÷ ï – ô )ñ ó ò ò ó + ( ô cosh÷ ï – õ )ñ ó ò + õ cosh÷ ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ quadratic equation Œ 2 + Œ +  = 0.

ñóò ò ó ò ó

+ ( ô cosh÷

ï

õ ï )ñ óò ò ó

+

1

cosh÷

ï

+ õ (ô

= exp  − 12 ' 2  ,

ï

+ 2)ñó ò +

ôƒõ

cosh÷

ï ñ

= exp  − 12 ' 2  X exp 

2

= 0.

and Œ

1

1 2

ï ñóò ò ó ò ó

are roots of the

2

= 0.

' 2   . M

+ ï [ ô cosh(2 ï ) + õ ]ñ„ó ò – 2[ ô cosh(2 ï ) + õ ]ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.4.9: ¨2

 + [  cosh(2 ) +  ]¨ = 0.

ï ñóò ò ó ò ó

+ ï ( ô cosh2 ï + õ )ñó ò – 2( ô cosh2 ï + õ )ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.4.10: ¨2

 + (  cosh2  +  )¨ = 0.

= (cosh÷ ï – ô ï )ñó ò ò ó + ( ô cosh÷ Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ +  = 0.

ï ñóò ò ó ò ó

– õ ï )ñó ò + õ cosh÷ ï = exp( Œ 2  ), where Œ

ï

2

+ ( ô ï 2 cosh÷ ï + õ ï )ñó ò ò ó + [ ô ( õ – 2)ï cosh ÷ Particular solutions: 1 =  + 1 , 2 =  + 2 , where ¡ equation 2 + (  − 3) + ( −  + 2 = 0.

ï 2 ñ óò ò ó ò ó

¡

¡

¡

¡

+ ï 2 ( ô cosh÷ Particular solutions: equation 2 + (  − 1)

ï 3 ñ óò ò ó ò ó

cosh ÷ ï ñó ò ò ó ò ó cosh ÷

ï ñóò ò ó ò ó

ï

+õ 1 = +(

1

= exp 

+ ôñó ò ò ó + õ cosh÷

– õ cosh÷

õ 2 (ô

− 12

'  cos 

ï ñóò

2 b . Particular solutions with  < 0:

1

1

=



,

2

= 

3 2

ï )ñ

'  ,

1

and Œ

are roots of the

2

+ ö ]ñó ò + ô ( ö – õ + 2) cosh÷ ï ñ = 0. 1 and 2 are roots of the quadratic

¡

= 0. 2

= exp  − 21 '  sin 

= 0.

= cos   £   ,

2

= exp  − £ −   ,

+ ôñó ò ò ó – õ (2 ô + 3 õ cosh ÷



2

ôƒõÿñ

+ 1

ï ñóò ò ó ò ó

.

óò ò ó

1 b . Particular solutions with  > 0: cosh ÷

ï

ñ

+ ï ( ôƒõ cosh÷ ï + ö – õ )ñ ó ò + ö ( ô cosh÷ ï – 2)ñ = 0.  + 1 , 2 =  + 2 , where 1 and 2 are roots of the quadratic ¡ ¡ = 0. )ñ

+ ôñó ò ò ó + ôƒõÿñó ò +

Particular solutions:

45.

– 2ô

õ ï (ô ï

+

ñ óò ò ó ò ó

Particular solutions:

43.

ï )ñ óò

£  , 1 = cos  

1

Particular solutions: 37.

– ô cosh÷

2

1 2

ï )ñ óò ò ó

Particular solutions:

35.

ï

+ (õ





ï )ñ óò

+

õ 2 (ô

2

3 2

'  .

= sin   £   . 2

= exp   £ −   .

+ 2 õ cosh÷

ï )ñ

= 0.

.

cosh ÷ ï ñó ò ò ó ò ó + ñó ò ò ó + [( õ – ô 2 ) cosh÷ ï + ô ]ñó ò + õ (1 – ô cosh ÷ ï )ñ = 0.   Particular solutions: 1 =  1 , 2 =  2 , where Œ 1 and Œ 2 are roots of the quadratic 2 equation Œ + Œ +  = 0. cosh ÷ (

ï )ñ óò ò ó ò ó

+ô

Particular solutions:

ï 2 ñ óò ò ó 1

ï ñóò

– 2ô

= ,

2

2

+ 2 ôñ = 0.

= .

© 2003 by Chapman & Hall/CRC

Page 519

520 47.

THIRD-ORDER DIFFERENTIAL EQUATIONS cosh÷

ï ñóò ò ó ò ó

+ ( ô cosh÷

Particular solutions: 48.

cosh ÷

ï ñóò ò ó ò ó

ï

+ (ô

1

cosh÷

ï

ï ñóò ò ó ò ó

1

ï

3

50.

cosh÷

ï ñ óò ò ó ò ó

+ô

1

ï

3

cosh÷

ï ñ óò ò ó ò ó

+ô

1

ï

3

52.

cosh÷

ï ñ óò ò ó ò ó

+ô

1

ï

3

cosh÷

ï ñ óò ò ó ò ó

+ï

ï

= =

2

+ ï )ñó ò ò ó + ( ô – 2 ï cosh÷ −1

,

2

= .

– 6 ï cosh÷ −2

,

2

ï ñ óò

=

= 0.

ï )ñ óò

+ ôñ = 0.

= exp  − 21   X exp 

−1

=

ï

1 2

 2   . M

+ 2)ñó ò + ô (cosh÷

ï

sin  £   .

+ 2(2 cosh ÷

ï

– ô )ñ = 0.

+ 6(2 cosh ÷

ï

– ô )ñ = 0.

= .

ï

1

ñ

+ ï )ñ = 0.

3

(cosh÷

2 −

2

2

1 = cos  ln |  |  ,

Particular solutions:

–ô

2

2

ï ñ óò

+ ï ( ô – cosh÷

2

ï ñóò

cosh÷

ï

cos  £   ,

−1

ï 2 ñ óò ò ó

Particular solutions: 53.

= exp  − 12   ,

=

2

−  . 2 = 

2

ï 2 ñ óò ò ó

Particular solutions:

+ 1)ñó ò ò ó + ô

+ 1)ñó ò ò ó + ô ( ï + 2 cosh ÷

ï

ï 2 ñ óò ò ó

Particular solutions: 51.

= ,

+ (3 cosh÷

Particular solutions:

ô ï

+

cosh÷

Particular solutions: 49.

ï

+ ô )ñ

óò ò ó

,

=

2

ï )ñ ó ò

+ ( ô – 3 cosh÷

2 = sin  ln |  |  .

+ ï [ ô – ( õ + 1) cosh ÷

2

ï )ñ

ï ]ñ ó ò

= 0. + õ (2 cosh÷

ï

– ô )ñ = 0.

.

3.1.4-3. Equations with hyperbolic sine and cosine. 54.

ñóò ò ó ò ó

+ [ ô sinh(2 ï ) + õ ]ñ„ó ò +

cosh(2ï )ñ = 0.

ñ óò ò ó ò ó

+ [ ô sinh(2 ï ) + õ ]ñ

ñóò ò ó ò ó

+ [ ô cosh(2 ï ) + õ ]ñ„ó ò +

ñóò ò ó ò ó

+ [ ô cosh(2 ï ) + õ ]ñ„ó ò + 2 ô sinh(2 ï ) ñ = 0.

ñóò ò ó ò ó

+ ( õ cosh ï – ô

ô

This is a special case of equation 3.1.9.26 with  ( ) = 12 [  sinh(2 ) +  ]. 55.

56.

57.

58.

óò

+ 2 ô cosh(2 ï ) ñ = 0.

Integrating yields a second-order linear equation: "

 + [  sinh(2 ) +  ] = L .

ô

sinh(2 ï )ñ = 0.

Solution: = L 1 ¨ 12 + L 2 ¨ 1 ¨ 2 + L 3 ¨ 22 . Here, ¨ 1 and ¨ 2 form a fundamental set of solutions of the modified Mathieu equation 2.1.4.9: 4¨u

 + [  cosh(2 ) +  ]¨ = 0. Integrating yields a second-order linear equation: "

 + [  cosh(2 ) +  ] = L (see 2.1.4.9 for the solution of the corresponding modified homogeneous Mathieu equation with L = 0). 2

)ñó ò + õ (sinh ï – ô cosh ï )ñ = 0.

This is a special case of equation 3.1.9.29 with  ( ) =  cosh  . 3.1.4-4. Equations with hyperbolic tangent. 59.

ñ óò ò ó ò ó

–ô

3

tanh( ô

ï )ñ

= 0.

= cosh(  ) XZH ( )  leads to a M

second-order linear equation of the form 2.1.4.43: H  + 3  tanh( ) H 
+ 3  2 H = 0. Particular solution:

0

= cosh(  ). The substitution

© 2003 by Chapman & Hall/CRC

Page 520

521

3.1. LINEAR EQUATIONS

60.

ñ óò ò ó ò ó

=

ñóò ò ó ò ó

– 3ô

+ (1 – ô ) tanh ï

ôñ óò

ñ

.

This is a special case of equation 3.1.9.30 with  ( ) =  and ( ) = cosh  . 61.

2

ñóò

+ 2ô

3

Particular solutions: 62. 63.

64.

tanh÷

1

67.

ô

ñóò ò ó ò ó

+

ôñóò ò ó

+ [ õ tanh(

ñóò ò ó ò ó

+

ôñóò ò ó

– [2 ô tanh(

ï

ñ óò ò ó ò ó

– tanh ï

ñ óò ò ó

1

ï

) + 3 ]ñ„ó ò +

= cosh( Π ),

óò

– ôñ

ñ

ñóò ò ó ò ó

+

ô

tanh÷ (

ï )ñ óò ò ó

+

ñóò ò ó ò ó

+

ô

tanh÷ (

ï )ñ óò ò ó

+ ôƒõ tanh÷ (

õ ïƒú ñóò

The substitution ¨ = 

 + ' +

ñóò ò ó ò ó

tanh÷ (

ô

+

1

ñóò ò ó ò ó

+

tanh÷ (

ô

ï )ñ óò ò ó



ñóò ò ó ò ó

+

ô ƒï ÷ ñ óò ò ó



,

ñóò ò ó ò ó + ô

tanh÷

1

= 

2

ñóò ò ó ò ó

+

ô ï

tanh÷

1

ï ñóò ò ó



73.

ñ óò ò ó ò ó

+

ô ï

2

tanh÷ (

ñ óò ò ó ò ó

= (tanh÷

ï

= cosh  ,

2

+ (õ

ï

2

ï

+1

= cosh  ,

ï )ñ ó ò ò ó

Particular solutions: 74.

ï 

– ô )ñ

1

= ,

óò ò ó

) – ô ]ñ = 0.

tanh÷ (

ï

[ô

ï

3

cosh  .

3

cosh  .

)+þ

]ñ = 0.

2

'  ,

3 2

– 2ô

)+ö –

2

ïƒ÷

=  cosh  .

õ 2 ]ñ óò

ï )ñ óò

2

'  sin 

tanh÷ (

ï

+

ï )ñ ó ò

– õ )ñ 2

+ ö [ ô tanh÷ ( 1

3 2

'  .

) + 2 õ ]ñ = 0.

óò

ï

) – õ ]ñ = 0.

and ý

õ ï (ô ï

+2

2

tanh÷

ï –ô

tanh÷

ï

2

are roots of the

1 2

ï

+ 2 tanh ï )ñ = 0.

+ 3)ñ = 0.

£  . 2 = sin  

= .

ï

) – õ ]ñ = 0.

− 21

(2 tanh2 ï – 1) + 2 tanh ï ]ñ = 0.

=  cosh  .

2

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0.

õ 2 [ô

+ 3)ñó ò + (2 ô tanh÷

tanh÷ (

+ ( ô tanh÷

= exp 

2

ï

= exp(ý 2  ), where ý

2

2

ï

tanh÷ (

) + 3 õ ]ñó ò +

ï

– ô tanh÷

1 2

õ 2[ô

+

.

£  , 1 = cos  

Particular solutions:

£ − ) + L 2 sin( 

tanh ï + 3)ñ„ó ò + [ ô

– (2 ô tanh÷

ï ñóò ò ó

2

'  cos 

− 12

+ [ ôƒõ tanh÷ (

ïƒ÷

– (2 ô

–1

exp( £  ) + L

2

ï )ñ óò

– õ [2 ô tanh ÷ ( =

ï

) + 2 tanh(

leads to a first-order linear equation: ¨ 
+  tanh ( Œ  )¨ = 0.

= exp 

ï )ñ óò ò ó 1

õ ïƒú

+

ï

[2 ô tanh2 (

= 0.

2 b . Solution for  < 0:

Particular solutions: 72.

) + ö ]ñ = 0.

=  cosh( Π ).

2

+ ô tanh ï

2

= L 1 exp(− £  ) + L = L 1 cos( £ −  ) + L

Particular solutions: 71.

ï

) + ö ]ñ„ó ò + ô [ õ tanh(

Particular solutions: 1 = exp(ý 1  ), quadratic equation ý 2 + 'ý + ( = 0. 70.

.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.4.22: ¨2

 + [  tanh( Œ  ) + ( ]¨ = 0.

Particular solutions: 69.

ï )ñ

This is a special case of equation 3.1.9.30 with  ( ) =  tanh   and ( ) = cosh  .

Particular solutions: 68.

=  cosh(  ).

2

+ tanh ï (1 – ô tanh ÷

ï ñ óò

=

1 b . Solution for  > 0:

66.

= 0.

= cosh(  ),

ñ óò ò ó ò ó

Particular solutions: 65.

ï )ñ

tanh( ô

2

+ 2 ô tanh÷ (

+ õ tanh÷

ï ñ

ï )ñ

= 0.

.

= exp( Œ 2  ), where Œ

1

and Œ

2

are roots of the

© 2003 by Chapman & Hall/CRC

Page 521

522 75.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ó

+ ( ô tanh÷

ï

õ ï )ñ ó ò ò ó

+

Particular solutions: 76.

ñóò ò ó ò ó

ïƒ÷

+ [ô

1

ñóò ò ó ò ó

1

ñóò ò ó ò ó

+ ( ô tanh÷

1

80.

81.

82.

83.

ï 2 ñ óò ò ó ò ó

+ (ô

ï

2

1

tanh÷

ï

2

=





,

ï 3 ñ óò ò ó ò ó

+

tanh÷

ï 2(ô

tanh ÷

ï ñóò ò ó ò ó

+

ôñóò ò ó

ï

óò ò ó

+ õ )ñ

+ ôƒõÿñó ò + = exp 

1

tanh ÷

+ õ tanh÷

ï ñóò ò ó ò ó

+

ôñóò ò ó

ï

2

– õ tanh÷

õ 2(ô

2

'  cos 

ï ñóò

3 2

and

1

and

1

¡

ï )ñ

'  ,

+ ôƒõÿñ = 0.

2 b . Particular solutions with  < 0:

1

= exp  − £ −   ,

+

ôñóò ò ó

tanh ÷

ï ñóò ò ó ò ó

– õ (2 ô + 3 õ tanh÷ 1





=

,

tanh ÷ ( tanh ÷ tanh ÷

ï ñ óò ò ó ò ó

ï

tanh÷

= 

+ ñó ò ò ó + [( õ – ô 2 ) tanh÷

ï )ñ óò ò ó ò ó

ï ñ óò ò ó ò ó

2

ô ï 2 ñ óò ò ó

+

1

+ ( ô tanh÷ + (ô

ï

ï ñóò ò ó ò ó

1

ï

1

,

ô ï

+

ï

2

ï ñóò 2

+ 1)ñ

óò ò ó

=

óò ò ó

2

= 0.

are roots of the quadratic

+ ö ( ô tanh÷

¡

ï ñ

ï

– 2)ñ = 0.

are roots of the quadratic

2

3 2

'  .

= sin  £   .

2

2

= exp  £ −   .

+ 2 õ tanh ÷

õ 2 (ô



, where Œ

+

ô 2 ï ñ óò

and Œ

1

–ô

+ ô ( ï + 2 tanh ÷

+ ï )ñó ò ò ó + ( ô

−1

+ 1)ñ = 0.

ï )ñ

= 0.

2

ï )ñ

= 0.

are roots of the quadratic

+ 2 ôñ = 0.

= exp  − 12  2  ,

ï

2

= . + 1)ñ

2

+ ô ]ñó ò + õ (1 – ô tanh÷

= 

2

+

ï

.

−  . 2 = 

= ,

+ (3 tanh÷

Particular solutions:



– 2ô

= ,

tanh÷ 1

1

ï



ï )ñ óò 

+ ô tanh÷

ï

= exp  − 21 '  sin 

2

= cos  £   ,

ï ñóò ò ó ò ó

= 0.

= 0.

1

tanh ÷

(1 – õ tanh ï ) + 1]ñ = 0.

+1

¡

óò

+ ö – õ )ñ

=  + 2 , where

' 2   . M

+ ö ]ñó ò + ô ( ö – õ + 2) tanh ÷

ï

1 b . Particular solutions with  > 0:

Particular solutions: 88.

+ ï ( ôƒõ tanh÷

1 2

= 0.

ô 2 ƒï ÷ –1 ñ

+

– 1]ñó ò + õ (– ôƒõ tanh ÷

+ [ ô ( õ – 2)ï tanh ÷

− 12

Particular solutions:

Particular solutions: 87.

ïƒ÷ ñóò

= cosh  .

2

Particular solutions: 1 =  + 1 , equation 2 + (  − 1) + ( = 0. ¡ ¡

Particular solutions: 86.

2

– õ )ñó ò ò ó

ï )ñ óò ò ó

+õ

]ñó ò ò ó – ô

ï

ï ñ

ïƒ÷

– 1) ïƒ÷ – 1]ñó ò + õ [ ô

2

= cosh( Π ).

ï

tanh÷

ôƒõ

+

= exp  − 12 ' 2  X exp 

2

–1

– 1) tanh ÷

2

Particular solutions: 1 =  equation Π2 + Π+  = 0.

85.

ïƒ÷

óò

+ 2)ñ

Particular solutions: 1 =  + 1 , 2 =  + 2 , where ¡ equation 2 + (  − 3) + ( −  + 2 = 0. ¡ ¡

Particular solutions: 84.

= ,

ï

= cosh  .

2

– 1) – ô

+ [ô ( õ

Particular solutions: 79.

,

– ôƒõ tanh÷

ï

+1





=

ï )(ô ïw÷

+ [ tanh(

Particular solutions: 78.

= exp  − 12 ' 2  ,

(tanh ï – õ ) – õ ]ñ„ó ò ò ó + [ ô ( õ

Particular solutions: 77.

tanh÷

ï

+ õ (ô

cos   £   ,

2

ñ

= 0.

ï )ñ ó ò

+ ôñ = 0.

= exp  − 21  2  X exp 

2

tanh÷

ï 2

=

−1

ï

1 2

 2   . M

+ 2)ñó ò + ô (tanh÷

sin   £   .

ï

+ ï )ñ = 0.

© 2003 by Chapman & Hall/CRC

Page 522

523

3.1. LINEAR EQUATIONS

ï

3

89.

tanh÷

ï ñóò ò ó ò ó

+

Particular solutions:

ï

3

90.

tanh÷

ï ñ óò ò ó ò ó

+

1

ï

3

tanh÷

ï ñ óò ò ó ò ó

+

1

ï

3

tanh÷

ï ñ óò ò ó ò ó

=

−2

,

2

= .

2

– 6 ï tanh÷ ,

3

ï ñóò

+ 2(2 tanh÷

ï

– ô )ñ = 0.

ï ñ óò

+ 6(2 tanh÷

ï

– ô )ñ = 0.

= .

2

+ ï ( ô – tanh÷

ô ï 2 ñ óò ò ó

1 = cos  ln |  |  ,

Particular solutions: 92.

=

−1

ô ï 2 ñ óò ò ó

Particular solutions: 91.

– 2 ï tanh÷

ô ï 2 ñ óò ò ó

+ï

2

(tanh÷

Particular solutions:

1

2 −

=

ï

ï )ñ

2 = sin  ln |  |  .

+ ï [ ô – ( õ + 1) tanh÷

+ ô )ñ

óò ò ó

,

=

2

+ ( ô – 3 tanh÷

ï )ñ ó ò

2

ï ]ñ ó ò + õ

= 0. (2 tanh÷

ï

– ô )ñ = 0.

.

3.1.4-5. Equations with hyperbolic cotangent. 93.

ñ óò ò ó ò ó

–ô

3

ï )ñ

coth( ô

= 0.

= sinh(  ) XZH ( )  leads to a M

second-order linear equation of the form 2.1.4.52: H  + 3  coth( ) H 
+ 3  2 H = 0. Particular solution:

94.

ñóò ò ó ò ó

=

ñ óò ò ó ò ó

– 3ô

0

= sinh( ). The substitution

+ (1 – ô ) coth ï

ôñóò

ñ

.

This is a special case of equation 3.1.9.30 with  ( ) =  and ( ) = sinh  . 95.

2

ñ óò

+ 2ô

3

Particular solutions: 96.

97.

98.

coth÷

ï ñóò

1

ô

ñ óò ò ó ò ó

+

ôñ óò ò ó

+ [ õ coth(

ñ óò ò ó ò ó

+

ôñ óò ò ó

– [2 ô coth(

ï )ñ

.

ï

óò

) + ö ]ñ

+ ô [ õ coth(

ï

) + ö ]ñ = 0.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.4.44: ¨2

 + [  coth( Œ  ) + ( ]¨ = 0.

ñóò ò ó ò ó

– coth ï

ñóò ò ó

1

ñóò ò ó ò ó

ï

2

coth ï

ô

ñ

2

+

[2 ô coth2 (

ï

ï

) + 2 coth(

) – ô ]ñ = 0.

=  sinh( Π ). = 0.

= L 1 exp(− £  ) + L 2 exp( £  ) + L 3 sinh  . = L 1 cos( £ −  ) + L 2 sin( £ −  ) + L 3 sinh  .

+ ( ô coth ï – ôƒõ – õ )ñ„ó ò ò ó + ( ôƒõ 1

ñóò ò ó ò ó

+

ô

coth÷ (

ï )ñ óò ò ó

ñóò ò ó ò ó

+

ô

coth÷ (

ï )ñ óò ò ó

=





,

1

2

+

õ ïƒú ñóò

+

ôƒõ

The substitution ¨ = 

 + ' +

Particular solutions:

óò

) + 3 ]ñ

= sinh( Π ),

– ôñó ò +

Particular solutions:

102.

=  sinh( ).

This is a special case of equation 3.1.9.30 with  ( ) =  coth   and ( ) = sinh  .

2 b . Solution for  < 0:

101.

2

+ coth ï (1 – ô coth÷

=

1 b . Solution for  > 0:

100.

= 0.

= sinh( ),

ñóò ò ó ò ó

Particular solutions: 99.

ï )ñ

coth( ô

= exp 

2

– ô – 1)ñó ò + õ (– ôƒõ coth ï + ô + 1)ñ = 0.

= sinh  . +

õ ïƒú

–1

[ô

ï

coth÷ (

ï

)+þ

]ñ = 0.

leads to a first-order linear equation: ¨ 
+  coth ( Œ  )¨ = 0.

coth÷ ( − 12

ï )ñ óò

'  cos 

2

+ 3 2

õ 2 [ô

'  ,

coth÷ ( 2

ï

) – õ ]ñ = 0.

= exp  − 21 '  sin 

2

3 2

'  .

© 2003 by Chapman & Hall/CRC

Page 523

524 103.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+

coth÷ (

ô

Particular solutions: 104.

ñóò ò ó ò ó

+

ô ï

coth÷

ï ñóò ò ó

Particular solutions: 105.

ñóò ò ó ò ó

+

ô ï

2

coth÷ (

107. 108.

110.

111.

112.



=

+ ô ï ÷ ñ ó ò ò ó – (2 ô Particular solutions:

,

ï

+ (õ

1 2

1

ï

+

+ [ coth( ï )( ô Particular solutions:

1

¡

¡

¡

¡

+ ï 2 ( ô coth÷ Particular solutions: equation 2 + (  − 1)

ï 3 ñ óò ò ó ò ó

coth ÷ ï ñó ò ò ó ò ó

+

ôñóò ò ó

Particular solutions:

113. coth ÷

ï ñóò ò ó ò ó

+

ôñóò ò ó

ï

+1

ï

+ õ (ô

ï

1

2

coth÷

ôƒõÿñóò

+

= exp 

− 12

+ õ coth÷

'  cos 

ï ñóò

2

ïƒ÷ ñóò

Particular solutions:

=

ï





,

2

2

3 2

ï )ñ

ï ñ

+ ö ]ñ 1 and

1 2

= 0.

' 2   . M

ô 2 ƒï ÷ –1 ñ óò ¡

+ 2 coth ï )ñ = 0.

ï

= 0.

+ ô ( ö – õ + 2) coth÷ ï ñ = 0. 2 are roots of the quadratic

= 0.

'  ,

2

= cos  £   ,

= exp  − 21 '  sin 

= 

2

= exp  − £ −   ,

– õ (2 ô + 3 õ coth÷ 1

– ô coth÷

ï

+

+ ôƒõÿñ = 0. 1

ôñóò ò ó

+2

= exp  − 12 ' 2  X exp 

2

– õ coth÷

õ 2 (ô

2 b . Particular solutions with  < 0: +

= 0.

)ñó ò ò ó + ï ( ôƒõ coth÷ ï + ö – õ )ñó ò + ö ( ô coth÷ ï – 2)ñ = 0.  + 1 , 2 =  + 2 , where 1 and 2 are roots of the quadratic ¡ ¡ = 0.

1

ï ñóò ò ó ò ó

ï )ñ

+ 2)ñó ò + ôƒõ coth÷

ï

– 1) – ô ïƒ÷ –1 ]ñó ò ò ó – ô =  , 2 = sinh( Œ  ).

1 b . Particular solutions with  > 0: 114. coth ÷

+ 3)ñ = 0.

2

+ 3)ñó ò + (2 ô coth÷ =  sinh  .

= exp  − 12 ' 2  ,

+õ 1 = +( +

ï

) + 2 õ ]ñ = 0.

coth ï + 3)ñ ó ò + [ ô ï ÷ (2 coth2 ï – 1) + 2 coth ï ]ñ = 0. = sinh  , 2 =  sinh  .

+ ( ô ï 2 coth÷ ï + õ ï )ñ ó ò ò ó + [ ô ( õ – 2)ï coth ÷ Particular solutions: 1 =  + 1 , 2 =  + 2 , where ¡ equation 2 + (  − 3) + ( −  + 2 = 0.

ï 2 ñ óò ò ó ò ó

coth÷

+ 2 ô coth÷ (

ïw÷

ñóò ò ó ò ó

2

ï

= .

2

õ ï )ñ óò ò ó 1

1 2

ï )ñ óò

2

+ ô coth÷ ï ñó ò ò ó – (2 ô coth÷ Particular solutions: 1 = sinh  , + ( ô coth÷

õ ï (ô ï

+

coth÷ (

õ 2[ô

£  . 2 = sin  

coth÷ (

ï

ñóò ò ó ò ó ñóò ò ó ò ó

ï )ñ óò

2

– 2ô

= ,

) + 3 õ ]ñó ò +

.

– ô coth÷

2

ï ÷

ñ óò ò ó ò ó



= 

2

ï 

£  , 1 = cos  

1

Particular solutions: 109.

1



ï )ñ óò ò ó

Particular solutions: 106.

– õ [2 ô coth÷ (

ï )ñ óò ò ó





ï )ñ óò

+

õ 2(ô

2

3 2

'  .

= sin  £   . 2

= exp  £ −   .

+ 2 õ coth÷

ï )ñ

= 0.

.

115. coth ÷ ï ñ ó ò ò ó ò ó + ñ ó ò ò ó + [( õ – ô 2 ) coth÷ ï + ô ]ñ ó ò + õ (1 – ô coth ÷ ï )ñ = 0.   Particular solutions: 1 =  1 , 2 =  2 , where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + Œ +  = 0. 116. coth ÷ (

ï )ñ óò ò ó ò ó

+

ô ï 2 ñ óò ò ó

Particular solutions:

1

ï ñóò

– 2ô

= ,

117. coth ÷ ï ñó ò ò ó ò ó + ( ô coth÷ ï + ô Particular solutions: 1 =  ,

ï

2

+ 2 ôñ = 0.

2

= .

2

+ 1)ñó ò ò ó + ô  =  − .

2

ï ñóò

–ô

2

ñ

= 0.

© 2003 by Chapman & Hall/CRC

Page 524

525

3.1. LINEAR EQUATIONS

118. coth ÷

ï ñ óò ò ó ò ó

ï

+ (ô

coth÷

Particular solutions: coth÷

ï

119.

1

ï

3

coth÷

ï ñóò ò ó ò ó

+ô

1

ï

3

121.

coth÷

ï ñóò ò ó ò ó

+ô

1

ï

3

coth÷

ï ñóò ò ó ò ó

+ô

1

ï

3

coth÷

ï ñ óò ò ó ò ó

+ï

=

−2

1 2

1

ï ñóò

= .

2

,

2

3

ï ñóò

= .

2

+ ï ( ô – coth÷

= cos  ln   ,

(coth÷

Particular solutions:

,

2

– 6 ï coth÷

ï 2 ñ óò ò ó

Particular solutions: 123.

=

ï 2 ñ óò ò ó

Particular solutions: 122.

– 2 ï coth÷ −1

ï

,

2

+ ôñ = 0.

−1

=

=

 2   . M

+ 2(2 coth÷

ï

– ô )ñ = 0.

+ 6(2 coth÷

ï

– ô )ñ = 0.

+ ( ô – 3 coth÷

ï )ñ óò

+ ï )ñ = 0.

ï

sin   £   .

+ ï [ ô – ( õ + 1) coth÷

2

1 2

+ 2)ñó ò + ô (coth÷

ï

ï )ñ

= sin  ln   .

2

óò ò ó

+ ô )ñ

2 −

=

coth÷

ï

cos   £   ,

−1

=

ï )ñ ó ò

= exp  − 21  2  X exp 

2

+ ï )ñó ò ò ó + ( ô

ï

ï 2 ñ óò ò ó

Particular solutions:

+ ô ( ï + 2 coth÷

óò ò ó

+ 1)ñ

= exp  − 12  2  ,

+ (3 coth÷

ï ñóò ò ó ò ó

Particular solutions: 120.

ï

ï ]ñ ó ò

.

= 0. + õ (2 coth÷

ï

– ô )ñ = 0.

3.1.5. Equations Containing Logarithmic Functions 3.1.5-1. Equations with logarithmic functions. 1.

2.

ñóò ò ó ò ó

ln ÷ (

ô

+

ï )ñ óò ò ó

5.

6.

+

ln÷ (

ôƒõ

1

2 b . Particular solutions with  < 0:

1

ñóò ò ó ò ó

ln ÷

ô

+

ï ñóò ò ó

+

ñóò ò ó ò ó

+

ô

ln ÷ (

ï )ñ óò ò ó

ln ÷

ï ñóò

= exp 

− 12

ôƒõ 1

Particular solutions: 4.

õÿñóò

1 b . Particular solutions with  > 0:

Particular solutions: 3.

+

+

1

=



,

ï

ï

+ õ )ñó ò ò ó +

ö“ñóò

ï

õ 2[ô





2

ï

+ õ )ñ = 0.

= exp  − £ − (  ,

ln ÷

ï ñ óò ò ó ò ó

+

ôñ óò ò ó

Particular solutions:

+

– õ )ñó ò + õ ln ÷

1

2

+

õ 2 (ô

– õ ln÷

= exp 

− 12

'  cos 

ôƒõÿñ óò

'  .

) + 2 õ ]ñ = 0.

+ ö ( ô ln÷ ï – õ )ñ = 0. = exp( Œ 2  ), where Œ 1 and Œ

1

ï

ï

3 2

õ 2 )ñ óò

2 b . Particular solutions with ( < 0: – ô )ñó ò ò ó + ( ô ln÷

2

.

= cos  £ (  ,

ï

= exp  £ −   .

ln÷ (

1

= (ln÷

2

= exp  − 21 '  sin 

1 b . Particular solutions with ( > 0:

ñóò ò ó ò ó

= sin   £   .

– õ )ñ = 0.

) + 3 õ ]ñó ò +

3 2

+ ö ( ô ln÷

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0.

7.

2

2

+ ô ln ÷ ï ñó ò ò ó + ( ôƒõ ln÷ ï + ö – Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0. + ( ô ln÷

ln ÷

'  ,

ñóò ò ó ò ó ñóò ò ó ò ó

2

= exp  − £ −   ,

õ 2(ô

= 

2

= 0.

= cos   £   ,

'  cos 

– õ [2 ô ln÷ (



ï )ñ

2

ï ñ

= sin  £ (  . 2

= exp   £ − (  .

.

= exp( Œ 2  ), where Œ

2

ï )ñ 3 2

1

and Œ

= 0.

'  ,

2

are roots of the

2

= exp  − 21 '  sin 

2

2

are roots of the

3 2

'  .

© 2003 by Chapman & Hall/CRC

Page 525

526 8.

THIRD-ORDER DIFFERENTIAL EQUATIONS ln ÷

ï ñóò ò ó ò ó

+ ñó ò ò ó + [( ô –

õ

2

) ln÷

Particular solutions: 1 =  equation Œ 2 + Œ +  = 0. 9.

10.

ln ÷

ï ñ óò ò ó ò ó





=

1

ln ÷ (

+ õ ln÷ (

+

,

,

Particular solutions:

ï )ñ óò ò ó ò ó

+ õ ]ñó ò + ô (1 – õ ln÷

ï

ôñóò ò ó

2

 2

= 

2

– õ (2 ô + 3 õ ln ÷

ôñ óò ò ó

+

 1

ï )ñ ó ò

ï )ñ óò

+





= 

, where Œ

1

ï )ñ

and Œ

+ 2 õ ln÷

õ 2 (ô

= 0. 2

ï )ñ

are roots of the quadratic = 0.

.

ôƒõÿñ

+

= 0.

1 b . Particular solutions with  > 0:

1

= cos   £   ,

2 b . Particular solutions with  < 0:

1

= exp  − £ −   ,

= sin   £   .

2

2

= exp   £ −   .

3.1.5-2. Equations with power and logarithmic functions. 11.

ñóò ò ó ò ó

+ õ ) ln ÷ (

ï

+ (ô

Particular solution: 12.

ñ óò ò ó ò ó

Particular solution: 13.

– ô ln÷ (

ï )ñ ó ò

– 2 ô ln÷ (

+

ô

ln ÷ (

ï )ñ óò ò ó

+

+

ô ï

ln÷

ï ñóò ò ó

+ (õ

ñóò ò ó ò ó ñóò ò ó ò ó

ï ñ óò ò ó

ï )ñ

= 0.

= (  +  ) .

0

ln ÷

+

= 0.

2

ô

ñ óò ò ó ò ó

ï )ñ

=  +  .

0

+ õ ) ln ÷ (

ï

+ (ô

ï )ñ óò

ï

+ (õ

+ ö )ñ

óò

+ ( ôƒõ

ln÷

ï

ï

ln÷

ô„ö

+

ï

+ õ )ñ = 0.





Integrating yields a second-order linear equation:  + ( ' + ( ) = L exp O − lX ln   M P (see 2.1.2.2 for the solution of the corresponding homogeneous equation with L = 0). 14.

15.

ñóò ò ó ò ó

+ ( ô ln÷

ï

+

18.

ln÷

ï

ñóò ò ó ò ó

+ (ô

ñóò ò ó ò ó

+ ( ôƒõ

ï

2

– ô ln÷ 1 2

ï

ln÷ (

ï

+

õ ï (ô ï

2

[ô

ï )ñ óò

)+þ

]ñ = 0.

ln÷

ï

ï

ï

= exp  − 12 '  ,

2

– 1) ln ÷

+ 3)ñ = 0.

2

+ 2)ñó ò + ôƒõ ln÷

2

ï

ln÷

£  . 2 = sin   1 2

2

+ õ (ô

+ õ )ñó ò ò ó + ô ( õ

ï ñ

= 0.

= exp  − 12 '  X exp  2

ï ñóò

– ôƒõ ln ÷

ï ñ

' 2   . M

1 2

= 0.

The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.5.13: ¨2

 +  ln  ¨2
−  ln   ¨ = 0.

ï

ln÷

+ ô ln÷

ï

ñóò ò ó ò ó

+

ô ï

2

ln÷ (

ï ñóò ò ó ò ó

+

ô ï ñóò ò ó

1

ï )ñ óò ò ó

Particular solutions: 20.

–1

£  , 1 = cos  

1

Particular solutions: 19.

ï

õ ï )ñ óò ò ó

Particular solutions: 17.

õ ïƒú

+

¨ 
+  ln ( Œ  )¨ = 0. The substitution ¨ = 

 + ' +* leads to a first-order linear equation: u Particular solutions:

16.

õ ïƒú ñóò

– õ (õ

ï

+ õ )ñó ò ò ó +

= , – 2ô

2

=

−

ln÷ (

ï





ôƒõ 2 ï

ï )ñ óò

– ôƒõ

2

ln ÷

ï ñ

= 0.

+ 2 ô ln÷ (

ï )ñ

= 0.

2

= ,

ï

ln2 ï + 1)ñó ò – ôƒõ ( õ

The substitution ¨ = 
+   ¨2

 − (  2  ln2  +  )¨ = 0.

ï ñóò

.

1

2

ln÷

= .

ï

ln2 ï + 1)ñ = 0.

leads to a second-order linear equation of the form 2.1.5.3:

© 2003 by Chapman & Hall/CRC

Page 526

527

3.1. LINEAR EQUATIONS

21.

ï ñ óò ò ó ò ó

+

ln÷ (

ô

ï )ñ ó ò ò ó

õ ï ñ óò

+

+ ôƒõ ln÷ (

2 b . Particular solutions with  < 0:

23.

ï ñ óò ò ó ò ó

+

ô ï

ln ï

ñ óò ò ó

+ ( ôƒõ =

Particular solutions:

1

= (ln÷

ï )ñ óò ò ó

ï ñóò ò ó ò ó

ï

–ô

ï −





ln ï –

õ 2ï

ï ñóò ò ó ò ó

ln ÷

ï

+ (ô



–

õ ï )ñ óò

,

+ ( ô ln÷

Particular solutions: 25.

ï ñóò ò ó ò ó

ln ÷

ï

+ (ô

ï

1

ï

ï ñóò ò ó ò ó

1

ï 2 ñ óò ò ó ò ó

,

2

=

+ [ ô ( õ – ln ï ) ïw÷ + 2]ñó ò ò ó + ô

Particular solutions: 27.

=

−1

ln÷ (

ô

+

1

ï )ñ óò ò ó

= , +

2

+

1 b . Particular solutions with  > 0: 2 b . Particular solutions with  < 0: 28.

29.

30.

31.

32.

33.

ï

+ õ ln÷

õ ï )ñ óò

+

2

ï

=

−1

M

 .

ï ñ

.

ïƒ÷ –1 ñ óò

–ô

ln ÷ (

ï

–

ïƒ÷ –2 ñ

ï )ñ

ln ÷

ï

ï

1

and Œ

2

are roots of the

+ 1)ñ = 0.

sin   £   .



  .

−1 −

ôƒõ

= 0.



+ õ (ô

+ 2 ô ln ÷

= ln  −  + 1.

õ ï 2 ñ óò



+2 )

= exp( Œ 2  ), where Œ

2

ln÷

ï

ôƒõÿñ

−

cos   £   ,

+ 3)ñó ò ò ó + ( ôƒõ

Particular solutions: 26.

=

−1

+

   ( 2 =   XZ

+ 3)ñó ò ò ó + (2 ô ln ÷

ï

óò

+ ô )ñ

−

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0. 24.

= 0.

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp   £ −   . 1 = exp  − 

1 b . Particular solutions with  > 0:

22.

ï )ñ

õ 2 ï )ñ óò

+ õ ( ô ln÷

ï

– õ )ñ = 0.

= 0.

= 0.

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp   £ −   . 1 = exp  − 

+ ï 2 ( ô ln ï + õ )ñ ó ò ò ó + 2 ô ï ñ ó ò – ôñ = 0. Integrating the equation twice, we arrive at a first-order linear equation:
 + (  ln  +  ) = L 1 + L 2 .

ï 2 ñ óò ò ó ò ó ï 2 ñ óò ò ó ò ó

ï [ô ï

– 3ô

ln2 (

ï

ï

+ ô [2 ô

2 2

1

ï 2 ñ óò ò ó ò ó

+

ln ï +

õ ï )ñ óò ò ó

ï 2 ñ óò ò ó ò ó

+ (ô

ln÷

+

Q

+ 2ï ( õ

ï

ï

ln3 (

= exp lX ln( Π )  S ,

Particular solutions:

ï 2(ô

óò

) + 1]ñ

2

M

) + 1]ñ = 0.

=  exp lX ln( Π )  S .

Q

M

+ ô )ñó ò – ôñ = 0.

Integrating the equation twice, we arrive at a first-order linear equation:
 + (  ln  + ' ) = L 1 + L 2 .

ï

2

ï

õ ï )ñ óò ò ó

+ [ ô ( õ – 2)ï ln ÷

ï

Particular solutions: 1 =  + 1 , 2 =  + 2 , where ¡ equation 2 + (  − 3) + ( −  + 2 = 0. ¡ ¡

+ ö ]ñó ò + ô ( ö – 1

and

¡

2

õ

+ 2) ln ÷

ï ñ

= 0.

are roots of the quadratic

+ ï 2 ( ô ln ï + õ )ñ ó ò ò ó + 2 ô ï ñ ó ò – ôñ = 0. Integrating the equation twice, we obtain a first-order linear equation: 
 + (  ln  +  − 2) = L 1 + L 2 .

ï 3 ñ óò ò ó ò ó ï 3 ñ óò ò ó ò ó

+

ô

ln÷ (

ï )ñ ó ò ò ó

+

õ ï 3 ñ óò

1 b . Particular solutions with  > 0: 2 b . Particular solutions with  < 0:

+

ôƒõ

ln ÷ (

ï )ñ

= 0.

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp   £ −   . 1 = exp  − 

© 2003 by Chapman & Hall/CRC

Page 527

528 34.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ï 3 ñ óò ò ó ò ó

ô ï

+

2

ln÷

Particular solutions: 35.

36.

37.

38.

ï 3 ñ óò ò ó ò ó

ô ï

+

2

ln÷

1

−2

=

1

ï 3 ñ óò ò ó ò ó

+

ï 2(ô

ln ï +

õ ï )ñ óò ò ó

ï 3 ñ óò ò ó ò ó

+

ï 2(ô

ln÷

+ õ )ñ

+ 2(2 – ô ln ÷

ñóò ,

– 6ï

Particular solutions:

2

,

2

= 0.

ï )ñ

= 0.

2

= . + 6(2 – ô ln ÷

ñ óò

ï )ñ

3

= .

+ 2ï ( õ

ï

óò

+ ô )ñ

– ôñ = 0.

Integrating the equation twice, we obtain a first-order linear equation: 
 +(  ln  + ' −2) = L 1 + L 2 .

ï

óò ò ó

+ ï ( ôƒõ ln÷

Particular solutions: 1 =  + 1 , equation 2 + (  − 1) + ( = 0. ¡ ¡

ln ÷

ï ñ óò ò ó ò ó

ln ÷

ï

+ (ô

ï

ln ÷

ï ñóò ò ó ò ó

+ ( ô ln÷

ln ÷ (

ï )ñ ó ò ò ó ò ó

+

+ 1)ñ

1

ï

+ô

ï

= ,

ô ï 2 ñ óò ò ó

– 2ô

Particular solutions:

1

ï

= ,

2

+ ô ( ï + 2 ln÷

óò ò ó

2

+ 1)ñó ò ò ó + ô

2

 . 2 =  −

ï ñ óò 2

ö

+

– õ )ñ

=  + 2 , where

= exp  − 12  2  ,

1

Particular solutions: 40.

−1

=

ï ñ óò ò ó

Particular solutions: 39.

– 2ï

ï ñóò ò ó

¡

ï )ñ ó ò

óò

+ ö ( ô ln ÷

1

and

¡

2

–ô

2

– 2)ñ = 0.

are roots of the quadratic

+ ôñ = 0.

= exp  − 21  2  X exp 

ï ñóò

ï

ñ

1 2

 2   . M

= 0.

+ 2 ôñ = 0.

=  2.

3.1.6. Equations Containing Trigonometric Functions 3.1.6-1. Equations with sine. 1.

2.

3.

4.

5.

ñ óò ò ó ò ó

+

ô

sin ï

ñóò ò ó ò ó

+

ô

sin2 ï

ñ óò ò ó ò ó

+ [ ô sin(

ñ óò ò ó ò ó

+

– õ ( ô sin ï +

ñ óò

õ 2 )ñ

= 0.

 The substitution ¨ =   J 2 (
 −  ) leads to a second-order linear equation of the form 2.1.6.2: ¨2

 +   sin  + 3  2  ¨ = 0. 4

ñóò

– õ ( ô sin2 ï +

õ 2 )ñ

= 0.

 The substitution ¨ =   J 2 (
 −  ) leads to a second-order linear equation of the form 2.1.6.3: ¨ 

 +  sin2  + 3  2  ¨ = 0. 4

ï

óò

) + õ ]ñ

– ö [ ô sin(

ï

) + õ + ö 2 ]ñ = 0.

 The substitution ¨ =  v J 2(
 − ( ) leads to a second-order linear equation of the form 2.1.6.2: ¨2

 + V  sin( Œ  ) +  + 3 ( 2 W ¨ = 0. 4

ôñ óò ò ó

+ [ õ sin(

ï

) + ö ]ñ

The substitution ¨ = 
+  ¨2

 + [  sin( Œ  ) + ( ]¨ = 0.

ñ óò ò ó ò ó

+

ôñ óò ò ó

+ õ sin2 (

ï )ñ ó ò

The substitution ¨ = 
+  ¨2

 +  sin2 ( Œ  ) ¨ = 0.

óò

+ ô [ õ sin(

ï

) + ö ]ñ = 0.

leads to a second-order linear equation of the form 2.1.6.2: + ôƒõ sin2 (

ï )ñ

= 0.

leads to a second-order linear equation of the form 2.1.6.3:

© 2003 by Chapman & Hall/CRC

Page 528

529

3.1. LINEAR EQUATIONS

6.

7.

ñ óò ò ó ò ó

sin ÷ (

ô

+

ï )ñ ó ò ò ó

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

ñ óò ò ó ò ó

sin ÷ (

ô

+

+ ôƒõ sin÷ (

ï )ñ ó ò ò ó

Particular solutions: 8.

ñóò ò ó ò ó

+

sin ÷ (

ô

1

ï )ñ óò ò ó

10. 11.

= exp 

1





=

ï

+ ô sin ÷„ï ñó ò ò ó + ( ôƒõ sin ÷„ï + ö – Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0.

13.

18.

19.

ï

sin÷ (

õ 2 [ô

2

3 2

'  .

) + 2 õ ]ñ = 0.

. + ö ( ô sin÷„ï – õ )ñ = 0. = exp( Œ 2  ), where Œ 1 and Œ

õ 2 )ñ óò 2

2

are roots of the

+ ( ô sin÷„ï + õ )ñó ò ò ó + ö^ñó ò + ö ( ô sin÷„ï + õ )ñ = 0. 1 b . Particular solutions with ( > 0: 1 = cos   £ (  , 2 = sin   £ (  . = exp  − £ − (  ,

1

= exp   £ − (  .

2

= (sin ÷„ï – ô )ñó ò ò ó + ( ô sin ÷„ï – õ )ñó ò + õ sin ÷„ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ quadratic equation Œ 2 + Œ +  = 0.

ñóò ò ó ò ó ñóò ò ó ò ó

+ ( ô sin÷„ï +

ñ óò ò ó ò ó

+

ô ï

sin ÷

õ ï )ñ óò ò ó 1

ï ñ óò ò ó

ñóò ò ó ò ó

+ ( ôƒõ

ï

sin÷„ï +

ñóò ò ó ò ó

+

ô ï

sin ÷ (

2

Particular solutions: 17.

= exp  − 21 '  sin 

2

ñóò ò ó ò ó

Particular solutions: 16.



) – õ ]ñ = 0.

+ ô sin ÷ ( ï )ñó ò ò ó + õ ïƒú ñó ò + õ ïƒú –1 [ ô ï sin÷ ( ï ) + þ ]ñ = 0. The substitution ¨ = 

 + '  +2 leads to a first-order linear equation: ¨u
+  sin ( Œ  ) ¨ = 0.

ï

+ õ (ô

= exp 

+ (õ

ï

2

− 12

sin ÷„ï + 2)ñó ò + ôƒõ sin÷„ï

'  , 2

– ô sin÷

2

ï )ñ ó ò

£  , 1 = cos   1 2

Particular solutions: 15.

2

ï

ñóò ò ó ò ó

Particular solutions: 14.



'  ,

= exp   £   .

= sin   £ −   .

2

sin ÷ (

) + 3 õ ]ñó ò +

ñóò ò ó ò ó

2 b . Particular solutions with ( < 0:

12.

2

= cos   £ −   ,

= 

2

= 0.

= exp  − £   ,

'  cos 

− 12

,

ï )ñ

ï )ñ ó ò + õ 2 [ ô 2 3

– õ [2 ô sin ÷ (

Particular solutions: 9.

– ôƒõ sin ÷ (

óò

– õÿñ

2

sin ÷„ï + õ )ñó ò ò ó +

ô 1

= ,

ï )ñ óò ò ó 1

– 2ô

= ,

2

ï

2

=

−





sin÷ ( 2

= exp  +

− 12

õ ï (ô ï

ôƒõ 2 ï

sin ÷„ï

and Œ

2

are roots of the

= 0.

'  X exp  12 '  2   . M 2

2

sin÷

ï

£  . 2 = sin   1 2

ñ

1

+ 3)ñ = 0.

2

ñóò

– ôƒõ

+ 2 ô sin÷ (

ï )ñ

2

sin ÷„ï

ñ

= 0.

.

ï )ñ óò

= 0.

= .

+ ï [ ô sin( ï ) + õ ]ñ ó ò – 2[ ô sin( ï ) + õ ]ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.6.2: ¨2

 + [  sin( Œ  ) +  ]¨ = 0.

ï ñ óò ò ó ò ó

= (sin÷ ï – ô ï )ñ ó ò ò ó + ( ô sin÷ ï – õ ï )ñ ó ò + õ sin ÷ ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ quadratic equation Œ 2 + Œ +  = 0.

ï ñ óò ò ó ò ó ï ñóò ò ó ò ó

+ (ô

ï

sin ÷„ï + 3)ñó ò ò ó + (2 ô sin÷„ï +

Particular solutions:

1

=

−1

cos   £   ,

õ ï )ñ óò

2

=

−1

+ õ (ô

ï

1

and Œ

2

are roots of the

sin ÷„ï + 1)ñ = 0.

sin   £   .

© 2003 by Chapman & Hall/CRC

Page 529

530 20.

21.

THIRD-ORDER DIFFERENTIAL EQUATIONS = (sin ÷„ï – ô ï 2 )ñó ò ò ó + ( ô sin ÷„ï – õ ï 2 )ñó ò + õ sin÷„ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ 1 and Œ quadratic equation Œ 2 + Œ +  = 0.

ï 2 ñ óò ò ó ò ó

+ ô ï 2 sin ÷ ( ï )ñó ò ò ó + õ ï Particular solutions: 1 =  + 1 , equation 2 − +  = 0.

ï 3 ñ óò ò ó ò ó

¡

22.

23.

25.

27.

sin ÷

ï ñ óò ò ó ò ó

+

ôñ óò ò ó

sin ÷„ï

ñóò ò ó ò ó

+

29.

ôñóò ò ó

1

¡

= exp 

– õ sin ÷

õ 2(ô ñóò

+ ôƒõÿñ = 0.

2 b . Particular solutions with  < 0:

1

ñóò ò ó ò ó

+

ï )ñ

'  cos  1

sin ÷„ï

2

− 12

1 b . Particular solutions with  > 0:

¡

3 2

1

=





,

2

'  ,

2

= cos   £   ,

= exp  − 21 '  sin 

2

= exp  − £ −   ,

– õ (2 ô + 3 õ sin ÷„ï )ñó ò +

ôñóò ò ó

= 0.





= 

2

3 2

'  .

= sin   £   . 2

= exp   £ −   .

+ 2 õ sin ÷„ï )ñ = 0.

õ 2 (ô

.

sin ÷„ï ñó ò ò ó ò ó + ñó ò ò ó + [( õ – ô 2 ) sin ÷„ï + ô ]ñó ò + õ (1 – ô sin ÷„ï )ñ = 0.   Particular solutions: 1 =  1 , 2 =   2 , where Œ 1 and Œ 2 are roots of the quadratic 2 equation Œ + Œ +  = 0.

ï )ñ óò ò ó ò ó

sin ÷ (

ô ï 2 ñ óò ò ó

+

1

ï ñóò

– 2ô

= ,

2

+ 2 ôñ = 0.

=  2.

sin ÷„ï ñó ò ò ó ò ó + ( ô sin ÷„ï + ô ï + 1)ñó ò ò ó +  Particular solutions: 1 =  , 2 =  −  .

sin ÷„ï

ï

ñóò ò ó ò ó

sin ÷ï

30.

ï

+ (ô

ñóò ò ó ò ó

ï

3

31.

sin ÷„ï

ñóò ò ó ò ó

1

ï

3

sin ÷„ï

ñóò ò ó ò ó

1

1

sin ÷„ï ñó ò ò ó ò ó + ô ï Particular solutions: 3

sin ÷„ï

ñóò ò ó ò ó

+

=

ô ï 2 ñ óò ò ó

+

3

ï

=

ô ï 2 ñ óò ò ó

+

Particular solutions:

ï

= exp  − 12  2  ,

ï

2

Particular solutions:

1 2

=

−1

cos   £   ,

– 2 ï sin ÷„ï −1

,

2

–ô

2

ñ

= 0.

,

2

= exp  − 21  2  X exp 

2

=

−1

 2   . M

sin   £   .

+ 2(2 sin ÷„ï – ô )ñ = 0.

ñóò

+ 6(2 sin ÷„ï – ô )ñ = 0.

=  3.

1 2

sin ÷„ï + 2)ñó ò + ô (sin÷„ï + ï )ñ = 0.

ñóò

=  2.

– 6 ï sin ÷„ï −2

2

ï

+ (3 sin÷„ï + ï )ñó ò ò ó + ( ô

Particular solutions: 32.

ô 2 ï ñ óò

sin÷„ï + 1)ñó ò ò ó + ô ( ï + 2 sin ÷„ï )ñó ò + ôñ = 0.

Particular solutions:

34.

+

+ õ sin ÷„ï

Particular solutions:

33.

óò

+ ôƒõÿñ

Particular solutions:

28.

2

sin2 ï ñó ò ò ó ò ó + ô sin2 ï ñó ò ò ó + õÿñó ò + ôƒõÿñ = 0. The substitution ¨ = 
+  leads to a second-order linear equation of the form 2.1.6.21: sin2  ¨ 

 +  ¨ = 0.

Particular solutions:

26.

+ õ [ ô sin ÷ ( ï ) – 2]ñ = 0. =  + 2 , where 1 and 2 are roots of the quadratic

ñóò

¡

Particular solutions:

24.

are roots of the

2

1

+ ï ( ô – sin÷„ï )ñó ò + ( ô – 3 sin ÷„ï )ñ = 0. = cos(ln  ), 2 = sin(ln  ).

1

=

ñóò ò ó

(sin÷„ï + ô )ñó ò ò ó + ï [ ô – ( õ + 1) sin ÷„ï ]ñó ò + õ (2 sin÷„ï – ô )ñ = 0. −

2

,

2

=

2

.

© 2003 by Chapman & Hall/CRC

Page 530

531

3.1. LINEAR EQUATIONS

3.1.6-2. Equations with cosine. 35.

36.

37. 38.

cos(2 ï )ñ

ñ óò ò ó ò ó

+

ñóò ò ó ò ó

+ [ ô cos(

ñ óò ò ó ò ó

+

ôñ óò ò ó

ñóò ò ó ò ó

+

ô

ô

– õ [ ô cos(2ï ) +

óò

õ 2 ]ñ

ï

+ ( õ cos 2 ï + ö )ñ

The substitution ¨ = 
+ 

ï )ñ óò ò ó

cos÷ (

ñ óò ò ó ò ó

cos÷ (

ô

+

+ ô ( õ cos 2 ï + ö )ñ = 0.

óò

ñ óò ò ó ò ó

cos÷ (

ô

+

ñóò ò ó ò ó

cos÷„ï

ô

+

= exp 

ï )ñ ó ò

1





=

,

2

ï

= 



ö

–

+ ( ôƒõ cos÷„ï +

ñóò ò ó

Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0.

43.

44.

cos÷ (

ï )ñ ó ò ò ó

õ ï ú ñ óò

ñ óò ò ó ò ó

+

ñóò ò ó ò ó

+ ( ô cos÷„ï + õ )ñó ò ò ó +

ô

+

óò

2

cos÷ (

õ 2[ô

+

) – õ ]ñ = 0.

2

= exp  − 21 '  sin 

ï

3 2

'  .

) + 2 õ ]ñ = 0.

. + ö ( ô cos÷„ï – õ )ñ = 0.

õ 2 )ñ óò

= exp( Œ 2  ), where Œ

–1

[ô

cos÷ (

ï

ï

)+þ

1

and Œ

are roots of the

2

]ñ = 0.

+ ö ( ô cos÷„ï + õ )ñ = 0.

ö^ñóò

1 b . Particular solutions with ( > 0:

1

= cos  £ (  ,

2 b . Particular solutions with ( < 0:

1

= exp  − £ − (  ,

ñ óò ò ó ò ó

= (cos÷

ï

– ô )ñ

+ ( ô cos÷

óò ò ó

ñóò ò ó ò ó

+ ( ô cos÷„ï +

õ ï )ñ óò ò ó

ñóò ò ó ò ó

+

ô ï

cos÷„ï

ñóò ò ó ò ó

+ ( ôƒõ

ï

1

ï

– õ )ñ

cos÷„ï +

Particular solutions:

ï

+ õ (ô

2

ï

2

+ õ cos÷

óò

2

 2£   ,

= cos 

ô

cos÷„ï + õ )ñó ò ò ó + ôƒõ

1

= ,

=

2

= exp   £ − (  .

.

−





1

ñ

õ ï (ô ï

2

= sin  2

ï

1 2

2

and Œ

2

are roots of the

= 0.

= exp  − 12 ' 2  X exp 

1

2

ï ñ

= sin  £ (  .

= exp( Œ 2  ), where Œ

– ô cos÷„ï )ñó ò + 1 2

2

cos÷„ï + 2)ñó ò + ôƒõ cos÷„ï

= exp  − 12 ' 2  ,

+ (õ

ñóò ò ó

Particular solutions: 47.

'  ,

) + 3 õ ]ñ

õï ú

+

3 2

ï

¨ 
+  cos ( Œ  ) ¨ = 0. The substitution ¨ = 

 + ' +2 leads to a first-order linear equation: u

Particular solutions: 46.



2

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0. 45.

2

cos÷ (

õ 2[ô

+

'  cos 

− 12

= 0.

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp  £ −   . 1 = exp  − 

– õ [2 ô cos÷ (

ï )ñ ó ò ò ó

ï )ñ

+ ôƒõ cos÷ (

õÿñóò

+ ôƒõ cos÷ ( 1

Particular solutions:

42.

+ ö 2 ]ñ = 0.

¨ 

 +(  cos 2 + ( )¨ = 0. leads to the Mathieu equation 2.1.6.29: u

+

ï )ñ ó ò ò ó

Particular solutions:

41.

õ

)+

 The transformation ˆ = 12 Œ  , ¨ =  v J 2 (
 − ( ) leads to the Mathieu equation 2.1.6.29: ¨2Šª

Š + 4 Œ −2 V  cos(2ˆ ) +  + 3 ( 2 W ¨ = 0. 4

2 b . Particular solutions with  < 0:

40.

ï

) + õ ]ñ„ó ò – ö [ ô cos(

1 b . Particular solutions with  > 0:

39.

= 0.

 The substitution ¨ =   J 2 (
 −  ) leads to a Mathieu equation of the form 2.1.6.29: ¨2

 + V  cos(2 ) + 3  2 W ¨ = 0. 4

1 2

' 2   . M

cos÷ï + 3)ñ = 0.

 2£   .

cos÷„ï

ñóò

– ôƒõ

2

cos÷„ï

ñ

= 0.

.

© 2003 by Chapman & Hall/CRC

Page 531

532 48.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+

ô ï

ï )ñ óò ò ó

cos÷ (

2

Particular solutions: 49. 50.

51.

52.

53.

1

ï ñóò ò ó ò ó

ï

– 2ô

= ,

ï )ñ óò

cos÷ (

2

2

55.

+ ï ( ô cos 2 ï + õ )ñ„ó ò – 2( ô cos 2ï + õ )ñ = 0. ¨ 

 +(  cos 2 +  )¨ = 0. The substitution ¨ =  
−2 leads to the Mathieu equation 2.1.6.29: u

ï ñóò ò ó ò ó

+ ï ( ô cos2 ï + õ )ñó ò – 2( ô cos2 ï + õ )ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.6.30: ¨2

 + (  cos2  +  )¨ = 0.

= (cos÷ ï – ô ï )ñ ó ò ò ó + ( ô cos÷ ï – õ ï )ñ ó ò + õ cos÷ ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ quadratic equation Œ 2 + Œ +  = 0.

ï ñ óò ò ó ò ó ï ñóò ò ó ò ó

+ (ô

ï

cos÷„ï + 3)ñó ò ò ó + (2 ô cos÷ï + 1

ï 2 ñ óò ò ó ò ó

ï 2 )ñ óò ò ó

= (cos÷„ï – ô

cos   £   ,

−1

=

Particular solutions:

+ ô ï 2 cos÷ ( ï )ñó ò ò ó + õ Particular solutions: 1 =  + 1 , equation 2 − +  = 0.

cos2 ï

¡

¡

cos÷„ï

ñóò ò ó ò ó

+ ô cos2 ï

ñóò ò ó ò ó

+ ôñó ò ò ó + ôƒõÿñó ò +

Particular solutions: 57.

58.

cos ÷„ï

ñóò ò ó ò ó

60.

1

õ 2 (ô

sin   £   .

ï 2 )ñ óò

+ õ cos÷„ï

ñ

= exp( Œ 2  ), where Œ

. 1

2

3 2





,

2

'  ,

2

= cos   £   ,

= 

2





õ 2(ô

Šª

Šª
Š +  sin2 ˆ

sin 2 ˆ

= exp  − 21 '  sin 

= exp  − £ −   ,

+ ôñó ò ò ó – õ (2 ô + 3 õ cos÷„ï )ñó ò + =

are roots of the

2

¡

+ ôƒõÿñ = 0. 1

1

and Œ

– õ cos÷„ï )ñ = 0.

= exp  − 12 '  cos 

ñóò

are roots of the

2

cos÷„ï + 1)ñ = 0.

¡

2 b . Particular solutions with  < 0:

ñóò ò ó ò ó

ï

and Œ

+ õ [ ô cos÷ ( ï ) – 2]ñ = 0. =  + 2 , where 1 and 2 are roots of the quadratic

1

2

3 2

Šª

Š +

'  .

= sin   £   . 2

= exp   £ −   .

+ 2 õ cos÷„ï )ñ = 0.

.

cos÷„ï ñó ò ò ó ò ó + ñó ò ò ó + [( õ – ô 2 ) cos÷„ï + ô ]ñó ò + õ (1 – ô cos÷„ï )ñ = 0.   Particular solutions: 1 =  1 , 2 =   2 , where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + Œ +  = 0. cos÷ (

ï )ñ óò ò ó ò ó

+ô

ï 2 ñ óò ò ó

Particular solutions: 61.

2

−1

+ õ (ô

1

ñóò ò ó + õÿñóò + ôƒõÿñ = 0. z + 2 leads to an equation of the form 3.1.6.22:

+ ôñó ò ò ó + õ cos÷„ï

Particular solutions: 59.

ï ñóò

=

2

2

1 b . Particular solutions with  > 0: cos ÷„ï

õ ï )ñ óò

+ ( ô cos÷„ï – õ

ï 3 ñ óò ò ó ò ó

The substitution  = ˆ  
Š +  = 0. 56.

= 0.

= .

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0. 54.

ï )ñ

+ 2 ô cos÷ (

1

– 2ô

= ,

ï ñóò

2

+ 2 ôñ = 0.

=  2.

cos÷„ï ñó ò ò ó ò ó + ( ô cos÷„ï + ô ï + 1)ñó ò ò ó + ô  Particular solutions: 1 =  , 2 =  −  .

2

ï ñóò

–ô

2

ñ

= 0.

© 2003 by Chapman & Hall/CRC

Page 532

533

3.1. LINEAR EQUATIONS

62.

cos ÷

ï ñ óò ò ó ò ó

cos÷

ï

+ (ô

Particular solutions:

ï

cos÷„ï

63.

ñóò ò ó ò ó

1

ï

3

cos÷„ï

ñóò ò ó ò ó

1

ï

3

cos÷„ï

ñóò ò ó ò ó

1

cos÷„ï ñó ò ò ó ò ó + ô ï Particular solutions:

ï ï

3

3

67.

cos÷

ï ñ óò ò ó ò ó

+

= =

ï

2

Particular solutions:

1 2

=

−1

,

ï

,

+

ôñ

= 0.

= exp  − 21  2  X exp 

2

=

−1

1 2

 2   . M

sin   £   .

ñóò

+ 2(2 cos÷„ï – ô )ñ = 0.

ñóò

+ 6(2 cos÷„ï – ô )ñ = 0.

=  3.

2

ï )ñ ó ò

cos÷„ï + 2)ñó ò + ô (cos÷„ï + ï )ñ = 0.

=  2.

2

– 6 ï cos÷„ï −2

2

cos   £   ,

−1

– 2 ï cos÷„ï

ô ï 2 ñ óò ò ó

+

Particular solutions:

66.

+ ô ( ï + 2 cos÷

óò ò ó

= exp  − 12  2  ,

ô ï 2 ñ óò ò ó

+

Particular solutions: 65.

+ 1)ñ

+ (3 cos÷„ï + ï )ñó ò ò ó + ( ô

Particular solutions: 64.

ï

+ ï ( ô – cos÷„ï )ñó ò + ( ô – 3 cos÷„ï )ñ = 0. = cos(ln | |), 2 = sin(ln | |).

ñóò ò ó 1

(cos÷ 1

ï

= | |

2 −

+ ï [ ô – ( õ + 1) cos ÷

óò ò ó

+ ô )ñ

,

2

= | |

2

.

ï ]ñ ó ò

+ õ (2 cos÷

ï

– ô )ñ = 0.

3.1.6-3. Equations with sine and cosine. 68.

69.

70.

71. 72.

73.

+ [ ô sin( ï ) + õ ]ñ ó ò + ô cos( ï ) ñ = 0. Integrating yields a second-order linear equation: "

 + [  sin( Π ) +  ] = L the solution of the corresponding homogeneous equation with L = 0).

ñ óò ò ó ò ó

+ [ ô sin( ï ) – õ 2 ]ñ ó ò + ô [ cos( ï ) – õ sin( ï )]ñ = 0.  By integrating and substituting ¨ =   J 2 , we obtain a second-order nonhomogeneous  3 ¨

 + V  sin( Œ  ) − 14  2 W ¨ = L  3  J 2 (see 2.1.6.2 for the solution of the linear equation: corresponding homogeneous equation).

ñ óò ò ó ò ó

+ [ ô cos(2ï ) + õ ]ñ„ó ò – ô sin(2ï ) ñ = 0. = L 1 ¨ 12 + L 2 ¨ 1 ¨ 2 + L 3 ¨ 22 . Here, ¨ 1 , ¨ 2 is a fundamental set of solutions of Solution: the Mathieu equation 2.1.6.29: 4¨u

 + [  cos(2 ) +  ]¨ = 0.

ñóò ò ó ò ó

+ [ ô cos(2ï ) + õ ]ñ ó ò – 2 ô sin(2 ï ) ñ = 0. Integrating yields a nonhomogeneous Mathieu equation: "

 + [  cos(2 ) +  ] = L .

ñ óò ò ó ò ó

+ [ ô cos(2ï ) – õ 2 ]ñó ò – ô [ õ cos(2ï ) + 2 sin(2 ï )]ñ = 0.  By integrating and substituting ¨ =   J 2 , we obtain a nonhomogeneous Mathieu equation:  ¨2

 + [  cos(2 ) − 1  2 ]¨ = L  3  J 2 . 4

ñóò ò ó ò ó ñóò ò ó ò ó

– 3 ô [ ô sin2 ( õ

ï

Particular solutions: 74. 75.

(see 2.1.6.2 for

) + õ cos( õ 1

ï )]„ñ óò

= exp −



Q 

+ ô sin( õ

cos( ' )S ,

ï )[õ

2

2

+ 2ô

=  exp −

2



Q 

sin2 ( õ

ï )]ñ

= 0.

cos( ' )S .

sin( ï )ñ„ó ò ò ó ò ó + õÿñó ò ò ó + 3 ô 2 sin( ï )ñó ò + 2 ô 3 cos( ï )ñ = 0. This is a special case of equation 3.1.9.105 with  ( ) = 0.

ô

sin( ï )ñ ó ò ò ó ò ó + [ ô + (2 + 1) cos( ï )]ñ ó ò ò ó – ( 2 + 2 ) sin( This is a special case of equation 3.1.9.106 with  ( ) = 0.

ï )ñ ó ò

–

2

cos(

ï )ñ

= 0.

© 2003 by Chapman & Hall/CRC

Page 533

534 76.

THIRD-ORDER DIFFERENTIAL EQUATIONS sin2 ï

+ 3 sin ï cos ï

ñóò ò ó ò ó

+ [cos 2 ï + 4 b ( b + 1) sin2 ï ]ñó ò + 2 b ( b + 1) sin 2 ï

ñ„óò ò ó

ñ

= 0.

=L +L 2 1 2+L Here, 1 , 2 form a fundamental set of solutions of Solution: the Legendre equation 2.1.2.154, with the argument  of the functions 1 and 2 substituted by cos  . 2 1 1

2 3 2.

3.1.6-4. Equations with tangent. 77.

78.

ñóò ò ó ò ó

+

ñ óò ò ó ò ó

–ô

ô

3

tan( ô

ï )ñ

= 0.

tan( ô

ï )ñ

= 0.

Integrating yields a second-order nonhomogeneous linear equation:
Šª
Š + tan ˆ 
Š − = L cos ˆ , where ˆ =   (see 2.1.6.53 for the solution of the corresponding homogeneous equation). 3

Particular solution: 0 = cos(  ). The substitution = cos(  ) XZH ( )  leads to a secondM order linear equation of the form 2.1.6.53: H Šª

Š − 3 tan ˆtH Š
− 3 H = 0, where ˆ =  . 79.

ñ óò ò ó ò ó

+ 3ô

2

ñ óò

+ 2ô

3

tan( ô

Particular solutions: 80.

ñ óò ò ó ò ó

ôñ óò

+

1

ï )ñ

= 0.

= cos(  ),

+ ( ô – 1) tan ï

ñ

2

=  cos(  ).

= 0. = cos pXvH ( )  leads to a second-order

Particular solution: 0 = cos  . The substitution linear equation: H 

 − 3 tan 2H 
+ (  − 3) H = 0. 81.

ñ óò ò ó ò ó

ôñ óò

+

+ (ô –

2

) tan(

ï )ñ

M

= 0.

= cos( Π ) XZH ( )  leads to a secondM

Š ª Š − 3 tan ˆtH Š
+ ( Œ −2 − 3) H = 0, where ˆ = Œ  . order linear equation of the form 2.1.6.53: H Particular solution:

82.

83.

84.

ô

tan2 ï

ñóò ò ó ò ó

+

ñóò ò ó ò ó

+ [ ô tan2 (

ñóò ò ó ò ó

+

0

= cos( Π ). The substitution

– õ ( ô tan2 ï +

ñóò

õ 2 )ñ

= 0.

 The substitution ¨ =   J 2 (
 −  ) leads to a second-order linear equation of the form 2.1.6.51: ¨3

 +   tan2  + 34  2  ¨ = 0.

ï

) + õ ]ñó ò – ö [ ô tan2 (

ï

)+

õ

+ ö 2 ]ñ = 0.

 The transformation ˆ = Œ  , ¨ =  v J 2 ( 
− ( ) leads to an equation of the form 2.1.6.51: ¨2Šª

Š + Œ −2 (  tan2 ˆ +  + 3 ( 2 )¨ = 0. 4 tan÷„ï

ô

ñóò

+ tan ï ( ô tan÷„ï – 1)ñ = 0.

Particular solution: 0 = cos  . The substitution = cos  X linear equation: H 

 − 3 tan 2H 
+ (  tan  − 3) H = 0. 85.

86.

ñóò ò ó ò ó

+

ôñóò ò ó

+ ( õ tan2 ï + ö )ñó ò + ô ( õ tan2 ï + ö )ñ = 0.

The substitution ¨ = 
+  ¨2

 + (  tan2  + ( )¨ = 0.

ñóò ò ó ò ó

+

ôñóò ò ó

H ( )  leads to a second-order M

leads to a second-order linear equation of the form 2.1.6.51:

+ [3 + 2 ô tan(

Particular solutions:

1

ï )]„ñ óò

= cos( Π ),

2

+

2

{ ô [1 + 2 tan2 (

=  cos( Π ).

ï

)] + 2 tan(

ï )}ñ

= 0.

© 2003 by Chapman & Hall/CRC

Page 534

535

3.1. LINEAR EQUATIONS

87.

ñ óò ò ó ò ó

+

ï )ñ ó ò ò ó

tan(

2 b . Solution for  < 0:

ñóò ò ó ò ó

ô

+

ï )„ñ óò ò ó

tan(

+

Particular solution:

ï )ñ

– ô tan(

= 0.

= L 1 exp  − £   + L 2 exp   £   + L 3 cos( Œ  ). = L 1 cos   £ −   + L 2 sin   £ −   + L 3 cos( Œ  ).

1 b . Solution for  > 0:

88.

óò

– ôñ

õÿñóò

+ ( ô +

õ

2

–

ï )ñ

) tan(

= 0.

H

= cos( Π ). The transformation  =

0

= cos( Œ  ) X ¨

,

Œ to a second-order linear equation of the form 2.1.6.131: ¨p‘'

‘ +  Œ  Œ −2 − 3 − 2 Œ −1 tan2 H  ¨ = 0. 89.

ñ óò ò ó ò ó

tan÷ (

ô

+

ï )ñ ó ò ò ó

+

õÿñ óò

tan÷ (

ôƒõ

+

2 b . Particular solutions with  < 0:

ñ óò ò ó ò ó

tan÷ (

ô

+

ï )ñ ó ò ò ó

Particular solutions: 91.

ñóò ò ó ò ó

1

ï )ñ óò ò ó

tan÷ (

ô

+

Particular solutions: 92.

ñóò ò ó ò ó

tan÷„ï

ô

+

+

tan÷ (

ôƒõ

= exp 

ï )ñ ó ò

1



=

,

= 

2

+ ( ôƒõ tan÷„ï +

ñóò ò ó

ï

– õ [2 ô tan÷ (



ö

Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0. 93. 94.

95.

ï )ñ óò ò ó

tan÷ (

ô

õ ïƒú ñóò



3 2

'  ,

2

) + 3 õ ]ñó ò +

ï

= exp  − 21 '  sin 

2

3 2

'  .

) + 2 õ ]ñ = 0.

. = exp( Œ 2  ), where Œ

2

õ ïƒú

[ô

ï

ï

tan÷ (

+ ( ô tan÷„ï + õ )ñó ò ò ó + ö^ñó ò + ö ( ô tan÷„ï + õ )ñ = 0.

+

ï

tan÷ (

õ 2 [ô

ñóò ò ó ò ó

+

) – õ ]ñ = 0.

–1

)+þ

1

and Œ

2

are roots of the

]ñ = 0.

¨ 
+  tan ( Œ  ) ¨ = 0. The substitution ¨ = 

 + ' +2 leads to a first-order linear equation: u 1 b . Particular solutions with ( > 0:

1

= cos   £ (  ,

2 b . Particular solutions with ( < 0:

1

= exp  − £ − (  ,

ñóò ò ó ò ó ñ óò ò ó ò ó ñóò ò ó ò ó ñóò ò ó ò ó ñóò ò ó ò ó

2

= (tan÷„ï – ô )ñó ò ò ó + ( ô tan÷„ï – õ )ñó ò + õ tan÷„ï

+ ( ô tan÷

ï

+

+

ô ï

tan÷„ï

õ ï )ñ ó ò ò ó 1

+ ( ôƒõ

ï

+

ô ï

2

1

tan÷„ï + tan÷ (

Particular solutions:

ô 1

ï

2

= cos 

ï

tan÷

ï 2

 2£   ,

= , – 2ô

= ,

2

ï 2

=

−





= exp   £ − (  .

.

ôƒõ

+

tan÷

1

ï ñ

õ ï (ô ï

2

= sin  2

ï

1 2

2

and Œ

2

are roots of the

= 0. 1 2

' 2   . M

tan÷„ï + 3)ñ = 0.

 2£   .

tan÷„ï

ñóò

– ôƒõ

+ 2 ô tan÷ (

ï )ñ

2

tan÷„ï

ñ

= 0.

.

tan÷ ( 2

ñ

2

= exp  − 12 ' 2  X exp 

– ô tan÷„ï )ñó ò + 1 2

óò

+ 2)ñ

tan÷„ï + õ )ñó ò ò ó + ôƒõ

ï )ñ óò ò ó 1

+ õ (ô

= sin   £ (  .

= exp( Œ 2  ), where Œ

2

= exp  − 12 ' 2  ,

+ (õ

ñóò ò ó

Particular solutions: 99.

õ 2 [ô

+

Particular solutions: 98.



tan÷ (

+

ñóò ò ó ò ó

Particular solutions: 97.

= 0.

– õ 2 )ñó ò + ö ( ô tan÷„ï – õ )ñ = 0.

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0. 96.

2

'  cos 

− 12

M

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp   £ −   . 1 = exp  − 

1 b . Particular solutions with  > 0:

90.

ï )ñ

 leads − 3  tan H ¨2‘
+

−1

ï )ñ óò

= 0.

= .

© 2003 by Chapman & Hall/CRC

Page 535

536

THIRD-ORDER DIFFERENTIAL EQUATIONS

100.

– [ õ ( ô + tan ï ) ïw÷ + ô ]ñó ò ò ó + [ õ ( ô 2 + 1) ïƒ÷ + 1]ñó ò + ô [ õ ( ô tan ï – 1) ïw÷ – 1]ñ = 0.  Particular solutions: 1 =   , 2 = cos  .

101.

ñóò ò ó ò ó

ñóò ò ó ò ó

– ( ôƒõ tan÷„ï + õ tan÷

Particular solutions: 102. 103.

104.

105.

106.

+ [ tan( ï )( ô Particular solutions:

ñóò ò ó ò ó

1

ïw÷

1

ï ñ óò ò ó ò ó

ï

+1

+ ô )ñó ò ò ó + [ õ ( ô

 =  ,

= cos  .

2

+ 1) + ô ïƒ÷ –1 ]ñó ò ò ó – ô =  , 2 = cos( Œ  ).

2

ïƒ÷ ñóò

+ ô

2

ïƒ÷ –1 ñ

= (tan÷„ï – ô ï )ñó ò ò ó + ( ô tan÷„ï – õ ï )ñó ò + õ tan÷„ï ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ quadratic equation Œ 2 + Œ +  = 0.

ï ñóò ò ó ò ó ï ñóò ò ó ò ó

ï

+ (ô

tan÷„ï + 3)ñó ò ò ó + (2 ô tan÷„ï + −1

=

Particular solutions:

1

ï 2 ñ óò ò ó ò ó

ï 2)ñ óò ò ó

= (tan÷„ï – ô

ï 3 ñ óò ò ó ò ó

+

ô ï

ï )ñ óò ò ó

tan÷ (

2

¡

tan ÷ ï ñ ó ò ò ó ò ó

¡

+

ôñ óò ò ó

2

+ ( ô tan÷„ï –

õ ï 2 )ñ óò

+

õ ï ñóò

=  + 1,

Particular solutions: 109. tan ÷„ï

ñóò ò ó ò ó

+

ôñóò ò ó

+

ôƒõÿñ óò 1

+

2

− 12

= exp 

+ õ tan÷„ï

=  + 2 , where

+

ôƒõÿñ 1

2 b . Particular solutions with  < 0:

1

ñóò ò ó ò ó

+

ôñóò ò ó

Particular solutions: 111. tan ÷„ï

ñóò ò ó ò ó

= 0.

1

=





ï )ñ óò ò ó ò ó

ñóò ò ó ò ó

3 2

,

2

= 

ï

ñ

. 1

and Œ

are roots of the

2

) – 2]ñ = 0. and

1

¡

¡

2

are roots of the quadratic

= 0.

'  ,

= exp  − 21 '  sin 

2

= 0.

õ 2 (ô

are roots of the

2

tan÷„ï + 1)ñ = 0.

+ õ tan÷„ï

2

= exp  − £ −   ,



and Œ

sin   £   .

= cos   £   ,



ï

1

2

3 2

'  .

= sin   £   . 2

= exp   £ −   .

+ 2 õ tan÷„ï )ñ = 0.

.

+ ñó ò ò ó + [( õ – ô 2 ) tan÷„ï + ô ]ñó ò + õ (1 – ô tan÷„ï )ñ = 0.

+

ô ï 2 ñ óò ò ó

Particular solutions: 113. tan ÷„ï

ï )ñ

– õ (2 ô + 3 õ tan ÷„ï )ñó ò +

Particular solutions: 1 =   equation Π2 + Π+  = 0.

112. tan ÷ (

2

'  cos 

ñóò

−1

+ õ [ ô tan÷ (

– õ tan÷

õ 2 (ô

=

+ õ (ô

= exp( Œ 2  ), where Œ

2

1 b . Particular solutions with  > 0: 110. tan ÷„ï

õ ï )ñ óò

cos   £   ,

Particular solutions: 1 equation 2 − +  = 0.

108.

+ 1) tan÷„ï + 1]ñó ò + ô ( ôƒõ tan÷ +1 ï – õ tan÷„ï – 1)ñ = 0.

+ ï ( ô tan2 ï + õ )ñ ó ò – 2( ô tan2 ï + õ )ñ = 0. The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.6.51: ¨2

 + (  tan2  +  )¨ = 0.

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0. 107.

2

1

= ,

+ ( ô tan÷„ï +

Particular solutions:

1

– 2ô

ô ï

= ,

1



,

ï ñóò 2

=  

2

2



, where Œ

1

and Œ

2

are roots of the quadratic

+ 2 ôñ = 0.

2

= .

+ 1)ñó ò ò ó + ô

 . 2 =  −

2

ï ñóò

–ô

2

ñ

= 0.

© 2003 by Chapman & Hall/CRC

Page 536

537

3.1. LINEAR EQUATIONS

114. tan ÷

ï ñ óò ò ó ò ó

tan÷

ï

+ (ô

Particular solutions:

ï

tan÷„ï

115.

1

ï

3

tan÷ï

ñóò ò ó ò ó

1

ï

3

tan÷ï

ñóò ò ó ò ó

1

ï

3

tan÷

ï ñ óò ò ó ò ó

1

ï

3

tan÷ï

ñóò ò ó ò ó

+

– 2 ï tan÷„ï

ï

1 2

−1

=

2

ï

,

2

ñóò

−2

=

,

2

ï )ñ ó ò

+

ôñ

= 0.

= exp  − 21  2  X exp 

1 2

 2   . M

tan÷„ï + 2)ñó ò + ô (tan÷„ï + ï )ñ = 0. 2

=

−1

sin   £   .

+ 2(2 tan ÷„ï – ô )ñ = 0.

=  2.

– 6 ï tan÷„ï

ñóò

+ 6(2 tan ÷„ï – ô )ñ = 0.

=  3.

+ ï ( ô – tan÷

ô ï 2 ñ óò ò ó

+

Particular solutions: 119.

+ ô ( ï + 2 tan÷

cos   £   ,

−1

=

ô ï 2 ñ óò ò ó

+

Particular solutions: 118.

óò ò ó

= exp  − 12  2  ,

ô ï 2 ñ óò ò ó

+

Particular solutions: 117.

+ 1)ñ

+ (3 tan÷„ï + ï )ñó ò ò ó + ( ô

ñóò ò ó ò ó

Particular solutions: 116.

ï

= cos(ln  ),

ï )ñ ó ò

+ ( ô – 3 tan ÷

= sin(ln  ).

2

ï )ñ

= 0.

(tan÷„ï + ô )ñó ò ò ó + ï [ ô – ( õ + 1) tan ÷„ï ]ñó ò + õ (2 tan÷„ï – ô )ñ = 0.

Particular solutions:

1

−

=

2

,

2

=

2

.

3.1.6-5. Equations with cotangent. 120.

ñóò ò ó ò ó

ô

3

+

cot( ô

ï )ñ

= 0.

The substitution  = a + 121.

ñ óò ò ó ò ó

–ô

3

cot( ô

ï )ñ

ñóò ò ó ò ó

+ 3ô

2

ñóò

– 2ô

3

ñ óò ò ó ò ó

ôñ óò

+

125.

126.

cot2 ï

ñóò ò ó ò ó

+

ô

ñóò ò ó ò ó

+

ôñóò ò ó

ñóò

tan( a ) = 0.

leads to a linear equation of the form 3.1.6.77: "
g
g
g + 

3

tan( a ) = 0.

cot( ô 1

ï )ñ

0

= 0.

= sin(  ),

ñ

2

=  sin(  ).

= 0.

= sin  .

– õ ( ô cot2 ï +

õ 2 )ñ

= 0.

 The substitution ¨ =   J 2 (
 −  ) leads to a second-order linear equation of the form 3 ¨

2 2.1.6.81:  +   cot  + 34  2  ¨ = 0. + ( õ cot2 ï + ö )ñó ò + ô ( õ cot2 ï + ö )ñ = 0.

The substitution ¨ = 
+  ¨2

 + (  cot2  + ( )¨ = 0.

ñóò ò ó ò ó

+

cot÷„ï

ô

ñóò

Particular solution: 127.

3

+ (1 – ô ) cot ï

Particular solution: 124.



2

Particular solutions: 123.

leads to a linear equation of the form 3.1.6.78: "
g
g
g − 

= 0.

The substitution  = a + 122.



2

ñóò ò ó ò ó

+

ôñóò ò ó

leads to a second-order linear equation of the form 2.1.6.81:

+ cot ï (1 – ô cot÷„ï )ñ = 0. 0

= sin  .

+ [3 – 2 ô cot(

Particular solutions:

1

ï )]„ñ óò

= sin( Π ),

2

+

2

{ ô [1 + 2 cot2 (

=  sin( Π ).

ï

)] – 2 cot(

ï )}ñ

= 0.

© 2003 by Chapman & Hall/CRC

Page 537

538 128.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ó

–

ï )ñ óò ò ó

cot(

1 b . Solution for  > 0:

– ôñ

2 b . Solution for  < 0: 129.

ñóò ò ó ò ó

ï )ñ óò ò ó

cot÷ (

ô

+

Particular solutions: 130.

ñóò ò ó ò ó

1

ï )ñ óò ò ó

cot÷ (

ô

+

Particular solutions: 131.

ñóò ò ó ò ó

cot÷„ï

ô

+

+

óò

ï )ñ óò

cot÷ (

ôƒõ

ï





=

,

2

= 

cot÷ (

ï )ñ óò ò ó

2

õ ïƒú ñóò

3 2





ï

cot÷ (

'  ,

) – õ ]ñ = 0.

2

= exp  − 21 '  sin 

2

) + 3 õ ]ñó ò +

ï

cot÷ (

õ 2 [ô

3 2

'  .

) + 2 õ ]ñ = 0.

. + ö ( ô cot÷ï – õ )ñ = 0.

õ 2 )ñ óò

Particular solutions: 1 = exp( Œ 1  ), quadratic equation Œ 2 + Œ + ( = 0.

= exp( Œ 2  ), where Œ

2

õ ïƒú

ï

and Œ

are roots of the

2

ï

cot÷ (

ñóò ò ó ò ó

+

) + þ ]ñ = 0. The substitution ¨ = 

 + ' +* leads to a first-order linear equation: ¨u
+  cot ( Œ  ) ¨ = 0.

133.

ñóò ò ó ò ó

+ ( ô cot÷„ï + õ )ñó ò ò ó +

+

+

–1

+ ö ( ô cot÷„ï + õ )ñ = 0.

ö^ñóò

1 b . Particular solutions with ( > 0:

1

= cos   £ (  ,

2 b . Particular solutions with ( < 0:

1

= exp  − £ − (  ,

ñóò ò ó ò ó ñóò ò ó ò ó

+

ô ï

cot÷„ï

ñóò ò ó

Particular solutions: 136.

ñóò ò ó ò ó

+ ( ôƒõ

ï

1

ñ óò ò ó ò ó

+

ô ï

cot÷ (

2

1

ô ï ñóò ò ó ò ó

140.

 2£   ,

1

2

ï

– 2ô

= ,

2

1

= ,

=

−





cot÷ ( 2

2

= sin 

2

ôƒõ 2 ï

ñ

= exp   £ − (  .

.

1 2

2

1

and Œ

2

are roots of the

cot÷„ï + 3)ñ = 0.

 2£   .

cot÷„ï

ñóò

– ôƒõ

+ 2 ô cot÷ (

ï )ñ

cot÷„ï

2

ñ

= 0.

.

ï )ñ ó ò

= .

ï )]„ñ óò ò ó

õ ï (ô ï

2

–

= sin( Π ).

ï ñóò

2

+

ï ñóò ò ó ò ó

+ ï ( ô cot2 ï + õ )ñó ò – 2( ô cot2 ï + õ )ñ = 0.

ï ñóò ò ó ò ó

+ (ô

ñ

= 0.

= 0.

The substitution ¨ =  
− 2 leads to a second-order linear equation of the form 2.1.6.81: ¨2

 + (  cot2  +  )¨ = 0.

ï

cot÷„ï + 3)ñó ò ò ó + (2 ô cot÷„ï +

Particular solutions: 141.

= ,

+ [1 – ( ô + 1) ï cot(

Particular solutions: 139.

1 2

2

= exp( Œ 2  ), where Œ

– ô cot÷„ï )ñó ò +

2

= cos 

ï )ñ ó ò ò ó

Particular solutions: 138.

ï

cot÷„ï + ô cot÷„ï + õ )ñó ò ò ó +

Particular solutions: 137.

+ (õ

2

= sin   £ (  .

2

= (cot÷„ï – ô )ñó ò ò ó + ( ô cot÷„ï – õ )ñó ò + õ cot÷„ï

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0. 135.

[ô

1

132.

134.

ô

õ 2 [ô

+

= exp  − 12 '  cos 

+ ( ôƒõ cot÷„ï + ö –

ñóò ò ó

= 0.

= L 1 exp  − £   + L 2 exp  £   + L 3 sin( Œ  ). = L 1 cos   £ −   + L 2 sin   £ −   + L 3 sin( Œ  ).

– õ [2 ô cot÷ ( 1

ï )ñ

+ ô cot(

ï 2 ñ óò ò ó ò ó

= (cot÷„ï – ô

1

=

−1

ï 2)ñ óò ò ó

õ ï )ñ óò

cos   £   ,

2

+ ( ô cot÷ï –

õ ï 2 )ñ óò

Particular solutions: 1 = exp( Π1  ), quadratic equation Π2 + Π+  = 0.

2

=

−1

+ õ (ô

ï

cot÷„ï + 1)ñ = 0.

sin   £   . + õ cot÷„ï

ñ

= exp( Œ 2  ), where Œ

. 1

and Œ

2

are roots of the

© 2003 by Chapman & Hall/CRC

Page 538

539

3.1. LINEAR EQUATIONS

ï 3 ñ óò ò ó ò ó

142.

+

ô ï

ï )ñ óò ò ó

cot÷ (

2

+

õ ï ñóò

=  + 1,

Particular solutions: 1 equation 2 − +  = 0.

¡

143.

cot÷ ï ñ ó ò ò ó ò ó

¡

óò ò ó

+ ôñ

+

Particular solutions: 144. cot÷

ï ñ óò ò ó ò ó

1

+

– õ cot÷

õ 2 (ô

ï ñ óò

ï )ñ

2

'  cos 

3 2

ï

) – 2]ñ = 0. and

1

¡

'  ,

2

+ ôƒõÿñ = 0.

= cos  £   ,

2 b . Particular solutions with  < 0:

1

= exp  − £ −   ,

+ ôñó ò ò ó – õ (2 ô + 3 õ cot÷„ï )ñó ò +

Particular solutions: 146. cot÷

ï ñ óò ò ó ò ó

=





,

Particular solutions: 1 =  equation Π2 + Π+  = 0.

ï )ñ óò ò ó ò ó

147. cot÷ (

ô ï 2 ñ óò ò ó

+

Particular solutions: 148. cot÷„ï

ñóò ò ó ò ó

1

149. cot÷„ï

ñóò ò ó ò ó

+ (ô

ï

1

ï

cot÷„ï

ñóò ò ó ò ó

1

ï

3

cot÷„ï

+ô

ñóò ò ó ò ó

1

ï

3

cot÷„ï

+ô

ñóò ò ó ò ó

1

ï

3

cot÷

ï ñ óò ò ó ò ó

+ô

1

3

cot÷

ï ñ óò ò ó ò ó

+ï

=

2

1

= =

  ,

óò

1

−1

ï

,

2

= sin  £   .

,

2

ï

+ õ (1 – ô cot÷

, where Œ

=

−

2

,

ï )ñ

and Œ

1

= 0. 2

are roots of the quadratic

–ô

2

ñ

= 0. = 0.

  X exp  12   2   . M

cot÷„ï + 2)ñó ò + ô (cot÷„ï + ï )ñ = 0. 2

=

−1

sin   £   .

+ 6(2 cot÷„ï – ô )ñ = 0.

ï )ñ ó ò

+ ( ô – 3 cot÷

= sin(ln  ).

2

+ ï [ ô – ( õ + 1) cot÷

óò ò ó

ôñ

2

ñóò

=  3.

2

= exp 

− 21

+ 2(2 cot÷„ï – ô )ñ = 0.

+ ï ( ô – cot÷ + ô )ñ

= exp   £ −   .

2

ñóò

=  2.

– 6 ï cot÷„ï −2

2

cos   £   ,

= cos(ln  ),

(cot÷

Particular solutions:

−1

'  .

+ 2 õ cot÷„ï )ñ = 0.

õ 2 (ô

ô 2 ï ñ óò

2

– 2 ï cot÷„ï

ï 2 ñ óò ò ó

Particular solutions:

ï

−  . 2 = 

= exp 

ï 2 ñ óò ò ó

Particular solutions: 153.

=  2.

− 12

3 2

+ 2 ôñ = 0.

+ 1)ñó ò ò ó +

= ,

ï 2 ñ óò ò ó

Particular solutions: 152.

2

 2

= 

2

2

.

+ ô ]ñ

+ (3 cot÷„ï + ï )ñó ò ò ó + ( ô

Particular solutions: 151.

,

2

cot÷„ï + 1)ñó ò ò ó + ô ( ï + 2 cot÷„ï )ñó ò +

Particular solutions: 150.

= ,

ô ï

ï

ï ñóò

– 2ô

+ ( ô cot÷„ï +

Particular solutions:

 1





= 

2

+ [( õ – ô 2 ) cot÷

óò ò ó

+ñ

1

are roots of the quadratic

= exp  − 21 '  sin 

1

ñóò ò ó ò ó

2

¡

= 0.

1 b . Particular solutions with  > 0: 145. cot÷„ï

154.

=  + 2 , where

2

− 12

= exp 

+ õ cot÷

óò ò ó

+ ôñ

ôƒõÿñ óò

+ õ [ ô cot÷ (

=

2

.

ï )ñ ï ]ñ ó ò

= 0. + õ (2 cot÷

ï

– ô )ñ = 0.

3.1.7. Equations Containing Inverse Trigonometric Functions 1.

ñ óò ò ó ò ó

+

ôñ óò ò ó

+

õÿñ óò

+

ö^ñ

= arcsin

ï

.

This is a special case of equation 5.1.6.26.

© 2003 by Chapman & Hall/CRC

Page 539

540 2.

THIRD-ORDER DIFFERENTIAL EQUATIONS + arcsin ï ñ ó ò ò ó + ôñ ó ò + ô arcsin ï ñ = 0. 1 b . Particular solutions with  > 0: 1 = cos   £   ,

ñ óò ò ó ò ó

2 b . Particular solutions with  < 0:

The substitution ¨ = 

 + 

1

= sin   £   .

2

= exp  − £ −   ,

= exp   £ −   .

2

¨ 
+ arcsin   ¨ = 0. leads to a first-order linear equation: u

3.

+ arcsin¦ï ñó ò ò ó + ô ïƒ÷ ñó ò + ô ïƒ÷ –1 ( ï arcsin³ï + ù )ñ = 0. ¨ 
+ arcsin   ¨ = 0. The substitution ¨ = 

 +   leads to a first-order linear equation: u

4.

ñóò ò ó ò ó

ñóò ò ó ò ó

+ arcsin¦ï

ñóò ò ó

6.

ô 2 2(arcsin¦ï 3

ô

1

= exp  − 12   cos 

Particular solutions: 5.

arcsin ³ï

+

ñóò

+

  ,

2

– ô )ñ = 0. 2

2

= exp  − 12   sin 

  .

3 2

+ arcsin¦ï ñó ò ò ó – ô (2 arcsin ¦ï + 3 ô )ñó ò + ô 2 (arcsin¦ï + 2 ô )ñ = 0.   Particular solutions: 1 =   , 2 =    .

ñóò ò ó ò ó

+ arcsin¦ï ñó ò ò ó + ( ô arcsin¦ï + õ – ô 2 )ñó ò + õ (arcsin³ï – ô )ñ = 0.   Particular solutions: 1 =  1 , 2 =   2 , where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + Œ +  = 0.

ñóò ò ó ò ó

7.

= (arcsin¦ï – ô )ñó ò ò ó + ( ô arcsin ³ï – õ )ñó ò + õ arcsin¦ï ñ . The substitution ¨ = 

 +  
+  leads to a first-order linear equation: ¨ 
= arcsin   ¨ .

8.

ñóò ò ó ò ó

ñóò ò ó ò ó

+

ï

arcsin ³ï

Particular solutions: 9.

ñóò ò ó ò ó

ô ï )ñ óò ò ó

Particular solutions: 10.

+ ï 2 arcsin³ï Solution: =L

11. 12.

1

 + L 2 2 + L

3

2

 2£   ,

2

−3

= sin 

1 2

arcsin ³ï + 3)ñ = 0.

2

 2£   .

j M

= exp  − 21   X exp  2

2

– 2 ï arcsin¦ï

X 

ô ï (ï

+ ô ( ï arcsin¦ï + 2)ñó ò + ô arcsin¦ï

= exp  − 12   ,

ñóò ò ó O

1 2

– arcsin³ï )ñó ò +

2

1

ñóò ò ó ò ó

2

= cos 

1

+ (arcsin³ï +

ï

+ (ô

ñóò ò ó

+ 2 arcsin¦ï

ñóò

 − X 

−2

j

+ ( ô ï arcsin ï + arcsin ï + ô )ñ ó ò ò ó + ô  Particular solutions: 1 =  , 2 =  −  .

ñ óò ò ó ò ó

 , M P 2

ï

ñ

arcsin 

+ arccos¦ï ñó ò ò ó + ôñó ò + ô arccos¦ï ñ = 0. 1 b . Particular solutions with  > 0: 1 = cos   £   ,

The substitution ¨ = 

 + 

1

1 2

= 0.

 2   . M

= 0.

where j = exp O − X 

ñóò ò ó ò ó

2 b . Particular solutions with  < 0:

ñ

2

= exp  − £ −   ,

ï ñ óò

–ô

2

2

arcsin

arcsin  

ï ñ

 . M P

= 0.

= sin   £   . 2

= exp  £ −   .

leads to a first-order linear equation: ¨ 
+ arccos   ¨ = 0.

13.

+ arccos¦ï ñó ò ò ó + ô ïƒ÷ ñó ò + ô ïƒ÷ –1 ( ï arccos¦ï + ù )ñ = 0. The substitution ¨ = 

 +   leads to a first-order linear equation: ¨ 
+ arccos   ¨ = 0.

14.

ñóò ò ó ò ó

ñóò ò ó ò ó

+ arccos¦ï

ñóò ò ó

Particular solutions: 15.

+ 1

ô

arccos³ï

= exp 

− 12

ñóò

+

  cos 

ô 2 2 (arccos³ï 3 2

+ arccos ï ñ ó ò ò ó – ô (2 arccos ï + 3 ô )ñ   Particular solutions: 1 =   , 2 =    .

ñ óò ò ó ò ó

  ,

óò

+

ô

2

2

– ô )ñ = 0. = exp  − 12   sin 

(arccos

ï

2

3 2

  .

+ 2 ô )ñ = 0.

© 2003 by Chapman & Hall/CRC

Page 540

541

3.1. LINEAR EQUATIONS

16.

+ arccos ï ñ ó ò ò ó + ( ô arccos  Particular solutions: 1 =  1 , equation Œ 2 + Œ +  = 0.

ñ óò ò ó ò ó

ï 2

+ õ – ô 2 )ñ ó ò + õ (arccos  =  2 , where Œ 1 and Œ

ï 2

– ô )ñ = 0. are roots of the quadratic

17.

= (arccos¦ï – ô )ñó ò ò ó + ( ô arccos¦ï – õ )ñó ò + õ arccos¦ï ñ . The substitution ¨ = 

 +  
+  leads to a first-order linear equation: ¨ 
= arccos   ¨ .

18.

ñóò ò ó ò ó

ñóò ò ó ò ó

ï

arccos³ï

+

ñóò ò ó

Particular solutions: 19.

ñóò ò ó ò ó

1

+ ï 2 arccos Solution: =L

21. 22.

1

 + L 2 2 + L

3

2

= cos 

1

O

2

1 2

– arccos¦ï )ñó ò +

 2£   ,

− 12

  ,

−3

j M

−2

ñ

= 0.

  X exp   2   . M 2

1 2

ï ñ

= 0.

 , where j = exp O − Xç 2 arccos    . M P M P

j

+ ( ô ï arccos¦ï + arccos¦ï + ô )ñó ò ò ó + ô  Particular solutions: 1 =  , 2 =  −  .

2

ï

arccos¦ï

+ arctan³ï ñó ò ò ó + ôñó ò + ô arctan³ï ñ = 0. 1 b . Particular solutions with  > 0: 1 = cos   £   ,

ñóò ò ó ò ó

1

arccos¦ï + 3)ñ = 0. arccos¦ï

ô

+ 2 arccos

ñóò ò ó ò ó

2

= exp  − £ −   ,

ñóò

–ô

2

arccos¦ï

ñ

= 0.

= sin   £   . 2

= exp  £ −   .

The substitution ¨ = 

 + 

¨ 
+ arctan   ¨ = 0. leads to a first-order linear equation: u

ñ óò ò ó ò ó

óò

23.

+ arctan ï ñ ó ò ò ó + ô ï ÷ ñ The substitution ¨ = 

 +  

24.

ñóò ò ó ò ó

+ arctan³ï

ñóò ò ó

Particular solutions:

26.

− 21

= exp 

2

ï ñ óò

 − 3X\

2 b . Particular solutions with  < 0:

25.

2

 2£   .

1 2

= sin 

2

2

– 2 ï arccos

Xe

ô ï (ï

+ ô ( ï arccos³ï + 2)ñó ò +

= exp 

ï ñ óò ò ó

ñ óò ò ó ò ó

ï

ô ï )ñ óò ò ó

+ (arccos³ï +

Particular solutions: 20.

+ (ô

+ 1

ô

+ ô ï ÷ –1 ( ï arctan ï + ù )ñ = 0. ¨ 
+ arctan   ¨ = 0. leads to a first-order linear equation: u

arctan¦ï

ñóò

+ ô 2 (arctan³ï – ô )ñ = 0.

2

= exp  − 12   cos 

3 2

  ,

+ arctan³ï ñó ò ò ó – ô (2 arctan ³ï + 3 ô )ñó ò +   Particular solutions: 1 =   , 2 =    .

ñóò ò ó ò ó

ô

= exp  − 12   sin 

2 2

2

3 2

  .

(arctan¦ï + 2 ô )ñ = 0.

+ arctan³ï ñó ò ò ó + ( ô arctan³ï + õ – ô 2 )ñó ò + õ (arctan¦ï – ô )ñ = 0.   Particular solutions: 1 =  1 , 2 =   2 , where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + Œ +  = 0.

ñóò ò ó ò ó

27.

= (arctan ï – ô )ñ ó ò ò ó + ( ô arctan ï – õ )ñ ó ò + õ arctan ï ñ . ¨ 
= arctan   ¨ . The substitution ¨ = 

 +  
 +  leads to a first-order linear equation: u

28.

ñóò ò ó ò ó

ñ óò ò ó ò ó

+

ï

arctan³ï

ñóò ò ó

Particular solutions: 29.

ñ óò ò ó ò ó

+ (arctan

ï

+

Particular solutions:

1

+ (ô

2

= cos 

ô ï )ñ ó ò ò ó 1

ï

1 2

– arctan¦ï )ñó ò + ô

 2£   ,

+ ô ( ï arctan

= exp  − 12  2  ,

= sin 

2

2

ï

+ 2)ñ

ï (ï

2

arctan³ï + 3)ñ = 0.

 2£   .

1 2

óò

+

ô

arctan

= exp  − 21  2  X exp 

ï ñ 1 2

= 0.

 2   . M

© 2003 by Chapman & Hall/CRC

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542 30.

THIRD-ORDER DIFFERENTIAL EQUATIONS

=L 31. 32.

ï ñ óò ò ó

+ ï 2 arctan Solution:

ñ óò ò ó ò ó

1

 + L 2 2 + L

3

2

O

+ ( ô ï arctan Particular solutions:

ñ óò ò ó ò ó

ï

– 2 ï arctan

XZ

−3

j

ï ñ óò −2

 − 3XZ M

+ arctan ï + ô )ñ −  . 1 =  , 2 = 

ï ñ

+ 2 arctan

j

 , M P

óò ò ó

+

ô 2ï

where j = exp O − XÈ

+ arccot ï ñ ó ò ò ó + ôñ ó ò + ô arccot ï ñ = 0. 1 b . Particular solutions with  > 0: 1 = cos   £   , 1

ï ñ óò

arctan

ñ óò ò ó ò ó

2 b . Particular solutions with  < 0:

= 0.

–ô

2

2

arctan 

arctan

ï ñ

 . M P

= 0.

= sin   £   .

2

= exp  − £ −   ,

2

= exp   £ −   .

¨ 
+ arccot   ¨ = 0. leads to a first-order linear equation: u

The substitution ¨ = 

 +  33.

+ arccot³ï ñó ò ò ó + ô ïƒ÷ ñó ò + ô ïƒ÷ –1 ( ï arccot³ï + ù )ñ = 0. ¨ 
+ arccot   ¨ = 0. The substitution ¨ = 

 +   leads to a first-order linear equation: u

34.

ñ óò ò ó ò ó

ñóò ò ó ò ó

+ arccot

ï ñ óò ò ó

Particular solutions: 35. 36.

ï ñ óò

+ ô arccot 1

= exp 

− 12

+

  cos 

ô 2 2(arccot ï 3 2

  ,

– ô )ñ = 0.

2

= exp  − 12   sin 

2

3 2

+ arccot³ï ñó ò ò ó – ô (2 arccot¦ï + 3 ô )ñó ò + ô 2 (arccot³ï + 2 ô )ñ = 0.   Particular solutions: 1 =   , 2 =    .

  .

ñóò ò ó ò ó

+ arccot³ï ñó ò ò ó + ( ô arccot¦ï + õ – ô 2 )ñó ò + õ (arccot¦ï – ô )ñ = 0.   Particular solutions: 1 =  1 , 2 =   2 , where Œ 1 and Œ 2 are roots of the quadratic 2 equation Œ + Œ +  = 0.

ñóò ò ó ò ó

37.

= (arccot³ï – ô )ñó ò ò ó + ( ô arccot³ï – õ )ñó ò + õ arccot³ï ñ . The substitution ¨ = 

 +  
+  leads to a first-order linear equation: ¨ 
= arccot   ¨ .

38.

ñóò ò ó ò ó

ñóò ò ó ò ó

+

ï

arccot¦ï

Particular solutions: 39.

1

Particular solutions: 40.

+ ï 2 arccot¦ï Solution:

ñóò ò ó ò ó =L

41. 42.

1

 + L 2 2 + L

O

2

2

 2£   ,

1 2

= exp 

− 12

2

1 2

= sin 

2

arccot¦ï + 3)ñ = 0.

 2£   .

+ ô ( ï arccot¦ï + 2)ñó ò + ô arccot³ï

  , 2

– 2 ï arccot¦ï

Xç

ô ï (ï

– arccot³ï )ñó ò +

= cos 

−3

j M

2

= exp 

 − 3Xe

−2

− 21

j

ñ

= 0.

  X exp   2   . M 2

+ 2 arccot¦ï

ñóò

+ ( ô ï arccot¦ï + arccot¦ï + ô )ñó ò ò ó +  Particular solutions: 1 =  , 2 =  −  .

ñóò ò ó ò ó

1 2

ñ

= 0.

 , where j = exp O − Xç 2 arccot   . M P M P

ô 2ï

arccot¦ï

ñóò

–ô

2

arccot³ï

ñ

= 0.

+ ( ô ï 2 + õ )ñó ò ò ó + 4 ô ï ñó ò + 2 ôñ = arcsin ¦ï . Integrating the equation twice, we arrive at a first-order linear equation:

ï ñóò ò ó ò ó

 
+ (  43.

1

ñóò ò ó 3

ï

ï )ñ óò ò ó

+ (arccot¦ï + ô

ñóò ò ó ò ó

+ (ô

ñóò ò ó

ï ñóò ò ó ò ó

2

+  − 2) = L

1+L

 2 + X  ( − a ) arcsin  a a , M 0

+ ( ï arcsin¦ï + 3)ñó ò ò ó + (2 arcsin ¦ï + ô

Particular solutions:

1

=

−1

cos   £   ,

2

=

ï )ñ óò

−1



0

is any number.

+ ô ( ï arcsin¦ï + 1)ñ = 0.

sin   £   .

© 2003 by Chapman & Hall/CRC

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543

3.1. LINEAR EQUATIONS

44.

ï ñ óò ò ó ò ó

ï

+ ( ï arccos

+ 3)ñ

Particular solutions: 45.

ï ñóò ò ó ò ó

1

ï ñóò ò ó ò ó

1

1

ï 3 ñ óò ò ó ò ó


+ ( 

3

ï

2

+

2

51.

arcsin³ï

53.

ï 3 ñ óò ò ó ò ó

ï

2

+

1

arcsin³ï

1

ï 3 ñ óò ò ó ò ó

+ ï 2 arcsin³ï Particular solutions:

ï 3 ñ óò ò ó ò ó

ï

2

+

56.

58.

cos   £   ,

2

=

−1

cos   £   ,

2

=

+ ô ( ï arctan¦ï + 1)ñ = 0.

ï )ñ óò

+ ô ( ï arccot¦ï + 1)ñ = 0.

−1

−1

sin  £   .

+ 1)ñ = 0.

sin   £   . sin   £   .

=

 2 + X  ( − a ) arcsin a a , M 0

1+L

– 2ï −1

ñóò

,

2

– 6ï −2

ñóò

,

2



0

is any number.

+ 2(2 – arcsin ³ï )ñ = 0.

=  2.

+ 6(2 – arcsin ³ï )ñ = 0.

=  3.

1

+ ï (arcsin¦ï – 1)ñó ò + (arcsin¦ï – 3)ñ = 0. = cos(ln  ), 2 = sin(ln  ).

1

=

(arcsin³ï + 1)ñó ò ò ó + ï (arcsin¦ï – ô – 1)ñó ò – ô (arcsin¦ï – 2)ñ = 0.

ï 3 ñ óò ò ó ò ó

−

2 

,

2

=

2 

.

+ ï 2 (arcsin³ï + ô )ñó ò ò ó + ï ( ô arcsin ³ï + õ – ô )ñó ò + õ (arcsin³ï – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 1)B +  = 0.

ï 3 ñ óò ò ó ò ó

ï

2

+

ï ñ óò ò ó

arccos

ï 3 ñ óò ò ó ò ó

ï

2

+

arccos¦ï

ï 3 ñ óò ò ó ò ó

+ ï 2 arccos¦ï Particular solutions:

ï 3 ñ óò ò ó ò ó

ï

2

+

1

=

ñóò ò ó 1

=

– 2ï

−1

,

– 6ï −2

,

+ 2(2 – arccos

ñ óò 2

ñóò 2

2

ï )ñ

= 0.

= .

+ 6(2 – arccos¦ï )ñ = 0.

=  3.

1

+ ï (arccos³ï – 1)ñó ò + (arccos³ï – 3)ñ = 0. = cos(ln  ), 2 = sin(ln  ).

1

=

ñóò ò ó

(arccos³ï + 1)ñó ò ò ó + ï (arccos¦ï – ô – 1)ñó ò – ô (arccos³ï – 2)ñ = 0.

Particular solutions: 57.

−1

ô ï )ñ óò

−1

ï

ñóò ò ó

Particular solutions: 55.

=

ñóò ò ó

Particular solutions: 54.

=

ñóò ò ó

Particular solutions: 52.

=

+ ) =L

Particular solutions: 50.

=

+ ô ( ï arccos

+ [( ô + 6)ï 2 + õ ]ñó ò ò ó + 2(2 ô + 3) ï ñ„ó ò + 2 ôñ = arcsin ¦ï . Integrating the equation twice, we arrive at a first-order linear equation:

Particular solutions: 49.

2

ô ï )ñ ó ò

ï 3 ñ óò ò ó ò ó

 48.

cos  £   ,

+

+ ( ï arccot³ï + 3)ñó ò ò ó + (2 arccot¦ï + ô

Particular solutions: 47.

−1

ï

+ (2 arccos

+ ( ï arctan³ï + 3)ñó ò ò ó + (2 arctan³ï +

Particular solutions: 46.

=

óò ò ó

ï 3 ñ óò ò ó ò ó

−

2 

,

2

=

2 

.

+ ï 2 (arccos ï + ô )ñ ó ò ò ó + ï ( ô arccos ï + õ – ô )ñ ó ò + õ (arccos ï – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 1)B +  = 0.

ï 3 ñ óò ò ó ò ó

+

ï

2

arctan³ï

Particular solutions:

ñóò ò ó 1

=

– 2ï

−1

,

ñóò 2

+ 2(2 – arctan¦ï )ñ = 0.

=  2.

© 2003 by Chapman & Hall/CRC

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544 59.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ï 3 ñ óò ò ó ò ó

ï

2

+

ï ñ óò ò ó

arctan

Particular solutions: 60. 61.

1

ï 3 ñ óò ò ó ò ó

+ ï 2 arctan³ï Particular solutions:

ï 3 ñ óò ò ó ò ó

ï

2

+

63.

ï

(arctan

ï 3 ñ óò ò ó ò ó ï 3 ñ óò ò ó ò ó

ï

2

+

66.

ï 3 ñ óò ò ó ò ó

ï

2

+

= .

2

= 0.

+ ï (arctan³ï – 1)ñó ò + (arctan³ï – 3)ñ = 0. = cos(ln  ), 2 = sin(ln  ).

+ 1)ñ

+ ï (arctan

óò ò ó

2 −  1 = 

2  2 = 

,

1

1

+ ï 2 arccot¦ï Particular solutions: +

ï

2

(arccot

ï 3 ñ óò ò ó ò ó

−1

=

ñóò

,

2

– 6ï

ï ñ óò ò ó

ï 3 ñ óò ò ó ò ó ï 3 ñ óò ò ó ò ó

– 2ï

ñóò ò ó

arccot

Particular solutions: 67.

1

arccot¦ï

Particular solutions: 65.

,

3

ï )ñ

ï

óò

– ô – 1)ñ

ï

– ô (arctan

– 2)ñ = 0.

.

+ ï 2 (arctan¦ï + ô )ñó ò ò ó + ï ( ô arctan¦ï + õ – ô )ñó ò + õ (arctan¦ï – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 1)B +  = 0. Particular solutions:

64.

=

−2

+ 6(2 – arctan

ñ óò

ñóò ò ó

Particular solutions: 62.

– 6ï

=

−2

ñ óò

,

2

+ 2(2 – arccot³ï )ñ = 0.

=  2.

+ 6(2 – arccot 3

= .

ï )ñ

= 0.

+ ï (arccot¦ï – 1)ñó ò + (arccot³ï – 3)ñ = 0. = cos(ln  ), 2 = sin(ln  ).

ï

ñóò ò ó 1

+ 1)ñ

+ ï (arccot

óò ò ó

2 −  1 = 

2  2 = 

,

ï

– ô – 1)ñ

óò

– ô (arccot

ï

– 2)ñ = 0.

.

+ ï 2 (arccot¦ï + ô )ñó ò ò ó + ï ( ô arccot¦ï + õ – ô )ñó ò + õ (arccot¦ï – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 1)B +  = 0.

3.1.8. Equations Containing Combinations of Exponential, Logarithmic, Trigonometric, and Other Functions 1. 2. 3. 4. 5. 6.

ó

= tan ï ñ + ô (ñó ò + tan ï Particular solution: 0 = cos  .

ñóò ò ó ò ó

ó

ñ

).

ó

+ ô ñó ò ò ó + (2 ô tan ï + 3)ñ„ó ò + [ ô Particular solutions: 1 = cos  , 2 =  cos  .

ñóò ò ó ò ó

ó

ó

+ ô ñó ò ò ó + (3 – 2 ô cot ï )ñó ò + [ ô Particular solutions: 1 = sin  , 2 =  sin  .

ñóò ò ó ò ó

cosh÷ ï ñ ó ò ò ó + (2 ô cosh÷ Particular solutions: 1 = cos  ,

ï

ñ óò ò ó ò ó + ô

tan ï + 3)ñ 2 =  cos  .

óò

ó

ó

(2 tan2 ï + 1) + 2 tan ï ]ñ = 0. (2 cot2 ï + 1) – 2 cot ï ]ñ = 0.

+ [ ô cosh÷

cosh÷ ï ñó ò ò ó + (3 – 2 ô cosh ÷ ï cot ï )ñó ò + [ ô cosh÷ Particular solutions: 1 = sin  , 2 =  sin  .

ï

ñóò ò ó ò ó + ô

sinh÷ ï ñó ò ò ó + (2 ô sinh ÷ Particular solutions: 1 = cos  ,

ñóò ò ó ò ó + ô

ï

tan ï + 3)ñ„ó ò + [ ô sinh ÷ 2 =  cos  .

(2 tan2 ï + 1) + 2 tan ï ]ñ = 0.

ï

ï

(2 cot2 ï + 1) – 2 cot ï ]ñ = 0. (2 tan2 ï + 1) + 2 tan ï ]ñ = 0.

© 2003 by Chapman & Hall/CRC

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545

3.1. LINEAR EQUATIONS

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

+ ô sinh÷ ï ñó ò ò ó + (3 – 2 ô sinh ÷ ï cot ï )ñó ò + [ ô sinh÷ Particular solutions: 1 = sin  , 2 =  sin  . tanh÷ ï ñó ò ò ó + (2 ô tanh ÷ Particular solutions: 1 = cos  ,

ï

ñóò ò ó ò ó + ô

(2 cot2 ï + 1) – 2 cot ï ]ñ = 0.

ï

ñóò ò ó ò ó

tan ï + 3)ñ„ó ò + [ ô tanh÷ 2 =  cos  .

tanh÷ ï ñó ò ò ó + (3 – 2 ô tanh ÷ ï cot ï )ñó ò + [ ô tanh÷ Particular solutions: 1 = sin  , 2 =  sin  .

ï

ñóò ò ó ò ó + ô

coth÷ ï ñó ò ò ó + (2 ô coth÷ Particular solutions: 1 = cos  ,

ï

ñóò ò ó ò ó + ô

tan ï + 3)ñ„ó ò + [ ô coth÷ 2 =  cos  .

+ ô coth÷ ï ñ ó ò ò ó + (3 – 2 ô coth ÷ ï cot ï )ñ Particular solutions: 1 = sin  , 2 =  sin  .

ñ óò ò ó ò ó

óò

+ [ ô coth÷

+ ô ln÷ ï ñó ò ò ó – (2 ô ln ÷ ï tanh ï + 3)ñ„ó ò + [ ô ln÷ Particular solutions: 1 = cosh  , 2 =  cosh  . + ô ln ÷ ï ñó ò ò ó – (2 ô ln÷ ï coth ï + 3)ñ„ó ò + [ ô ln÷ Particular solutions: 1 = sinh  , 2 =  sinh  .

ï

ñóò ò ó ò ó

+ ô ln ÷ ï ñó ò ò ó + (2 ô ln ÷ ï tan ï + 3)ñ„ó ò + [ ô ln÷ Particular solutions: 1 = cos  , 2 =  cos  .

ï

ñóò ò ó ò ó

+ ô ln ÷ ï ñó ò ò ó + (3 – 2 ô ln ÷ Particular solutions: 1 = sin  ,

ñóò ò ó ò ó

ï

cot ï )ñó ò + [ ô ln÷ 2 =  sin  .

ï

ï

(2 cot2 ï + 1) – 2 cot ï ]ñ = 0.

ï

(2 tan2 ï + 1) + 2 tan ï ]ñ = 0. (2 cot2 ï + 1) – 2 cot ï ]ñ = 0.

(2 tanh2 ï – 1) + 2 tanh ï ]ñ = 0.

ï

ñóò ò ó ò ó

(2 tan2 ï + 1) + 2 tan ï ]ñ = 0.

ï

(2 coth2 ï – 1) + 2 coth ï ]ñ = 0. (2 tan2 ï + 1) + 2 tan ï ]ñ = 0. (2 cot2 ï + 1) – 2 cot ï ]ñ = 0.

cos÷ ï ñ ó ò ò ó – (2 ô cos÷ ï tanh ï + 3)ñ ó ò + [ ô cos÷ Particular solutions: 1 = cosh  , 2 =  cosh  .

(2 tanh2 ï – 1) + 2 tanh ï ]ñ = 0.

ï

ñ óò ò ó ò ó + ô

cos÷„ï ñó ò ò ó – (2 ô cos÷„ï coth ï + 3)ñ„ó ò + [ ô cos÷„ï (2 coth2 ï – 1) + 2 coth ï ]ñ = 0. Particular solutions: 1 = sinh  , 2 =  sinh  .

ñóò ò ó ò ó + ô

sin ÷„ï ñó ò ò ó – (2 ô sin ÷„ï tanh ï + 3)ñ„ó ò + [ ô sin ÷„ï (2 tanh2 ï – 1) + 2 tanh ï ]ñ = 0. Particular solutions: 1 = cosh  , 2 =  cosh  .

ñóò ò ó ò ó + ô

+ ô sin÷„ï ñó ò ò ó – (2 ô sin ÷„ï coth ï + 3)ñ„ó ò + [ ô sin÷„ï (2 coth2 ï – 1) + 2 coth ï ]ñ = 0. Particular solutions: 1 = sinh  , 2 =  sinh  .

ñóò ò ó ò ó

tan÷„ï ñó ò ò ó – (2 ô tan÷„ï tanh ï + 3)ñ„ó ò + [ ô tan÷„ï (2 tanh2 ï – 1) + 2 tanh ï ]ñ = 0. Particular solutions: 1 = cosh  , 2 =  cosh  .

ñóò ò ó ò ó + ô

tan÷ ï ñ ó ò ò ó – (2 ô tan÷ ï coth ï + 3)ñ ó ò + [ ô tan÷ Particular solutions: 1 = sinh  , 2 =  sinh  .

ï

ñ óò ò ó ò ó + ô

cot÷ ï ñ ó ò ò ó – (2 ô cot÷ ï tanh ï + 3)ñ ó ò + [ ô cot÷ Particular solutions: 1 = cosh  , 2 =  cosh  .

ñ óò ò ó ò ó + ô

ï

(2 coth2 ï – 1) + 2 coth ï ]ñ = 0.

(2 tanh2 ï – 1) + 2 tanh ï ]ñ = 0.

+ ô cot÷„ï ñó ò ò ó – (2 ô cot÷„ï coth ï + 3)ñ„ó ò + [ ô cot÷ï (2 coth2 ï – 1) + 2 coth ï ]ñ = 0. Particular solutions: 1 = sinh  , 2 =  sinh  .

ñóò ò ó ò ó

© 2003 by Chapman & Hall/CRC

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546 24. 25. 26. 27. 28.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ó

ó

+ ( õD + 2 ô ) cosh÷ ï ñó ò ò ó – ô ( õD cosh÷   Particular solutions: 1 =   , 2 =  −  +   .

ñóò ò ó ò ó

ó

ó

+ ( õD + 2 ô ) sinh ÷ ï ñó ò ò ó – ô ( õD sinh ÷   Particular solutions: 1 =   , 2 =  −  +   .

ñóò ò ó ò ó

ó

ï ï

ó

ó

+ ( õ D + 2 ô ) coth÷ ï ñ ó ò ò ó – ô ( õ D coth÷   Particular solutions: 1 =   , 2 =  −  +   .

ñ óò ò ó ò ó

ó

ï

+ ô )ñ

ó

+ ( õD + 2 ô ) ln÷ ï ñó ò ò ó – ô ( õD ln÷ ï + ô )ñó ò – 2 ô   Particular solutions: 1 =   , 2 =  −  +   .

ñóò ò ó ò ó

ó

ó

ó

29.

+( ô ln ÷ ï –2 õ )ñó ò ò ó – õ (2 ô ln÷ ï – õ +3)ñó ò + õ   Particular solutions: 1 = exp(   ), 2 =  exp(   ).

30.

ñ óò ò ó ò ó

ñóò ò ó ò ó

+ ( ô cos÷

ï

ó )ñ ó ò ò ó

– 2 õ

Particular solutions:

1

ñ óò ò ó ò ó + ( õ D ó

ó (2ô

– õ



= exp(   ),

2

cos÷

ó

32.

ñóò ò ó ò ó

Particular solutions:

34.



= exp(   ),

ï ñ

ln÷

ï ( õ  ó

ó [ô

óò

– 2ô

ñóò ò ó ò ó

ó

+ ( ô tan÷„ï – 2 õ

2

ó

Particular solutions:

ó

ó

+ 1) + ô ]ñó ò ò ó – ô 1 =  , 2 = cos  .

ó )ñ óò ò ó

1

–

õ ó (2ô 

= exp(   ),

2

ó



+ ô [

ï  ó ñóò

+

ô ó ñ

ó

ó

+ [   (cot ï + ô ) + ô ]ñ ó ò ò ó + [( ô 2 + 1)   Particular solutions: 1 =  −  , 2 = sin  .

ñ óò ò ó ò ó

ó

+ [ ô – cot ï ( ô ï  Particular solutions: 1 =  ,

ñóò ò ó ò ó

ó

+ 1)]ñó ò ò ó – ô 2 = sin  .



ó

óò

+ 1]ñ

ï  ó ñóò

+

+ ô [

ô ó ñ

ï ñ ï ñ

= 0. = 0.

–1)+2 õ

ó

ï ñ

ó

–1]ñ = 0.

– 1]ñ = 0.

= 0.

ó

– 1) + 2 õ

ñ

– 1]ñ = 0.

= 0.

tan ï – 1) – 1]ñ = 0.

= 0.

tan÷„ï – õ + 3)ñó ò ó + õ [ ô tan÷ ï ( õ  =  exp(   ).

ó

ó

ó (ô 

= 0.

= 0.

sin÷„ï

3

óò

+ 1]ñ

+ ( õD + 2 ô ) tan÷„ï ñó ò ò ó – ô ( õD tan÷„ï + ô )ñó ò – 2 ô   Particular solutions: 1 =   , 2 =  −  +   .

ñóò ò ó ò ó

cos÷

3

= 0.

– 1) + 2 õ

ó

ó

ñóò ò ó ò ó

ó

sin ÷„ï – õ + 3)ñó ò ó + õ [ ô sin ÷„ï ( õ  =  exp(   ).

– [   (tan ï + ô ) + ô ]ñ ó ò ò ó + [( ô 2 + 1)   Particular solutions: 1 =   , 2 = cos  .

ñ óò ò ó ò ó

36.

39.

ó (2ô

– õ

coth÷

ln÷

+ 2 ô ) sin ÷„ï ñó ò ò ó – ô ( õD sin ÷„ï + ô )ñó ò – 2 ô   Particular solutions: 1 =   , 2 =  −  +   .

+ [tan ï (ô ï - Particular solutions:

38.

1

ñóò ò ó ò ó + ( õD ó

35.

37.

ó )ñ óò ò ó

3

3

– õ + 3)ñ ó ò ó + õ [ ô cos÷„ï ( õ  =  exp(   ).

+ 2 ô ) cos÷ ï ñ ó ò ò ó – ô ( õ D cos÷ ï + ô )ñ   Particular solutions: 1 =   , 2 =  −  +   . + ( ô sin÷„ï – 2 õ

– 2ô

ï ñ ï ñ

tanh÷

3

ó

ï

31.

33.

óò

sinh ÷

3

+ ô )ñó ò – 2 ô

ï

cosh÷

3

+ ô )ñó ò – 2 ô

ó

+ ( õD + 2 ô ) tanh÷ ï ñó ò ò ó – ô ( õD tanh÷   Particular solutions: 1 =   , 2 =  −  +   .

ñóò ò ó ò ó

+ ô )ñó ò – 2 ô

3



ó

– 1) + 2 õ

tan÷„ï

ó

ñ

ó

– 1]ñ = 0.

= 0.

(1 – ô cot ï ) + 1]ñ = 0.

= 0.

© 2003 by Chapman & Hall/CRC

Page 546

547

3.1. LINEAR EQUATIONS

40.

ñ óò ò ó ò ó

+ ( ô cot÷

ï

– 2 õ

Particular solutions: 41. 42. 43.

ñóò ò ó ò ó + ( õD ó

ó )ñ ó ò ò ó 1

–



= exp(   ),

–[cosh÷ ï (tan ï + ô )+ ô ]ñ  Particular solutions: 1 =   ,

ñ óò ò ó ò ó

– õ + 3)ñ ó ò ó + õ [ ô cot÷„ï ( õ  =  exp(   ).

2

ó

óò ò ó

+[( ô 2 +1) cosh÷ 2 = cos  .

ñóò ò ó ò ó

ñ óò ò ó ò ó

+[sinh ÷

ï

–[tanh÷ +[tanh÷

−  , 1 = 

2

2

ï

 , 1 = 

2

(cot ï + ô )+ ô ]ñ„ó ò ò ó +[( ô

Particular solutions:

2

2

50.

ñ óò ò ó ò ó

ñóò ò ó ò ó

+ [ ô tan÷

ï

(tanh ï – õ ) – õ ]ñ

Particular solutions:

1





=

,

2

ñóò ò ó ò ó +[ô

+1]ñ

óò + ô

[sinh÷

ï (ô

ï

+1]ñó ò + ô [sinh ÷

óò ò ó

+ [ô ( õ

= cosh  .

2

ï

(1– ô cot ï )+1]ñ = 0.

ï

(1– ô cot ï )+1]ñ = 0.

ï

+1]ñó ò + ô [tanh÷

ï (ô

ï

+1]ñó ò + ô [tanh÷

ï

+1]ñó ò + ô [coth÷ +1]ñó ò + ô [coth÷

tan ï –1)–1]ñ = 0.

(1– ô cot ï )+1]ñ = 0.

ï ï

tan ï –1)–1]ñ = 0.

(tan ï –1)–1]ñ = 0. (1– ô cot ï )+1]ñ = 0.

– 1) tan÷ ï – 1]ñ ó ò + õ [ ô tan÷„ï (1 – õ tanh ï ) + 1]ñ = 0.

1

= exp  − £ − (  ,

2

= exp   £ − (  .

tan÷„ï (coth ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) tan÷„ï –1]ñó ò + õ [ ô tan÷„ï (1– õ coth ï )+1]ñ = 0. 1





=

,

2

= sinh  .

+ ( ô tan÷„ï + õ cothú ï )ñó ò ò ó + ö^ñó ò + ö ( ô tan÷„ï + õ cothú ï )ñ = 0. 1 b . Particular solutions with ( > 0: 1 = cos   £ (  , 2 = sin   £ (  .

ñóò ò ó ò ó

2 b . Particular solutions with ( < 0: 54.

ï

tan ï –1)–1]ñ = 0.

+ ( ô tan÷„ï + õ tanhú ï )ñó ò ò ó + ö^ñó ò + ö ( ô tan÷„ï + õ tanhú ï )ñ = 0. 1 b . Particular solutions with ( > 0: 1 = cos   £ (  , 2 = sin   £ (  .

Particular solutions: 53.

ï

ï

– 1]ñ = 0.

ñóò ò ó ò ó

2 b . Particular solutions with ( < 0: 52.

[cosh÷

ñóò ò ó ò ó

+[coth÷ ï (cot ï + ô )+ ô ]ñ„ó ò ò ó +[( ô 2 +1) coth÷  Particular solutions: 1 =  −  , 2 = sin  .

51.

óò + ô

= sin  .

ó

= 0.

+1]ñ

ï

+1) tanh ÷

49.

ñ ï (ô

+1) tanh ÷

–[coth÷ ï (tan ï + ô )+ ô ]ñ„ó ò ò ó +[( ô 2 +1) coth÷  Particular solutions: 1 =   , 2 = cos  .

cot÷ï

[cosh÷

= cos  .

2

−  , 1 = 

+1) sinh ÷

– 1) + 2 õ

óò + ô

= sin  .

(tan ï + ô )+ ô ]ñ„ó ò ò ó +[( ô

ï

Particular solutions:

ñóò ò ó ò ó

+[( ô 2 +1) sinh ÷ 2 = cos  .

(cot ï + ô )+ ô ]ñ„ó ò ò ó +[( ô

Particular solutions:

ñóò ò ó ò ó

óò ò ó

3

ó

+1]ñ

ï

+[cosh÷ ï (cot ï + ô )+ ô ]ñ ó ò ò ó +[( ô 2 +1) cosh÷  Particular solutions: 1 =  −  , 2 = sin  .

45.

48.

ó

ï

ñ óò ò ó ò ó

–[sinh÷ ï (tan ï + ô )+ ô ]ñ  Particular solutions: 1 =   ,

47.

cot÷

+ 2 ô ) cot÷„ï ñó ò ò ó – ô ( õD cot÷„ï + ô )ñó ò – 2 ô   Particular solutions: 1 =   , 2 =  −  +   .

44.

46.

õ ó (2ô

ñóò ò ó ò ó +[ô

1

= exp  − £ − (  ,

2

= exp  £ − (  .

cot÷„ï (tanh ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) cot÷„ï –1]ñó ò + õ [ ô cot÷„ï (1– õ tanh ï )+1]ñ = 0.

Particular solutions:

1

=





,

2

= cosh  .

© 2003 by Chapman & Hall/CRC

Page 547

548 55.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ó

cot÷

ô

+

tanhú

ï

ï ñ óò ò ó

– õÿñ

óò

1 b . Particular solutions with  > 0: 2 b . Particular solutions with  < 0: 56.

ñ óò ò ó ò ó

+[ ô cot÷

ï

(coth ï – õ )– õ ]ñ

Particular solutions: 57.

1





=

,

2

ï

ñóò ò ó ò ó

+[ ô ln÷

ñóò ò ó ò ó

ln ÷

ô

+

1

tanhú

ï

1





=

,

ï ñóò ò ó

2

–

= cosh  .

– ôƒõ ln ÷

õÿñóò

ñóò ò ó ò ó

+[ ô ln÷

Particular solutions:

1





=

,

2

2

ï

= exp   £   .

= sin   £ −   .

óò + õ [ô

cot÷

£  , 1 = exp  −  £ −  , 1 = cos 

ï

= exp   £ − (  .

2

ï ñ 2 2

= exp   £   .

= sin  £ −   .

–1]ñó ò + õ [ ô ln÷

1

= exp  − £ − (  ,

– [ln÷ ï (tan ï + ô ) + ô ]ñ ó ò ò ó + [( ô 2 + 1) ln÷  Particular solutions: 1 =   , 2 = cos  .

2

[ln÷

63.

+ [ln÷ ï (cot ï + ô ) + ô ]ñ„ó ò ò ó + [( ô 2 + 1) ln÷  Particular solutions: 1 =  −  , 2 = sin  .

ï

+ 1]ñó ò + ô [ln÷

64.

ñóò ò ó ò ó +[ô

ñóò ò ó ò ó

ñ óò ò ó ò ó

+

ô

cos÷

ï

tanhú

1





=

ï ñ óò ò ó

,

2

+

= cosh  .

õÿñ óò

1 b . Particular solutions with  > 0: 2 b . Particular solutions with  < 0:

ñóò ò ó ò ó +[ô

ï (ô ï

tan ï – 1) – 1]ñ = 0.

(1 – ô cot ï ) + 1]ñ = 0.

+

ôƒõ

cos÷

ï

tanhú

ï ñ

= 0.

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp  £ −   . 1 = exp  − 

cos÷„ï (coth ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) cos÷„ï –1]ñó ò + õ [ ô cos÷„ï (1– õ coth ï )+1]ñ = 0.

Particular solutions:

1





=

,

2

= sinh  .

+ ( ô cos÷ ï + õ cothú ï )ñ ó ò ò ó + ö^ñ ó ò + ö ( ô cos÷ 1 b . Particular solutions with ( > 0: 1 = cos  £ (  ,

ñ óò ò ó ò ó

2 b . Particular solutions with ( < 0:

ñóò ò ó ò ó

óò + ô

cos÷„ï (tanh ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) cos÷„ï –1]ñó ò + õ [ ô cos÷ï (1– õ tanh ï )+1]ñ = 0.

Particular solutions:

68.

(1– õ coth ï )+1]ñ = 0.

= exp   £ − (  .

+ 1]ñ

67.

ï

+ ( ô ln÷ ï + õ cothú ï )ñó ò ò ó + ö^ñó ò + ö ( ô ln÷ ï + õ cothú ï )ñ = 0. 1 b . Particular solutions with ( > 0: 1 = cos  £ (  , 2 = sin  £ (  .

ï

66.

(1– õ tanh ï )+1]ñ = 0.

ï

= 0.

ñ óò ò ó ò ó

65.

(1– õ coth ï )+1]ñ = 0.

ñóò ò ó ò ó

2 b . Particular solutions with ( < 0:

62.

ï

+ õ cothú ï )ñ = 0. £ ( . 2 = sin  

ï

tanhú

= sinh  .

= 0.

–1]ñó ò + õ [ ô ln÷

ï

(coth ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) ln÷

ï

2

= exp  − £ − (  ,

(tanh ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) ln÷

ï

ï ñ

–1]ñ

+ ( ô cot÷ ï + õ cothú ï )ñ ó ò ò ó + ö^ñ ó ò + ö ( ô cot÷ 1 b . Particular solutions with ( > 0: 1 = cos   £ (  ,

2 b . Particular solutions with  < 0:

61.

–1) cot÷

= sinh  .

1 b . Particular solutions with  > 0:

60.

2

ñ óò ò ó ò ó

Particular solutions: 59.

tanhú

ï

£  , 1 = exp  −  £ −  , 1 = cos  

óò ò ó +[ô ( õ

2 b . Particular solutions with ( < 0: 58.

– ôƒõ cot÷

1

ï

+ õ cothú ï )ñ = 0. £ ( . 2 = sin 

= exp  − £ − (  ,

2

= exp   £ − (  .

+[ ô sin ÷„ï (tanh ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) sin÷„ï –1]ñó ò + õ [ ô sin÷„ï (1– õ tanh ï )+1]ñ = 0.

Particular solutions:

1

=





,

2

= cosh  .

© 2003 by Chapman & Hall/CRC

Page 548

549

3.1. LINEAR EQUATIONS

69.

ñ óò ò ó ò ó

+

ô

sin ÷

ï

tanhú

ï ñ óò ò ó

+

õÿñ óò

1 b . Particular solutions with  > 0: 2 b . Particular solutions with  < 0: 70.

ñóò ò ó ò ó +[ô

sin ÷

ôƒõ

tanhú

ï

1

=





,

2

= sinh  .

+ ( ô sin÷„ï + õ cothú ï )ñó ò ò ó + ö^ñó ò + ö ( ô sin÷„ï + õ cothú ï )ñ = 0. 1 b . Particular solutions with ( > 0: 1 = cos   £ (  , 2 = sin   £ (  .

ó

1

= exp  − £ − (  ,

ó

72.

+ [ ô ï 2  ( õ – ln ï ) + 2]ñ„ó ò ò ó + ô ï  ñó ò – ô Particular solutions: 1 =  , 2 = ln  −  + 1.

73.

(   – 1)ñ ó ò ò ó ò ó – ( ô Particular solutions:

75. 76. 77. 78. 79. 80. 81. 82. 83. 84.

= 0.

£   , 2 = sin   £   . 1 = cos   £ −   , 2 = exp  £ −   . 1 = exp  − 

ñóò ò ó ò ó

2 b . Particular solutions with ( < 0:

74.

ï ñ

sin÷„ï (coth ï – õ )– õ ]ñ„ó ò ò ó +[ ô ( õ 2 –1) sin÷„ï –1]ñó ò + õ [ ô sin÷„ï (1– õ coth ï )+1]ñ = 0.

Particular solutions: 71.

+

ï ñóò ò ó ò ó ó



ó

ó

+ tan ï )ñ ó ò ò ó + (   +  , 1 =  2 = cos  .

óñ

2

= exp   £ − (  . = 0.

+ ô ( ô tan ï – 

ô 2 )ñ ó ò

ô

cosh÷ ï ñó ò ò ó ò ó + [tan ï (ô cosh ÷ ï + ï ) + 1]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = cos  .

ñóò

+ ñ = 0.

ô

cosh÷ ï ñó ò ò ó ò ó + [1 – cot ï ( ô cosh ÷ ï + ï )]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = sin  .

ñóò

+ ñ = 0.

ô

sinh ÷ ï ñó ò ò ó ò ó + [tan ï (ô sinh ÷ ï + ï ) + 1]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = cos  .

ñóò

+ ñ = 0.

ô

sinh ÷ ï ñó ò ò ó ò ó + [1 – cot ï ( ô sinh ÷ ï + ï )]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = sin  .

ñóò

+ ñ = 0.

tanh÷ ï ñ ó ò ò ó ò ó + [tan ï (ô tanh÷ ï + ï ) + 1]ñ Particular solutions: 1 =  , 2 = cos  .

–ï

ñ óò

+ ñ = 0.

tanh÷ ï ñó ò ò ó ò ó + [1 – cot ï ( ô tanh ÷ ï + ï )]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = sin  .

ñóò

+ ñ = 0.

–ï

ñ óò

+ ñ = 0.

coth÷ ï ñó ò ò ó ò ó + [1 – cot ï ( ô coth÷ ï + ï )]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = sin  .

ñóò

+ ñ = 0.

óò ò ó

ô ô

coth÷ ï ñ ó ò ò ó ò ó + [tan ï (ô coth÷ ï + ï ) + 1]ñ Particular solutions: 1 =  , 2 = cos  .

ô

óò ò ó

ô

ln÷ ï ñ ó ò ò ó ò ó + [tanh ï ( ï – ô ln ÷ ï ) – 1]ñ Particular solutions: 1 =  , 2 = cosh  .

–ï

ñ óò

+ ñ = 0.

ln÷ ï ñó ò ò ó ò ó + [coth ï (ï – ô ln ÷ ï ) – 1]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = sinh  .

ñóò

+ ñ = 0.

ô

óò ò ó

ô

ln÷ ï ñó ò ò ó ò ó + [tan ï (ô ln ÷ ï + ï ) + 1]ñó ò ò ó – ï Particular solutions: 1 =  , 2 = cos  .

ô

ñóò



ó )ñ

= 0.

+ ñ = 0.

© 2003 by Chapman & Hall/CRC

Page 549

550

THIRD-ORDER DIFFERENTIAL EQUATIONS ln÷

ô

85.

ï ñ óò ò ó ò ó

+ [1 – cot ï ( ô ln ÷

Particular solutions: cos÷„ï

ô

86.

cos÷„ï

ô ô ï

cos÷„ï sin÷„ï

ô

ñóò ò ó ò ó

sin÷

ô

ï ñ óò ò ó ò ó

ô ï

sin÷„ï tan÷

ô

ï ñ óò ò ó ò ó

tan÷„ï

ô ô ï

ñóò ò ó ò ó

tan÷„ï cot÷„ï

ô

cot÷„ï

ô

ñóò ò ó ò ó

ô ï

cot÷

= ,

2

= sinh  .

õ ï 2 )ñ óò ò ó

ln ï +

2

1

= ,

2

1

= ,

2

1

= ,

ñóò

ñóò

+ ñ = 0.

ñ óò

+ ñ = 0.

= ln  −  + 1. = cosh  .

ï

) – 1]ñ

= sinh  .

2 2

+ï

ln ï +

óò ò ó

–ï

õ ï 2 )ñ óò ò ó

= ln  −  + 1.

ï ñóò

+

– ñ = 0.

– ñ = 0.

ñóò

+ ñ = 0.

1

1

= , = ,

2

2

+ (2 ô tan÷„ï – ï 1

= ,

2

ï

) – 1]ñ

= cosh  . = sinh  . ln ï +

2

óò ò ó

õ ï 2 )ñ óò ò ó

1

= ,

2

= cosh  .

+ [coth ï (ï – ô cot÷„ï ) – 1]ñó ò ò ó – ï

ï ñ óò ò ó ò ó

1

= ,

+ (2 ô cot÷

Particular solutions:

1

ï

= ,

2

–ï 2

+ï

ñóò

ñóò

+ ñ = 0.

ñóò

+ ñ = 0.

= ln  −  + 1.

+ [tanh ï ( ï – ô cot÷„ï ) – 1]ñó ò ò ó – ï

Particular solutions: 97.

1

+ ñ = 0.

+ [coth ï (ï – ô tan ÷„ï ) – 1]ñó ò ò ó – ï

Particular solutions: 96.

2

ñóò

+ ñ = 0.

ñóò ò ó ò ó ñóò ò ó ò ó

= ,

+ ñ = 0.

ñ óò

Particular solutions: 95.

1

ñóò

–ï

Particular solutions: 94.

+ ñ = 0.

= cosh  .

+ [tanh ï ( ï – ô tan÷

Particular solutions: 93.

2

+ (2 ô sin÷„ï – ï

ñóò ò ó ò ó

Particular solutions: 92.

= ,

+ [coth ï (ï – ô sin ÷

Particular solutions: 91.

= sin  .

ñ óò

+ [tanh ï ( ï – ô sin ÷„ï ) – 1]ñó ò ò ó – ï

Particular solutions: 90.

1

+ (2 ô cos÷„ï – ï

ñóò ò ó ò ó

Particular solutions: 89.

2

–ï

+ [coth ï (ï – ô cos÷„ï ) – 1]ñó ò ò ó – ï

ñóò ò ó ò ó

Particular solutions: 88.

= ,

óò ò ó

+ [tanh ï ( ï – ô cos ÷„ï ) – 1]ñó ò ò ó – ï

ñóò ò ó ò ó

Particular solutions: 87.

1

+ ï )]ñ

ï

= sinh  .

2

ln ï +

õ ï 2 )ñ ó ò ò ó

= ln  −  + 1.

+

ï ñ óò

– ñ = 0.

– ñ = 0.

3.1.9. Equations Containing Arbitrary Functions ]

Notation:  =  ( ), = ( ), and % = % ( ) are arbitrary functions of  ;  ,  , ( , B , and Πare arbitrary parameters. 3.1.9-1. Equations of the form  3 ( )


 +  1 ( )
 +  0 ( ) = ( ). 1.

ñ óò ò ó ò ó

= ! ( ï )ñ .

The transformation  = a

−1

, = ‡"a

−2

leads to an equation of the same form: ‡ g

g
g = −a

−6

 (1 a )‡ .

© 2003 by Chapman & Hall/CRC

Page 550

551

3.1. LINEAR EQUATIONS

2.

3.

ñ óò ò ó ò ó

ô ï öï

=I ! O

+õ + ý P (ö

ñ

ï

. + ý )6   +  ¨ ¨ ªŠ

Šª
Š = ¢ , = leads to a simpler equation: u The transformation ˆ = (˜ + ( (˜ + )2 M M where ¢ =  − ˜( .

!„ñóò ò ó ò ó

–

!„ó ò ò ó ò óñ

= 0.

!„ñ óò ò ó ò ó

+

! óò òó ò ó ñ

= X H

=" .

6.

ñóò ò ó ò ó

+

!„ñóò

– ( ô! + ô 3 )ñ = 0.

 Particular solution: 0 =   . The substitution ¨ 2 ¨
3 ¨

equation:  +   + (  +  2 )¨ = 0.

ô ï (!

+

Particular solution:

0

ñóò ò ó ò ó

+

!„ñóò

+

ô 2ï

2

M

= 
− 

8.

9.

ñ óò ò ó ò ó

= exp  − 12  2  . The substitution

óò + ô!„ñ = 0.  Particular solution: 0 =  −  . The substitution ¨


¨ ¨ ¨ = 0. equation:  −   +  ñóò ò ó ò ó

+ ( ! – ô 2 )ñ

+

ï !„ñóò ï

+ (ô

11.

ñóò ò ó ò ó

+ (ô

2

=

ï

− 3  ) H = 0.

leads

leads to a second-order linear


− 2 leads to a second-order linear

+ õ ) !„ñó ò – 2 ô!„ñ = 0.

+ (! – ô

ï )ñ óò

2 2

+ (! – ô

ï ÷ )ñ ó ò

2 2

= (  +  )
 − 2 

+

ô ï (!

0

= exp  − 12  2  . The substitution

– ô [ï

leads to a

– 3 ô )ñ = 0.

to a second-order linear equation: H 

 − 3 "H 
+ (2  2 

ñ óò ò ó ò ó

M

=  +  .

0

Particular solution:

12.

= exp  − 12  2  XZH ( ) 

Particular solution: 0 = (  +  )2 . The substitution ¨ second-order linear equation: ¨u

 + (  +  )  ¨ = 0.

ñóò ò ó ò ó

 +L .

+ õ ) !„ñó ò – ô!„ñ = 0.

Particular solution: 10.

M

leads to a second-order linear

= 
+ 

– 2 !„ñ = 0.

Particular solution: 0 =  2 . The substitution ¨ equation: ¨ 

 +   ¨ = 0.

ñóò ò ó ò ó

= Xw

– 3 ô )ñ = 0.

to a second-order linear equation: H 

 − 3 "H 
+ (  + 3  2  7.

 leads to a second-order linear



 −  

+  



Integrating yields a second-order linear equation:  5.

 (ˆ )¨ ,

M

Particular solution: 0 =  . The substitution equation:  H 

 + 3  
H 
+ 3  

 H = 0. 4.

−3

÷ !



+ 3 ôù

–1

− 3  +  ) H = 0.

+ ù (ù – 1) ï

÷

–2

]ñ = 0.

M

leads



  +1 XZH ( )  P

B + 1 P M leads to a second-order linear equation: H  + 3   H 
+ (2  2  2  + 3 B/  −1 +  ) H = 0.

13.

Particular solution:

0

ï ñ óò ò ó ò ó

ï

+

ï !„ñ óò

– [( ô

Particular solution:

= exp O

B +1

+ 1) ! + ô

 . 0 =  

3

ï

 

ï 2÷

2

= exp  − 12  2  Z X H ( ) 

+1

. The substitution

= exp O

+ 3 ô 2 ]ñ = 0.

© 2003 by Chapman & Hall/CRC

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552 14.

15.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ï 2 ñ óò ò ó ò ó

+ (ï

ï 2 ñ óò ò ó ò ó

+ [ï (ô

–ô

!

2

– ô )ñó ò + ( ô – 1) !„ñ = 0.

Particular solution: 0 =  1−  . The substitution ¨ =  linear equation:  ¨3

 − (  + 1)¨2
+  ¨ = 0.

ï

Particular solution: 16.

ï (ï

+ 1)ñ„ó ò ò ó ò

ï 3 ñ óò ò ó ò ó

+

= +

0

!„ñ

+ −1

= 0.

.

+ ï ( ! – ï – 3)ñ„ó ò – ( ï + 1) !„ñ = 0.

ó

Particular solution: 17.

óò

+ 1) ! – 6]ñ


+ (  − 1) leads to a second-order

ï !„ñóò



=  .

0

+ ( ô – 1)( ! + ô

2

+ ô )ñ = 0.

Particular solution: 0 =   . The substitution ¨ =  
+ (  − 1) leads to a second-order linear equation:  2 ¨2

 − (  + 1) ¨2
+ (  +  2 +  )¨ = 0. 1−

18.

ï 6 ñ óò ò ó ò ó

+

ï 2 „! ñóò

+ (ô

Particular solution: 19.

ñóò ò ó ò ó

+ (! – ô

 

2 2

ô!

– 2ï

! )ñ

ó (!

+ 3 ô

J . 0 =  

ó )ñ óò

– ô

0

= exp O

= 0.

ñóò ò ó ò ó

+ [(1 +

õD ó ) !

ñóò ò ó ò ó

=

ñ óò ò ó ò ó

=

ñóò ò ó ò ó

+



– ô 2 ]ñó ò +



ó

+

2

)ñ = 0.

    . The substitution = exp O   X H ( )  leads to Œ

P ΠM 
P 2 2      + 3   H + (  + 2    + 3 Π ) H = 0. a second-order linear equation: H Particular solution:

21.

+

2

Particular solution:

20.

3

ô!„ñ

  + . 0 =  −

!„ñóò

+ tanh ï (1 – ! )ñ .

!„ñ óò

+ coth ï (1 – ! )ñ .

!„ñóò

+ tan ï ( ! – 1)ñ = 0.

= 0.

This is a special case of equation 3.1.9.30 with ( ) = cosh  . 22.

This is a special case of equation 3.1.9.30 with ( ) = sinh  . 23.

Particular solution: 0 = cos  . The substitution linear equation: H 

 − 3 tan 2H 
+ (  − 3) H = 0. 24.

ñóò ò ó ò ó

+

!„ñóò

ñóò ò ó ò ó

+

!„ñóò

+

!„óò ñ

0

= sin  .

=" .

Integrating yields a second-order linear equation: 26.

27.

ñóò ò ó ò ó

+ 2 !„ñó ò +

!„ó¿ò ñ



 + 

+

!„óò ñóò

=X

M

 +L .

= 0.

= L 1 ¨ 12 + L 2 ¨ 1 ¨ 2 + L 3 ¨ 22 . Here, ¨ 1 and ¨ Solution: of the second-order linear equation: 2¨u

 +  ¨ = 0.

ñóò ò ó ò ó

H ( )  leads to a second-order M

+ cot ï (1 – ! )ñ = 0.

Particular solution: 25.

= cos  X

+ ! (2 !„ó ò –

!

2

)ñ = 0.

Integrating yields a second-order linear equation:

2

are linearly independent solutions



 +  
+ 

2

= L exp O X



 . M P

© 2003 by Chapman & Hall/CRC

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553

3.1. LINEAR EQUATIONS

28.

ñóò ò ó ò ó

+ ( ô – 1) !

2

ñóò

– [ !„ó ò ò ó – (2 ô + 1) !

!ƒó ò

+ ô!

3

]ñ = 0.

Integrating yields a second-order equation:



 +  
+ ( 

2

−  
) = L exp O X



 . M P

29.

ñóò ò ó ò ó

+ ( ! – ô 2 )ñó ò + ( !„ó ò – ô! )ñ = 0. ¨ 
−  ¨ = 0. The substitution ¨ = 

 +  
 +  leads to a first-order linear equation: u

30.

ñóò ò ó ò ó

=

" óò ò ó ò ó ñ "

+ ! ( ï ) Oñó ò –


The substitution ¨ = 
−

" óò ñ " P

.

leads to a second-order linear equation.



3.1.9-2. Equations of the form  3 ( )


 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( ). 31.

+ ôñó ò ò ó + õÿñó ò + ö^ñ = ! ( ï ). This is a special case of equation 5.1.6.26 with B = 3.

32.

ñóò ò ó ò ó

+ ôñó ò ò ó + !„ñó ò + ô!„ñ = 0. ¨ 

 +  ¨ = 0. The substitution ¨ = 
+  leads to a second-order linear equation: u

33.

ñóò ò ó ò ó

34.

ñóò ò ó ò ó

+ !„ñó ò ò ó – ô 2 ( ! + ô )ñ Particular solution: 0 =  equation: ¨3

 + (  +  )¨2


ñóò ò ó ò ó

+

!„ñóò ò ó

+

ôñóò

= 0.

  . The substitution ¨ = 
−  +  (  +  )¨ = 0.

+ ô!„ñ = 0.

leads to a second-order linear

1 b . Particular solutions with  > 0: 1 = cos  £   , 2 = sin  £  2 b . Particular solutions with  < 0: 1 = exp  − £ −   , 2 = exp   The substitution ¨ = 

 +  leads to a first-order linear equation:

35.

+ !„ñó ò ò ó + ô ïƒ÷ ñó ò + ô ïƒ÷ The substitution ¨ = 

 +  

36.

ñóò ò ó ò ó

+ !„ñó ò ò ó + ô!„ñó ò + ô The substitution ¨ = 
+ 

37.

ñóò ò ó ò ó

ñóò ò ó ò ó

+

!„ñóò ò ó

+

ô!„ñóò

Particular solutions:

3

ñ

–1

£ −  . ¨u
+  ¨ = 0.

( ï ! + ù )ñ = 0. leads to a first-order linear equation: ¨ 
+  ¨ = 0.

= 0. ¨ 

 +(  −  )2 ¨ 
+  2 ¨ = 0. leads to a second-order linear equation: u

+ ô 2 ( ! – ô )ñ = 0. 1

.

= exp  − 12   cos 

2

3 2

  ,

2

= exp  − 12   sin 

2

3 2

  .

38.

ñóò ò ó ò ó

+ !„ñó ò ò ó + "Iñó ò + % = 0. ¨ 
+ ¨ + % = 0. ¨ 

 +  2 The substitution ¨ = 
leads to a second-order linear equation: u

39.

ñóò ò ó ò ó

+ !„ñó ò ò ó – ô (2 ! + 3 ô )ñ„ó ò + ô 2 ( ! + 2 ô )ñ = 0.   Particular solutions: 1 =   , 2 =    .

40.

+ !„ñó ò ò ó + ï "Iñó ò – "Iñ = 0. ¨ 

 +(  −1)¨2
+  2 ¨ = 0. The substitution ¨ =  
− leads to a second-order equation:  u

41.

ñóò ò ó ò ó

ñóò ò ó ò ó

+ !„ñó ò ò ó + (" – ô 2 )ñó ò – ô ( ô! + " )ñ = 0.  Particular solution: 0 =   . The substitution ¨ 3 ¨

2 ¨
  + )¨ = 0. equation:  + (  +  )  + ( 

= 
− 

leads to a second-order linear

© 2003 by Chapman & Hall/CRC

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554 42.

43.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+ !„ñó ò ò ó + ( ô! + õ – ô 2 )ñó ò + õ ( ! – ô )ñ = 0.   Particular solutions: 1 =  1 , 2 =  2 , where Œ equation Œ 2 + Œ +  = 0.

ñóò ò ó ò ó

+ ( ! – ô )ñó ò ò ó – ô 2 !„ñ Particular solution: 0 =  equation: ¨3

 +  ¨2
+ 

1

and Œ

2

are roots of the quadratic

= 0.

  . The substitution ¨ = 
− 

leads to a second-order linear

¨ = 0.

44.

ñóò ò ó ò ó = ( ! – ô )ñóò ò ó + (ô! – õ )ñóò + õ!„ñ . Particular solutions: 1 = exp( Œ 1  ), 2 = exp( Œ 2  ), where Œ 1 and Œ 2 are roots of the quadratic equation Œ 2 + Œ +  = 0. The substitution ¨ = 

 +  
 +  leads to a first-order linear equation: ¨3
=  ¨ .

45.

ñóò ò ó ò ó

+ ( ! – ô )ñó ò ò ó + "Iñó ò – ô ( ô! + " )ñ = 0.  Particular solution: 0 =   .

46.

ñóò ò ó ò ó

+ ( ! + ô )ñó ò ò ó + ( ô! + " )ñó ò + ô"Iñ = 0.  Particular solution: 0 =  −  .

47.

ñóò ò ó ò ó

+

ï !„ñóò ò ó

ï

+ (ô

Particular solutions: 48.

ñ óò ò ó ò ó

ï

+ (ô

1

Particular solution: 49.

ñ óò ò ó ò ó

+ (! +

ô ï )ñ óò ò ó

0

ñ óò ò ó ò ó

+

ï 2 „! ñ óò ò ó

– 2ï

Particular solutions: Solution: = L 1  + L 51.

ñ óò ò ó ò ó

+ (! +

ô ï )ñ óò ò ó

!„ñ óò

2

=

+ ô (ï

Particular solutions: 50.

= cos 

+ï

óò ò ó

+ õ ) !„ñ

– ! )ñó ò + ô

2

1

 £  ,

2

óò

+

ô!„ñ

= exp  − 12  2  ,

2

!„ñ óò

1 2

 2£   .

= 0. = exp  − 21  2  X exp 

2

+ (" + 2 ô ) ñ

óò

M

+ ô [ï

 − 3XZ + (1 – ô

"

−2

j

− 12

53.

ñóò ò ó ò ó

ñóò ò ó ò ó

ï

2

+

õï

Particular solution:

2

ô 2 ï „! ñóò

–ô

−  . 2 = 

2

!„ñ

ï 2 ) ! ]ñ

= 0.

0

2

= 

ï

= 0.

+ ' + ( .

+ ï ( ï ! + " )ñó ò ò ó – "Iñó ò – 2 !„ñ = 0. Particular solution: 0 =  2 . The substitution ¨   ¨ = 0. equation: ¨3

 +  (  + )¨2
+ – ï (ô

leads to a second-order

+ ö ) !„ñó ò ò ó – 2 ô!„ñ = 0.

ñóò ò ó ò ó ñóò ò ó ò ó

 2   . M

 , where j = exp O − Xw 2   . M P M P

  . The substitution ¨ = 
+  Particular solution: 0 = exp  2 ¨
3 ¨

linear equation:  +   + ( −   )¨ = 0.

+ (ô

1 2

+ 2 !„ñ = 0.

=  , 2 =  2. 2 XÈ −3 j 2 + L 3 O  1

= sin 

– 2 !„ñ = 0. + 2)ñ

!

+ ( ô ï ! + ! + ô )ñó ò ò ó + Particular solutions: 1 =  ,

55.

+ 3)ñ = 0.

2

+ 2  +  .

52.

54.

1 2

ï (ï 2 !

=


− 2 leads to a second-order linear

+ õ ) !„ñó ò ò ó + ( õ – ô 2 ) !„ñó ò + 2 ô!„ñ = 0.

Particular solution:

0

=

2

+  + 12 ( 

2

−  ).

© 2003 by Chapman & Hall/CRC

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555

3.1. LINEAR EQUATIONS

56.

– [(2 ï + ô ) ! + ( ï

ñóò ò ó ò ó

Particular solution: 57.

ï ñ óò ò ó ò ó

ï

+ ï (ô

óò ò ó

+ 3ñ

Particular solution: 58.

60.

2

64.

65.

ï ñóò ò ó ò ó

+ (ï

ï ñóò ò ó ò ó

+ (ï

ï ñ óò ò ó ò ó

+ (ï

−1

ï

ï

– (ô

2

– 1) !„ñ = 0.

.

1

=

+ (ô

ï !

1

=

+L

2

ô 2 )ñ óò

  + X  ( − a )  (a ) a , M 0

ô ï )ñ óò

+ ô (ï

cos( £  ),

−1

− 21

exp 

2

=

2

! =

ï ! 2

  cos 

+ 2! – ô ,

ï

– [ô (ô

+ 2) !„ñó ò + ô ( ô −1

1

óò ò ó

+ 3)ñ

!

2

ï )ñ ó ò

+ 2) ! + ( ô

ï



0

is any number.

+ 1)" ]ñ = 0.

+ 1)ñ = 0. −1

sin( £  ).

+

!

3 2

–ô

2

  ,

ï )ñ

2

=

= 0. −1

exp  − 12   sin 

2

3 2

  .

+ ô ( ! – ô )ñ = 0.

   .

−1 −

+ ( ï ! + 3)ñó ò ò ó + (2 ! + ô ïƒ÷ +1 )ñó ò + ô ïƒ÷ ( ï ! + ù + 1)ñ = 0. ¨ 

 +   2 ¨ 
+ The substitution ¨ =  leads to an equation of the form 3.1.9.35: ¨I


 +  2 ¨  −1  (  + B ) = 0.

ï ñóò ò ó ò ó ï ñóò ò ó ò ó

+ ( ï 2 ! + ô + 2)ñó ò ò ó – ô ( ô + 1) !„ñ = 0. Particular solution: 0 =  −  . The substitution ¨ =  equation: ¨ 

 +   ¨ 
− (  + 1)  ¨ = 0.

ï ñóò ò ó ò ó

+ [ï

2

ï

(ô

2

ï ñ óò ò ó ò ó

+ [ï (ô

ï

0

(ô

ï

– 1)ñ„ó ò ò ó ò

ó

ï 2 ñ óò ò ó ò ó

+

ï 2 ñ óò ò ó ò ó

+ ï [( ô

ï !„ñóò ò ó

+ ï [ï (ô

ï 2 ñ óò ò ó ò ó

+ ï (ï ! By integrating,  2 

 + (  − 2)

=  + 

– 1) ! + ï

2


+ 

leads to a second-order linear

+ 1) ! + 3]ñ„ó ò ò ó – 2 !„ñ = 0.

ï

0

ï

2

(ô

– 2) ! – ô 1

0

2

= ,

ï

−1

.

2

+ 1)" + 3]ñ

ï

=  + 

+ [ï (ô

Particular solution:

70.

=

+ 3)ñó ò ò ó + ( ô

!

Particular solution: 69.

=  + 

1

Particular solutions: 68.

−1

+ 3)ñó ò ò ó + (2 ! +

!

Particular solution: 67.

óò

+ 1) !„ñ

+ ï ( ! – 2 ô )ñó ò ò ó + ï (" +  Particular solution: 0 =    .

Particular solution: 66.

+  +  .

ï ñóò ò ó ò ó

Particular solutions:

63.

+ õ )" ]ñó ò ò ó + 2 !„ñó ò + 2"Iñ = 0.

+  − 2) = L

Particular solutions: 62.

2

ô ï

+ ( ô ï 2 + õ )ñó ò ò ó + 4 ô ï ñó ò + 2 ôñ = ! ( ï ). Integrating the equation twice, we arrive at a first-order linear equation:

Particular solutions: 61.

+

= 2

0

ï ñóò ò ó ò ó

 
+ (  59.

0

2

−1

óò ò ó

– 2 !„ñ

óò

– 2"Iñ = 0.

ï 2)!

+

ô 2]ñ óò

.

2

]ñó ò ò ó + [(2 – ô

 . 2 = 

2

+ 2ô (ô

ï

– 1) !„ñ = 0.

+ 1)" + 2 ! – 6]ñ„ó ò + "Iñ = 0.

= +

−1

.

+ 1) ! + 3]ñ„ó ò ò ó – 2 !„ñ = 0. 0

= +

−1

.

+ ô )ñ ó ò ò ó + [( ô – 2)ï ! + õ ]ñ ó ò + ( õ – ô + 2) !„ñ = 0. we obtain the second-order nonhomogeneous Euler equation 2.1.9.15: 
+ (  −  + 2) = L exp O − X   .

M P

© 2003 by Chapman & Hall/CRC

Page 555

556 71.

72.

73. 74.

THIRD-ORDER DIFFERENTIAL EQUATIONS ( ô ï + õ )ï ñ óò ò ó ò ó + (  ï +  )ñ óò ò ó + ï ñ óò + ñ = ! . Integrating yields a second-order linear equation: (  +  ) ( + 2  − ƒ ) = X¯  + L . + ô ï 2 ñ óò ò ó

ï 3 ñ óò ò ó ò ó

+ ( ô + 2) ï 2 ñó ò ò ó + ï !„ñó ò + ô!„ñ = 0. Particular solution: 0 =  −  .

ï 3 ñ óò ò ó ò ó

+ [( ô + 6)ï 2 + õ ]ñó ò ò ó + 2(2 ô + 3) ï ñ„ó ò + 2 ôñ = ! ( ï ). Integrating the equation twice, we arrive at a first-order linear equation: 3


+ ( 

−2

ï 3 ñ óò ò ó ò ó

 
+ (  +



−2 −1

ï 2 „! ñóò ò ó

ï 3 ñ óò ò ó ò ó

+

ï 2 „! ñóò ò ó

79.

81.

82.

83.

ï 3 ñ óò ò ó ò ó

1

+L

2

  + X  ( − a )  (a ) a , M 0



0

is any number.

1

ñóò

=

ñóò

=

1

+L

2

  + X  ( − a )a −2  0

−3

+ 2(2 – ! )ñ = 0. −1

,

2

 (a ) a , M 

0

is any number.

=  2.

+ 6(2 – ! )ñ = 0. −2

,

2

=  3.

+ ï 2 !„ñó ò ò ó + ï ( ! – 1)ñó ò + ( ! – 3)ñ = 0. Particular solutions: 1 = cos(ln | |), 2 = sin(ln | |).

ï 3 ñ óò ò ó ò ó

+

ï 2(!

+ 1)ñó ò ò ó + ï ( ! – ô – 1)ñ„ó ò – ô ( ! – 2)ñ = 0.

ï 3 ñ óò ò ó ò ó

1

=

−

2 

,

=

2

2 

.

+ ï 2 ( ! + ô )ñó ò ò ó + ï ( ô! + õ – ô )ñó ò + õ ( ! – 2)ñ = 0. Particular solutions: 1 =   1 , 2 =   2 , where B 1 and B 2 are roots of the quadratic equation B 2 + (  − 1)B +  = 0.

ï 3 ñ óò ò ó ò ó

+ ï 2 ( ! + ô )ñó ò ò ó + ï [" + ( ô – 1) ! ]ñ„ó ò + ( ô – 2)"Iñ = 0. Particular solution: 0 =  2−  . The substitution ¨ =  
+ (  − 2) leads to a second-order linear equation:  2 ¨2

 +   ¨2
+ ¨ = 0.

+ ï 2 ( ! + 2 ô ï )ñ ó ò ò ó + ï (2 ô ï ! + ô 2 ï 2 + õ )ñ ó ò + ( ô 2 ï 2 ! + õ! – 2 õ )ñ = 0.   Particular solutions: 1 =  −    1 , 2 =  −    2 , where B 1 and B 2 are roots of the quadratic equation B 2 − B +  = 0.

ï 3 ñ óò ò ó ò ó

ó

+ ô ñó ò ò ó – 3 2 ñó ò + 2 3 ñ = ! ( ï ). Integrating the equation twice, we arrive at a first-order linear equation:

ñóò ò ó ò ó

 − 84.

1

– 6ï

Particular solutions:

80.

+ ) =L

– 2ï

Particular solutions:

78.

+ ) =L

+ ï 2 ( õ ï 2 D +1 – 3 ô )ñó ò ò ó + 2 ô ( ô + 1)(2 ô + 1)ñ = ! ( ï ). Integrating the equation twice, we arrive at a first-order linear equation:

Particular solutions: 77.

2

ï 3 ñ óò ò ó ò ó 

76.

M

+ õ ï ñó ò + ö^ñ = ! ( ï ). Nonhomogeneous Euler equation. The substitution a = ln | | leads to an equation of the form g 3.1.9.31: g

g
g + (  − 3) g

g + (  −  + 2) g
+ ( =  (A  ).

ï 3 ñ óò ò ó ò ó

 75.



 + [( ƒ − 2  ) + ý −  ] 
+


  + ( + 2 Œ  − ) = L

ñ óò ò ó ò ó

1+L

+ ( ! + ô )ñ ó ò ò ó + [ ô! + (1 + õ  Particular solution: 0 =  −  +  .

D

 2 + X  ( − a )  −  g  (a ) a , M 0

ó )" ] ñ ó ò



0

is any number.

+ ô"Iñ = 0.

© 2003 by Chapman & Hall/CRC

Page 556

557

3.1. LINEAR EQUATIONS

85.

ñóò ò ó ò ó

ó

+ ( õD

 , 1 = 

Particular solutions: 86.

ñóò ò ó ò ó

+( ! –2 ô

ñóò ò ó ò ó

+ ô )ñó ò – 2 ô

+ ( ! – ô

= exp O

1

ó )ñ óò ò ó

3

−  +  . 2 = 

ó )ñ óò ò ó – ô  ó (2 ! – ô  ó

Particular solutions: 87.

ó !

+ 2 ô ) !„ñó ò ò ó – ô ( õD

Œ





  



=  exp O

2

ó )ñ óò

+ (" – 2 ô

= 0.

ó [(ô  ó

+3 )ñó ò + ô

,

P

!„ñ

Œ





 

ó

–

2

]ñ = 0.

.

P

ó [(ô  ó

– ô

– ) ! +2 ô

+ )! + " +

2





]ñ = 0.

X H ( )  leads to Z   . The substitution = exp O   Œ

P ΠP  M  
 2 2 a second-order linear equation: H  + (  + 2   ) H  + (2    + +    + Π ) H = 0. Particular solution:

88.

ñóò ò ó ò ó

– [( ô + ö +

õD ó ) !

ó

 ñóò ò ó ò ó

+ (2 

ó

– ô + ö ]ñó ò ò ó + [( ö

v , 1 = 

Particular solutions: 89.

= exp O

0

+ J



ó

–ô

2

2

−  + ( . 2 = 

+ )ñó ò ò ó + (

ó



2

+

õ^öD ó ) !

+ 2J C

– ô„ö ]ñó ò + ô„ö ( ô + ö ) !„ñ = 0.

ó )ñ óò 

+ J

2

ó

  ñ

= ! ( ï ).

Integrating the equation twice, we arrive at a first-order linear equation:

  90.

ñóò ò ó ò ó

+


  + (ý   + = ) = L

!„ñóò ò ó

1

+L

2

ï )("

+ "Iñó ò – [ #! + tanh(

Particular solution:

0

  + X  ( − a )  (a ) a , M 0 2

+

ñóò ò ó ò ó

+

!„ñóò ò ó

Particular solutions: 92.

ñ óò ò ó ò ó

+

!„ñ óò ò ó

1

ñ óò ò ó ò ó

1

ñóò ò ó ò ó ñóò ò ó ò ó ñóò ò ó ò ó

ï )(ï !

+ [ coth(

1

ï )(ï !

Particular solutions: 97.

ï ñ óò ò ó ò ó ñ óò ò ó ò ó

2

+ [( ô 2

+ [ï

2

+

!„ñ óò ò ó

= ,

Particular solution:

óò

2

= ,

1

= ,

2

óò ò ó 2

óò

– 1) ! – 1]ñ

2

M

leads to a

+ 2 Œ" tanh( Œ  )] H = 0.

) – 1] ! + 2 tanh( ) – 1] ! + 2 coth(

ï )}ñ

ï )}ñ

= 0. = 0.

+ ô [(1 – ô tanh ï ) ! + 1]ñ = 0.

– 1) ! – 1]ñ„ó ò + ô [(1 – ô coth ï ) ! + 1]ñ = 0.

= sinh  .

ï !„ñóò

+

2

!„ñ

= 0.

ï !„ñóò

+

2

!„ñ

= 0.

!„ñ

= 0.

2

= cosh( Π ). 2

= sinh( Π ). +

ï !„ñ óò

–

ï )("

–

= ln  −  + 1.

+ [ #! + tan( 0

ï

{[2 coth2 (

= cosh  . 2

ï

{[2 tanh2 (

=  sinh( Π ).

– 1) – ! ]ñ„ó ò ò ó –

( ô – ln ï ) ! + 2]ñ + "Iñ

2

2

2

+

– 1) – ! ]ñ„ó ò ò ó –

1

Particular solutions: 98.

óò ò ó

 , 1 = 

Particular solutions: 96.

= sinh( Π ),

+ [(coth ï – ô ) ! – ô ]ñ„ó ò ò ó + [( ô + [ tanh(

óò

2

=  cosh( Π ).

2

) + 3 ]ñ

 , 1 = 

Particular solutions: 95.

ï

+ [(tanh ï – ô ) ! – ô ]ñ

Particular solutions: 94.

) + 3 ]ñ„ó ò +

= cosh( Π ),

– [2 ! coth(

Particular solutions: 93.

ï

is any number.

= cosh( Π ) XZH ( ) 

= cosh( Π ). The substitution

– [2 ! tanh(

0

)]ñ = 0.

second-order linear equation: H 

 + [  + 3 Πtanh( Π )] H 
+ [ + 3 Π91.



2

)]ñ = 0.

= cos( Π ). The substitution

= cos( Π ) XZH ( )  leads to a second-

order linear equation: H 

 + [  − 3 Œ tan( Œ  )] H 
+ [ − 3 Œ

2

M

− 2 Œ" tan( Œ  )] H = 0.

© 2003 by Chapman & Hall/CRC

Page 557

558 99.

THIRD-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ó

+

!„ñóò ò ó

Particular solutions: 100.

ñóò ò ó ò ó

+

!„ñóò ò ó

 , 1 = 

−  , 1 = 

ñóò ò ó ò ó

+ [! +

ï )(ï !

tan(

ñóò ò ó ò ó

+ [! –

ï )(ï !

cot(

Particular solutions: 105.

ô

ï )„ñ óò ò ó ò ó

sin(

+

õÿñóò ò ó

2

+ 3 ô

2

– 2 cot(

= 0.

{[1 + 2 cot2 (

ï )}ñ

= 0.

+ 1) ! + 1]ñ„ó ò + ô [(1 – ô cot ï ) ! + 1]ñ = 0.

2

= sin  . 2

ï !„ñóò

!„ñ

= 0.

!„ñ

= 0.

+

2

+

2

= cos( Π ).

+ 1)]ñ„ó ò ò ó –

= ,

1

ï )] !

ï )}ñ

= cos  .

2

2

+ 2 tan(

=  sin( Π ).

+ 1)]ñ„ó ò ò ó –

= ,

1

ï )] !

{[1 + 2 tan2 (

+ 1) ! + 1]ñ„ó ò + ô [( ô tan ï – 1) ! – 1]ñ = 0.

2

+ [(cot ï + ô ) ! + ô ]ñ„ó ò ò ó + [( ô

ñóò ò ó ò ó

Particular solutions: 104.

2

2

+

2

– [( ô + tan ï ) ! + ô ]ñ„ó ò ò ó + [( ô

ñóò ò ó ò ó

Particular solutions: 103.

= sin( Π ),

1

2

=  cos( Π ).

2

ï )]„ñ óò

+ [3 – 2 ! cot(

Particular solutions: 102.

) + 3 ]ñ„ó ò +

= cos( Π ),

1

Particular solutions: 101.

ï

+ [2 ! tan(

2

ï !„ñóò

= sin( Π ).

sin(

ï )ñ óò

+ 2 ô

3

ï )ñ

cos(

= ! (ï ).

Integrating the equation twice, we arrive at a first-order linear equation:

 sin( Π ) 
+ [  − 2 Œ cos( Œ  )] = L 106. sin(

ï )ñ ó ò ò ó ò ó

1

+L

ï )]ñ óò ò ó

2

+ [ ô + (2 + 1) cos(

  + X  ( − a )  (a ) a , M 0

–(

2

+ 2 ) sin(

ï )ñ ó ò

–



0

2

cos(

is any number.

ï )ñ

= ! (ï ).

Integrating the equation twice, we arrive at a first-order linear equation: sin( Π ) 
+ [  + cos( Œ  )] = L 107. ( ! – 1)ñ

óò ò ó ò ó

– [ ô! +

Particular solutions: 108.

ñ óò ò ó ò ó

+

!„ñ óò ò ó

+ "Iñ

óò

ï )]ñ óò ò ó

tan(

 , 1 =  + ( !" + "

2

ó ò )ñ

+L

2

+(

2

1

  + X  ( − a )  (a ) a , M 0

!

+

ô 2 )ñ ó ò

110.

ñóò ò ó ò ó

+ 3 !„ñó ò ò ó + ( !„ó ò + 2 ! 2 1

+ L 2¨ 1¨

) – #! ]ñ = 0.

= 0.



 +

= L exp O − Xv

 . M P

+ 2" )ñó ò + (2 !" + "Ió ò )ñ = 0.

+ ( ! + " )ñó ò ò ó + ( !„ó ò +

!"

+ % )ñó ò + ( %„òó + "/% )ñ = 0.



 +  
+ %

Integrating yields a second-order linear equation: 111.

ï

is any number.

¨ 2 ¨ ¨ Solution: 2 + L 3 2 . Here, 1 , 2 is a fundamental set of solutions of the 2 ¨
u ¨

1 second-order linear equation:  +   + 2 ¨ = 0.

ñóò ò ó ò ó

= L 1¨

2

+ ô [ ô tan(

0

= cos( Π ).

Integrating yields a second-order linear equation: 109.



ñ óò ò ó ò ó

+ ( ! + " )ñ

óò ò ó

+ (2"

óò

The substitution ¨ = 
+

+

!"

+ % )ñ

óò

+ ("

óò ò ó

+

!" óò

= L exp O − Xw

 . M P

+ "/% )ñ = 0.

¨ 
+ % ¨ = 0. ¨ 

 +  2 leads to a second-order linear equation: u

© 2003 by Chapman & Hall/CRC

Page 558

3.2. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò˜Í Ò

=

 Ë Ÿ Ï¢¡

(Ï

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

559

3.2. Equations of the Form í ïwî²î²ïwî ï = † ì  í  ( í­ï î ) ( í­ïwî²î ï ) S 3.2.1. Classification Table Table 28 presents below all solvable equations whose solutions are outlined in Subsections 3.2.2– 3.2.4. Two-parameter families (in the space of the parameters ƒ , ý , = , and > ), one-parameter families, and isolated points are represented in a consecutive fashion. Equations are arranged in accordance with the growth of > , the growth of = (for identical > ), the growth of ý (for identical > and = ), and the growth of ƒ (for identical > , = , and ý ). The number of the equation sought is indicated in the last column in this table. In Subsections 3.2.2–3.2.4, the value of the insignificant parameter # is in many cases defined in the form of a function of two (one) auxiliary coefficients  and  ,

# = Y ( ,  )

(1)

and the corresponding solutions are represented in parametric form,

 =  1 (U , L 1 , L 2 , L 3 ,  ),

=  2 (U , L

1,

L 2 , L 3 ,  ),

(2)

where U is the parameter, L 1 , L 2 , and L 3 are arbitrary constants, and  1 and  2 are some functions. Having fixed the auxiliary coefficient sign  > 0 (or  > 0), one should express the coefficient  in terms of both # and  with the help of

 = j ( # ,  ). Substituting this formula into (2) yields a solution of the equation under consideration (where the specific numerical value of the coefficient  can be chosen arbitrarily). The case  < 0 (or  < 0), which may lead to a branch of the solution or to a different domain of definition of the variables  and in (2), should be considered in a similar manner. TABLE 28 Solvable equations of the form 


 = #* 9

>

=

arbitrary

arbitrary

arbitrary ( > ≠ 2) = + 4ý + 5 = + 2ý + 3 3= + 7 2(= + 2) 3= + 7 2(= + 2) arbitrary ( > ≠ 1, 2) arbitrary ( > ≠ 2) 3ý + 4 2ý + 3 arbitrary ( > ≠ 32 )

arbitrary (= ≠ −1) arbitrary (= ≠ −1)

: ( 
); ( 

 ) < ƒ

Equation

0

arbitrary

3.2.4.15*

0

0

3.2.4.1

arbitrary (ý ≠ −1)

0

3.2.4.174

arbitrary (= ≠ −2)

− 12

0

3.2.4.10

arbitrary (= ≠ −2)

1

0

3.2.4.7

−1

−1

0

3.2.4.175

−1

0

0

3.2.4.11

0

arbitrary (ý ≠ − 23 )

0

3.2.4.8

0

− 12

0

3.2.4.87

ý

* Given are formulas of reduction to the generalized Emden–Fowler equation.

© 2003 by Chapman & Hall/CRC

Page 559

560

THIRD-ORDER DIFFERENTIAL EQUATIONS TABLE 28 (Continued) Solvable equations of the form 


 = #* 9 : (
 ); (

 )


=

ƒ ý

Equation

1

arbitrary (ý ≠ −1)

0

3.2.4.2

1

−1

0

3.2.4.13

1

1

0

3.2.4.4

3

arbitrary (ý ≠ − 23 )

0

3.2.4.9

−1

3

− 75

0

3.2.4.168

−1

3 arbitrary (= ≠ −1)

0

0

3.2.4.164

0

0

3.2.4.3

0

arbitrary

− 41 (= + 5)

0

3.2.4.171

0

−2ý − 5

arbitrary (ý ≠ −2)

0

3.2.4.5

0

−13

1

0

3.2.4.153

0

−13

3

0

3.2.4.155

0

−7

0

0

3.2.4.141

0

−7

1

0

3.2.4.145

0

−4

− 12

0

3.2.4.127

0

−4

0

0

3.2.4.123

0

−3

−2

0

3.2.4.95

0

−3

−1

0

3.2.4.30

0

−3

0

0

3.2.4.26

0

−3

1

0

3.2.4.91

0

− 73 − 73 − 73 − 73 − 73 − 73 − 73 − 73 − 95 − 95

− 10 3 − 73 − 43 − 56 − 12

0

3.2.4.76

0

3.2.4.42

0

3.2.4.52

0

3.2.4.133

0

3.2.4.131

0

0

3.2.4.48

1

0

3.2.4.38

2

0

3.2.4.70

− 13 5

0

3.2.4.64

1

0

3.2.4.60

arbitrary ( > ≠ 1) arbitrary ( > ≠ 1) arbitrary ( > ≠ 2) 3ý + 4 2ý + 3

0

0 0 0 0 0 0 0 0 0

© 2003 by Chapman & Hall/CRC

Page 560

3.2. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò˜Í Ò

=

 Ë Ÿ Ï¢¡

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

(Ï

561

TABLE 28 (Continued) Solvable equations of the form 


 = #* 9 : (
 ); (

 )


= ý

0

−1

−2

0

3.2.4.22

0

−1

0

0

3.2.4.18

0

0

0

3.2.2.2

0

0

3

3.2.3.3

0

0

− 72 − 72 − 52

0

3.2.2.3

0

0

− 52

1

3.2.3.4

0

0

−2

0

3.2.2.6

0

0

− 43

− 43

3.2.3.5

0

0

− 43

0

3.2.2.4

0

0

− 32

3.2.3.7

0

0

0

3.2.2.8

− 53

3.2.3.6

ƒ

Equation

0

0

− 54 − 54 − 76

0

0

− 76

0

3.2.2.5

0

0

−3

3.2.3.8

0

0

− 12 − 12

− 32

3.2.3.9

0

0

− 12

0

3.2.2.7

0

0

0

arbitrary

3.2.3.1

0

0

0

0

3.2.2.1

0

0

1

arbitrary

3.2.3.2

0

1

1

0

3.2.4.35

0

2

− 72

0

3.2.4.165

0

2

0

3.2.4.161

0

3

0 arbitrary (ý ≠ −2)

0

3.2.4.85

0

3

−2

0

3.2.4.82

0

5

−5

0

3.2.4.105

0

5

− 20 7

0

3.2.4.117

0

5

− 15 7

0

3.2.4.111

0

5

0

0

3.2.4.101

1 2

0

− 52

0

3.2.4.74

1 2

3

− 15 8

0

3.2.4.114

1 2

3

20 − 13

0

3.2.4.120

© 2003 by Chapman & Hall/CRC

Page 561

562

THIRD-ORDER DIFFERENTIAL EQUATIONS TABLE 28 (Continued) Solvable equations of the form 


 = #* 9 : (
 ); (

 )
1 2 1 2 2 3 4 5

ƒ ý

Equation

3

− 54

0

3.2.4.108

3

0

0

3.2.4.104

0

− 76

0

3.2.4.157

−4 arbitrary (= ≠ 1)

− 12

0

3.2.4.137

−1

0

3.2.4.140

1

−3

− 12

0

3.2.4.32

1

−3

1

0

3.2.4.24

1

−1

0

3.2.4.177

1

1

−1 arbitrary (ý ≠ −1)

0

3.2.4.14

1

1

−1

0

3.2.4.17

1

1

1

0

3.2.4.21

3

0

3.2.4.159

0

3.2.4.151

0

3.2.4.109

0

3.2.4.57

3

− 34 − 12 − 23 − 12 − 12

0

3.2.4.148

3

0

0

3.2.4.144

− 94

1

0

3.2.4.66

0

− 13

0

3.2.4.169

0

1

0

3.2.4.62

0

− 52

0

3.2.4.80

0

0

3.2.4.121

0

− 23 − 72

0

3.2.4.68

−7

1

0

3.2.4.54

− 52 − 13 7 − 13

1

0

3.2.4.45

1

0

3.2.4.78

1

0

3.2.4.72

0

1

0

3.2.4.40

1

1

0

3.2.4.50

3

0

0

3.2.4.126

3

1

0

3.2.4.130

1

8 7 8 7 6 5 5 4 5 4 5 4 9 7 9 7 9 7 13 10 27 20 18 13 7 5 7 5 7 5 7 5 7 5 7 5 7 5 7 5

3 0 −4

© 2003 by Chapman & Hall/CRC

Page 562

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

=

 Ë Ÿ Ï¢¡

(Ï

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

563

TABLE 28 (Continued) Solvable equations of the form 


 = #* 9 : (
 ); (

 )


7 5 10 7 22 15 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 23 15 11 7 8 5 8 5 8 5 8 5 8 5 8 5 8 5 8 5 8 5 21 13 33 20 17 10 12 7 12 7 12 7 7 4 7 4

ƒ ý

Equation

11

1

0

3.2.4.135

0

− 52

0

3.2.4.43

0

− 23

0

3.2.4.115

= − 1)

0

3.2.4.173

−3

− 12

0

3.2.4.100

−3

1

0

3.2.4.97

0

−2

0

3.2.4.99

0

− 12

0

3.2.4.84

0

1

0

3.2.4.93

1

1

0

3.2.4.28

3

−2

0

3.2.4.98

3

− 12

0

3.2.4.94

3

0

0

3.2.4.29

− 13

− 12

0

3.2.4.116

−4

− 12

0

3.2.4.47

1

1

0

3.2.4.125

3

−4

0

3.2.4.55

3

0

3.2.4.46

0

3.2.4.79

0

3.2.4.73

3

− 74 − 10 7 − 23 − 12

0

3.2.4.41

3

0

0

3.2.4.51

3

1

0

3.2.4.129

3

5

0

3.2.4.136

−6

0

3.2.4.69

0

3.2.4.122

0

3.2.4.81

0

3.2.4.170

0

3.2.4.67

0

3.2.4.63

0

− 12 − 12 − 12 − 12 − 13 8 − 12 − 52

0

3.2.4.56

0

1

0

3.2.4.147

arbitrary

3 3

− 13 −4 1 3

3 3

1 2(

© 2003 by Chapman & Hall/CRC

Page 563

564

THIRD-ORDER DIFFERENTIAL EQUATIONS TABLE 28 (Continued) Solvable equations of the form 


 = #* 9 : (
 ); (

 )


7 4 9 5 13 7 13 7

2

ƒ ý

Equation

1

1

0

3.2.4.143

− 13

− 12

0

3.2.4.110

− 12

1

0

3.2.4.160

0 arbitrary (= ≠ −1)

1

0

3.2.4.152

0

0

3.2.4.12

2

−1

arbitrary (ý ≠ 0)

0

3.2.4.139

2

−1

−1

0

3.2.4.176

2

−1

0

0

3.2.4.16

2

0

−2

0

3.2.4.33

2

3

−2

0

3.2.4.25

2

3

0

0

3.2.4.19

11 5 7 3 5 2 5 2 5 2 5 2 5 2

0

− 52

0

3.2.4.138

− 43

0

3.2.4.158

−4

− 12 − 12

0

3.2.4.75

− 11 4

1

0

3.2.4.113

27 − 13 − 32

1

0

3.2.4.119

1

0

3.2.4.107

1 arbitrary (= ≠ −3)

1

0

3.2.4.103

1

0

3.2.4.86

3

−2ý − 5

arbitrary (ý ≠ −2)

0

3.2.4.6

3

arbitrary

−= − 2

0

3.2.4.172

3

−9

2

0

3.2.4.106

3

−6

0

3.2.4.59

3

−6

0

3.2.4.166

3

− 17 3

− 12 1 2 − 53

0

3.2.4.77

3

− 33 7 21 −5

2

0

3.2.4.118

− 75

0

3.2.4.65

− 12 − 53

0

3.2.4.37

0

3.2.4.44

2

0

3.2.4.112

3

3 3

−4

3

− 11 3 23 −7

3

© 2003 by Chapman & Hall/CRC

Page 564

3.2. EQUATIONS OF THE FORM

Ï Ò˜Í Í Ò˜Í Ò

=

 Ë Ÿ Ï¢¡

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

(Ï

565

TABLE 28 (Continued) Solvable equations of the form 


 = #* 9 : (
 ); (

 )


= ý

3

−3

−2

0

3.2.4.96

3

−3

−1

0

3.2.4.23

3

−3

− 21

0

3.2.4.90

3

−3

1

0

3.2.4.83

3

− 35

− 35

0

3.2.4.53

3

− 21 − 21

0

3.2.4.149

3

− 35 − 34

0

3.2.4.150

3

−1

−2

0

3.2.4.31

3

− 32

0

3.2.4.134

3

0

0

3.2.4.128

3

0

0

3.2.4.132

3

0

− 35 − 25 − 35 − 21

0

3.2.4.89

3

0

1

−3

3.2.4.88

3

1

−4

0

3.2.4.142

3

1

− 25

0

3.2.4.124

3

1

−2

0

3.2.4.27

3

1

− 35

0

3.2.4.49

3

1

−1

0

3.2.4.20

3

1

0

3.2.4.34

3

1

− 21 1 2

0

3.2.4.162

3

1

2

0

3.2.4.102

3

3

−7

0

3.2.4.154

3

3

−4

0

3.2.4.146

3

3

−2

0

3.2.4.92

3

3

0

3.2.4.39

3

3

0

3.2.4.61

3

3

− 35 − 57 − 21

0

3.2.4.58

3

3

0

0

3.2.4.36

3

5

− 35

0

3.2.4.71

3

7

−7

0

3.2.4.156

4

− 59

1

0

3.2.4.167

4

1

1

0

3.2.4.163

ƒ

Equation

© 2003 by Chapman & Hall/CRC

Page 565

566

THIRD-ORDER DIFFERENTIAL EQUATIONS

3.2.2. Equations of the Form 1. 2.

ñ óò ò ó ò ó =
. = 16 * #  Solution:

3

+L

2

−3 J 2 U + L 3, X V L 1  2 ‡ * + L 2  − ‡ * sin  £ 3 . U  W M −1 =  L 12 V L 1  2 ‡ * + L 2  − ‡ * sin  £ 3 . U  W , 3 1

 J . .

−3 9 2 3

=
ñ –5 ) 2 . Solution in parametric form: 7 1

X (U 3 − 3U + L 2 )−3 J

2

=
ñ –4 ) 3 . Solution in parametric form:

ñóò ò ó ò ó

7 1

 = L where ² =

A (4U ü

−1

Xv²

®

= L

6 3 1(

U − 3U + L 2 )−1 ,

(2UÈ®·Tç² )2 U + L

= XÈU ²

− 1),

3

U + L 3, M

M

−1

U + L 2 , # = A 18  M

=
ñ –7 ) 6 . Solution in parametric form:

9 1 (2

= L

3,

# = −6

where

 J .

−3 7 2

UÈ®·Te² )3 ,

 J .

−3 7 3

ñóò ò ó ò ó

where ² =

]

 + L 0.

ñóò ò ó ò ó

 = L

5.

1

=
ñ –7 ) 2 . Solution in parametric form:

where # = −8

4.

+L

2

ñ óò ò ó ò ó

 = L

3.



â$äãÑãÑäã ä = M âJ¥

ü

 = L

13 1

A (4U

− 1),

3

−1

X ²

®

(2UÈ®·Te² )−5 J

=X U ²

−1

M

2

M

U + L 3,

= L

U + L 2 , # = T 18 

18 1 (2

UÈ®·Tç² )−3 ,

 J .

−3 13 6

In the solutions of equations 6 and 7, the following notation is used:

:

ø

L

J U ) + L 2 n 1 J 3 (U ) for the upper sign, L 1 ® 1 J 3 (U ) + L 2 ; 1 J 3 (U ) for the lower sign, where ø 1 J 3 (U ) and n 1 J 3 (U ) are the Bessel functions, and ® 1 J 3 (U ) and ; 1 J 3 (U ) are the modified Bessel =

™

1 1 3(

functions. 6.

ñóò ò ó ò ó

=
ñ –2 . Solution in parametric form:

 = L 7.

1

:

−2

M

U + L 3,

= L

=
ñ –1 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

 = L 8.

−1

X U

1

X

:

M

U + L 3,

= L

1

U

J :

−2 3

−2

: U J 2,

2 2 3 1

,

where

where

# =A

# =T

4 −3 3 . 3





 J

4 −3 3 2 . 3



=
ñ –5 ) 4 . This is a special case of equation 3.2.4.171 with = = 0.

ñ óò ò ó ò ó

© 2003 by Chapman & Hall/CRC

Page 566

Ï Ò˜Í Í Ò˜Í Ò

3.2. EQUATIONS OF THE FORM

=

 Ë Ÿ Ï¢¡

(Ï

ÒÍ )£ (Ï Ò˜Í Í Ò )¤

567

â$äãÑãÑäã ä = Mx槦!âJ¥

3.2.3. Equations of the Form For ƒ = 0, see Subsection 3.2.2. 1.

ñ óò ò ó ò ó

=
Solution:

ï ^

. = #3 ( ) + L

2

3.

if ƒ ≠ −1, −2, −3;

( ƒ + 1)( ƒ + 2)( ƒ + 3) 1 2 ln | | − 34  2 2 − ln | | +  1 2 ln |  |








if ƒ = −1; if ƒ = −2; if ƒ = −3.

=
ï 3 ñ –7 ) 2 . Solution in parametric form:

1

3 1

J

−3 2

OXv

U +L M

−1 3

P

4 −1 1

= L

,

 2 ‡ * + L 2  − ‡ * sin  £ 3 . U  , # = 8 



J

−3 2

OXv

 J . .

M

U +L

−2 3

P

,

−6 9 2 3

=
ï ñ –5 ) 2 . Solution in parametric form:

ñ óò ò ó ò ó

= L where # = 6 

7 1

X (U Q

8 3 1(

U

3

− 3U + L

− 3U + L

2)

2)

−1

 J .

Q

J

−3 2

X (U

M 3

−1

U + L 3S − 3U + L

,

2)

J

−3 2

M

U + L 3S

−2

,

−4 7 2

In the solutions of equations 5 and 6, the following notation is used:

² =

A (4U ü

3

− 1),

®

−1

= XÈU ²

=
ï –4 ) 3 ñ –4 ) 3 . Solution in parametric form:

M

U + L 2.

ñ óò ò ó ò ó

 = L

7 1

Q

Xv²

−1

where # = T 18 6.

+3

ñ óò ò ó ò ó

 = L

5.

 + L 0 , where  9

=
ï ^ ñ . See equation 3.1.2.7.

where  = L

]





1

ñóò ò ó ò ó

 = L

4.

+L

2






 ( ) =

2.



(2UÈ®·Tç² )2 U + L

M

J  J .

3

S

−1

,

= L

5 1 (2

X ² UÈ®·Tç² )3 Z Q

−1

(2UÈ®·TC² )2 U + L

M

3

S

−2

,

−5 3 7 3

=
ï –5 ) 3 ñ –7 ) 6 . Solution in parametric form:

ñ óò ò ó ò ó

13 1

 = L = L where # = T 18

Q

8 1 (2

J  J .

XZ²

−1

(2UÈ®·Tç² )−5 J

X ² UÈ®·Tç² )−3 v Q

−1

2

M

U + L 3S

(2UÈ®lTç² )−5 J

−1

2

M

,

U + L 3S

−2

,

−4 3 13 6

© 2003 by Chapman & Hall/CRC

Page 567

568 7.

THIRD-ORDER DIFFERENTIAL EQUATIONS =
ï –3 ) 2 ñ –5 ) 4 . Solution in parametric form:

ñóò ò ó ò ó

 = L

3 1

O X U

where H = L 8.

2

J H

J  J

−1 2 −1 2 3 4

+ 14 U

2

−1

U +L M

3

P

2 1

= L

,

+ 4 $IU 1 J 2 ,  = exp O X H

J

−1 2

=
ï –3 ñ –1 ) 2 . Solution in parametric form:

J H

J  J

−1 2 −1 2 3 4

*O X U

U , # = 12 C $  M P

M

J  J .

−2

U +L

3

P

,

−3 2 9 4

ñóò ò ó ò ó

 =L where

:

O X

−1

U +L M

:

3

P

= 'U 2 J

,

:

3

2

O X

: M

−2

U +L

3

P

,

ø

L

J U ) + L 2 n 1 J 3 (U ) for the upper sign, L 1 ® 1 J 3 (U ) + L 2 ; 1 J 3 (U ) for the lower sign, ( U ) and n ( U ) are the Bessel functions, and ® 1 J 3 (U ) and ; 1 J 3 (U ) are the modified Bessel 1J 3 1J 3 # =A

ø

1

 J

4 3 2 , 3

™

=

1 1 3(

functions. 9.

=
ï –3 ) 2 ñ –1 ) 2 . Solution in parametric form:

ñóò ò ó ò ó

.

 = L

.

3

.

exp O 2 X

U , M P

 = L

3

.

2

.

exp O 2 X

U . M P

, L 2 ) is the general solution of the second Painlev´e transcendent: Here, .

= A"U . = + 2(. U ,3L, 1and # = A 14  −3 J 2  3 J 2 .

*+*

3.2.4. Equations with |¨ | + | © | ≠ 0 1.

ñóò ò ó ò ó

=
(ñó ò )` (ñó ò ò ó ) ª , Solution in parametric form:

 = L 1; + < where # = A 2.

−1

XçU

J O 1 AIU ;

= +1 2< 2−>

−2

 = L 1: + < X where # = A

]

ý +1 2< 1−>



Particular solutions: 44.

4

ü

Solution:

Œ

cos( @! ) + L

3

ú ñ

2

£ −(  , 1 = exp  − 

= 0. = sin  £ (  . 2

= exp  £ − (  .

¨ 
+ ' +*¨ = 0. ¨ 

 +   2 leads to a second-order linear equation: u

+ 4ñó ò ò ó ò ó + ô ï ñ = 0. The substitution ¨ ( ) =  leads to a constant coefficient linear equation of the form 4.1.2.1: ¨2



 +  ¨ = 0.

ï ñóò ò ó ò ò ó ó

– 4ùñó ò ò ó ò ó + ô ï ñ = 0, ù = 1, 2, 3, uuu Solution: =  4  +3 ( −3 c )  ( −3 ¨ ), where c = x x  and ¨ = ¨ ( ) is the general solution of a linear constant coefficient equation of the form 4.1.2.1: ¨p



  +  ¨ = 0.

ï ñóò ò ó ò ò ó ó

© 2003 by Chapman & Hall/CRC

Page 646

647

4.1. LINEAR EQUATIONS

51.

ï 2 ñ óò ò ó ò ò ó ó

+ 6 ï ñó ò ò ó ò ó + 6ñó ò ò ó – ô 2 ñ = 0. Equation of transverse vibrations of a pointed bar. 1 V L 1 ø 1  2 £   + L 2 n 1  2 £   + L = £ Solution:

®

 2 £   + L 4 ; 1  2 £   W , where ø 1 ( H ) and n 1 ( H ) are the Bessel functions, and ® 1 ( H ) and ; 1 ( H ) are the modified Bessel 

3 1

functions.

52.

53.

54. 55.

56.

57.

58.

ï 2 ñ óò ò ó ò ò ó ó

+ 2( ô + 2)ï ñ ó ò ò ó ò ó + ( ô + 1)(ô + 2)ñ ó ò ò ó – õ 4 ñ = 0. Solution: =  −  J 2 V L 1 ø   2  £   + L 2 n   2  £   + L 3 ®   2  £   + L 4 ;   2  £   W , where ø  (H ) and n  (H ) are the Bessel functions, and ®  (H ) and ;  (H ) are the modified Bessel functions. + 8 ï ñó ò ò ó ò ó + 12ñó ò ò ó + ô ï 2 ñ = 0. The substitution ¨ ( ) =  2 leads to a constant coefficient linear equation of the form 4.1.2.1: ¨2



 +  ¨ = 0.

ï 2 ñ óò ò ó ò ò ó ó

+ 8 ï ñó ò ò ó ò ó + 12ñó ò ò ó = ô ï 3 ñ + õ . ¨ 



 =  ¨ +  . The substitution ¨ ( ) =  2 leads to an equation of the form 4.1.2.3: u

ï 2 ñ óò ò ó ò ò ó ó

+ ô ï ñó ò ò ó ò ó + ( õ ïƒ÷ +1 + ö )ñó ò ò ó + ( ô – 4) õ ïƒ÷ ñó ò + õ ( ö – 2 ô + 6) ïw÷ –1 ñ = 0. The substitution ¨ ( ) =  2 

 + (  − 4) 
 + ( ( − 2  + 6) leads to a second-order equation of the form 2.1.2.7: ¨3

 + '  −1 ¨ = 0.

ï 2 ñ óò ò ó ò ò ó ó ï 3 ñ óò ò ó ò ò ó ó

+ 2 ï 2 ñó ò ò ó ò ó – ï ñó ò ò ó + ñó ò – ô 4 ï 3 ñ = 0. Solution: = L 1 ø 0 (  ) + L 2 n 0 (  ) + L 3 ® 0 (  ) + L 4 ; 0 (  ), where ø 0 ( H ) and n 0 ( H ) are the Bessel functions, and ® 0 ( H ) and ; 0 ( H ) are the modified Bessel functions.

ï 4 ñ óò ò ó ò ò ó ó

+
3 ï 3 ñ ó ò ò ó ò ó +
2 ï 2 ñ ó ò ò ó +
1 ï ñ ó ò +
0 ñ = 0. The Euler equation. The substitution a = ln | | leads to a constant coefficient linear equation of the form 4.1.2.41: g

g
g
g + ( # 3 − 6)
g
g
g + (11 − 3# 3 + # 2 )
g
g + (2 # 3 − # 2 + # 1 − 6)
g + # 0 = 0.

ï 4 ñ óò ò ó ò ò ó ó

+ 2 ï 3 ñó ò ò ó ò ó – (2 ô 2 + 1) ï 2 ñó ò ò ó + (2 ô 2 + 1) ï ñó ò – V õ 4 ï 4 – ô 2 ( ô 2 – 4)WNñ = 0. This equation governs free transverse vibration modes of a thin round elastic plate. The equation arises from separation of variables in the two-dimensional equation ¢e¢ ¨ −  4 ¨ = 0, where ¢ is the Laplace operator written in the polar coordinate system, with  being the polar radius. Solution: = L 1 ø  ( ' ) + L 2 n  ( ' ) + L 3 ®  ( ' ) + L 4 ;  ( ' ), where ø  ( H ) and n  ( H ) are the Bessel functions, and ®  ( H ) and ;  ( H ) are the modified Bessel functions. In applications, one usually sets  = B , where B = 0, 1, 2, 

^_ 59.

The solution is specified by Popov (1998).

– 2ù (ù + 1)ï 2 ñ ó ò ò ó + 4ù (ù + 1)ï ñ ó ò + [ ô ï 4 + ù (ù + 1)(ù + 3)(ù – 2)]ñ = 0. Here, B is a positive integer and  ≠ 0 (for  = 0, we have the Euler equation 4.1.2.57).

ï 4 ñ óò ò ó ò ò ó ó

Solution: equation Π60.

ï 4 ñ óò ò ó ò ò ó ó

4

74 . =  − LlÁ exp( Œ"Á ) Á ( ), where the Œ Á are four different roots of the Á

+  = 0, and

+ 2(2 – ù )ï

3

.

=1

ñ óò ò ó ò ó

Á ( ) is some definite polynomial of degree ≤ 4B . + (1 – ù )(2 – ù ) ï

2

ñ óò ò ó

–ô

4

ï 2÷ ñ

= 0.

Solution: = £  [ L 1 ø 1 J (ˆ ) + L 2 n 1 J (ˆ ) + L 3 ® 1 J (ˆ ) + L 4 ; 1 J  (ˆ )], where ˆ = 2(  B ) ´J 2; ø Á (ˆ ) and n Á (ˆ ) are the Bessel functions, and ®Á (ˆ ) and ; Á (ˆ ) are the modified Bessel functions.

© 2003 by Chapman & Hall/CRC

Page 647

648 61.

62.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ï 4 ñ óò ò ó ò ò ó ó

+6 ï 3 ñó ò ò ó ò ó +[4 ï 4 +(7– ô 2 – õ 2 ) ï 2 ]ñó ò ò ó + ï (16 ï 2 +1– ô 2 – õ 2 )ñó ò +(8 ï 2 + ô 2 õ 2 )ñ = 0. Solution for  ≠ 0: = L 1 ø ( ) ø Á ( ) + L 2 ø ( ) n Á ( ) + L 3 n  ( ) ø Á ( ) + L 4 n  ( ) n Á ( ), where ø ( ) and n  ( ) are the Bessel functions; “ = 12 (  +  ) and @ = 12 (  −  ).

ï 8 ñ óò ò ó ò ò ó ó

+ 4 ï 7 ñó ò ò ó ò ó = ôñ . The substitution ¨ ( ) =  leads to an equation of the form 4.1.2.8: 

8

¨2



 =  ¨ .

4.1.3. Equations Containing Exponential and Hyperbolic Functions 4.1.3-1. Equations with exponential functions.

ó

ñóò ò ó ò ò ó ó

ó

1.

+ ô 3 ñó ò + õD ( ô 2 – õD )ñ The substitution ¨ = 

 +  
 +    2.1.3.10: ¨3

 −  ¨2
+ (  2 −    )¨

2.

ñ óò ò ó ò ò ó ó

+

ô  ó ñ óò

+ ô

ó

 ñ

=

= 0. leads to a second-order linear equation of the form = 0.



õ ó

.

Integrating yields a third-order linear equation: 3.

ñóò ò ó ò ò ó ó

+

ô ó ñóò

Particular solution:

ó

ñóò ò ó ò ò ó ó

ó

– ( ôƒõ 0



=

+



.

4.

+ 2 ô ñó ò + ô ( 2   The substitution ¨ = 

 +   ¨2

 −    ¨ = 0.

5.

ñóò ò ó ò ò ó ó

+ ( ô

ó

õ 3 )ñ óò

+

Particular solution: 6. 7.

0

ñ óò ò ó ò ò ó ó

+

ô  ó ñ óò ò ó

0

   +L .

ó

óñ

= 0. = 0.

óñ

= 0.

– 3 ô

óñ

= 0.

2

= (  +  ) . 3

= (  +  ) .

– õ ( ô



ó

+ õ )ñ = 0.

1 b . Particular solutions with  > 0:

1

2 b . Particular solutions with  < 0:

1

ñóò ò ó ò ò ó ó

−1

= 0.

– 2 ô

The substitution ¨ = 

 −  ¨2

 + (    +  )¨ = 0. 10.

.

= '“

– ô 2  )ñ = 0. leads to a second-order linear equation of the form 2.1.3.1:

– ô =  +  .

ñóò ò ó ò ò ó ó + (ô ï + õ )  ó ñ óò

Particular solution: 9.

−

ñóò ò ó ò ò ó ó + (ô ï + õ )  ó ñ óò

Particular solution: 8.



0 =  

0

ó

ôƒõ ó ñ

+

ñóò ò ó ò ò ó ó + (ô ï + õ )  ó ñ óò

Particular solution:

õ 4 )ñ






 +   

ó

= exp  − £   , = cos   £ −   ,

2

= exp   £   .

= sin   £ −   .

leads to a second-order linear equation of the form 2.1.3.2:

ó

+ ( ô + õ )ñó ò ò ó + ôƒõ ñ = 0. 1 b . Particular solutions with  > 0: 1 = cos   £   , 2 b . Particular solutions with  < 0:

The substitution ¨ = 

 +  ¨2

 +    ¨ = 0.

2

1

2

= exp  − £ −   ,

= sin   £   . 2

= exp   £ −   .

leads to a second-order linear equation of the form 2.1.3.1:

© 2003 by Chapman & Hall/CRC

Page 648

649

4.1. LINEAR EQUATIONS

11.

ñ óò ò ó ò ò ó ó

+ 10 ô

ñóò ò ó ò ò ó ó

+

ó

 ñ óò ò ó

+ 10 ô

ó

 ñ óò

+ (3 ô

2

 

ó

+ 9ô

 

2 2



This is a special case of equation 4.1.6.25 with  ( ) =   . 12.

13.

ôñóò ò ó ò ó

+

õ ó ñóò

15.

16.

17.

Particular solution:

ñ óò ò ó ò ò ó ó

+

ô  ó ñ óò ò ó ò ó

=

+

ô ó ñóò ò ó ò ó

+ ( õ

+

ô ï 3   ó ñóò ò ó ò ó

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

20.

 ñ

.

õ ó ñ óò ò ó

ó

 ñ óò

+ ô„ö

+ ö ( õ

ó

– ö )ñ = 0.

1 b . Particular solutions with ( > 0: 1 = cos   £ (  , 2 = sin   £ ( 2 b . Particular solutions with ( < 0: 1 = exp  − £ − (  , 2 = exp  The substitution ¨ = 

 + ( leads to a second-order equation: ¨u

 + 

ó



ô„ö ó ñóò

+ ö )ñó ò ò ó +

+

õ^ö  ó ñ

ï  ó ñóò

– 6 ô

. £ −(  .    ¨2
+(    − ( )¨ = 0.

= 0.

1 b . Particular solutions with ( > 0: 1 = cos  £ (  , 2 = sin  £ (  . 2 b . Particular solutions with ( < 0: 1 = exp  − £ − (  , 2 = exp  £ − (  .  ¨
  +    ¨ = 0. The substitution ¨ = 

 + ( leads to a second-order equation: ¨u

 +   2

ï 2   ó ñóò ò ó

– 3ô

+ 6ô

óñ

Particular solutions: 1 =  , 2 =  , 3 =  . The substitution ¨ =   leads to a first-order linear equation: ¨u
+  3  ¨ = 0.

ï ñóò ò ó ò ò ó ó

ï  ó ñóò

+ô

ó

( ô

+ õ )ñó ò ò ó ò ò ó

ó

=

(ô

ïƒú

+

õ ó

ô ó ñ ó 0

óò ò ó ò ò ó ó

=

.

= 

0

+ ö )ñó ò ò ó ò ò ó

+ õ )ñ



ó

=  .

0

Particular solution:

ó (ô ï ú 

2

– [ ô ( ï + 1)-

Particular solution: 19.

= 0.

ó

– ôƒõ

= 0.

:ž¼  ( , = 1, 2, 3), where the ý are roots of the cubic equation   =

+

+

−

õÿñ óò

ô  ó ñ óò ò ó ò ó

ñ óò ò ó ò ò ó ó

Particular solution: 18.

ôƒõ ó ñ

 . 0 = 

Particular solutions: ý 3 −  = 0. 14.

+

ó )ñ



= 0. 3




 −3

2



 +6 
−6

+ ï + 4]ñ = 0.

+ .

õ ó ñ

, þ = 1, 2, 3.  =  + +   + ( . =

õÿñ

þ

,

= 0, 1, 2, 3.

   + +  − . 0 = 

Particular solution:

3

ïƒ÷ ) ñ óò ò ó

+ ô [ õ exp(

ïƒ÷

21.

ñóò ò ó ò ò ó ó

+ õ exp(

) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  exp( Œ   ).

22.

ñóò ò ó ò ò ó ó

+ [ ô + õ exp(

ïw÷ )]ñ óò ò ó

+ ôƒõ exp(

ïƒ÷ ) ñ

= 0.

This is a special case of equation 4.1.6.21 with  ( ) =  exp( Π  ).

4.1.3-2. Equations with hyperbolic functions. 23.

ñóò ò ó ò ò ó ó

+

ô

sinh ÷ (

Particular solution:

ï ) ñ óò 0

=

+ õ [ ô sinh÷ ( −





ï

)–

õ 3 ]ñ

= 0.

.

© 2003 by Chapman & Hall/CRC

Page 649

650 24.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ñóò ò ó ò ò ó ó

+ [ ô sinh÷ (

ï

Particular solution:

)+ 0

=

õ 3 ]ñ óò 



−

+

ôƒõ

sinh ÷ (

.

25.

+ ( ô ï + õ ) sinh ÷ ( ï )ñó ò – ô sinh ÷ ( Particular solution: 0 =  +  .

26.

ñ óò ò ó ò ò ó ó

+ õ ) sinh ÷ (

ï

+ (ô

ñ óò ò ó ò ò ó ó

0

29.

0

ï )ñ

= 0.

ï )ñ ó ò

– 3 ô sinh ÷ (

ï )ñ

= 0.

3

= (  +  ) .

+ õ sinh ÷ ( ï ) ñ ó ò ò ó + ô [ õ sinh ÷ ( ï ) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  sinh  ( Œ  ).

ñ óò ò ó ò ò ó ó

+ [ ô + õ sinh ÷ ( ï )]ñó ò ò ó + ôƒõ sinh ÷ ( ï ) ñ = 0. This is a special case of equation 4.1.6.21 with  ( ) =  sinh  ( Œ  ).

ñóò ò ó ò ò ó ó

( ô ïƒú + õ sinh ï )ñ„ó ò ò ó ò ò ó ó = õ sinh ï ñ , + +  sinh  . Particular solution: 0 = 

31.

ñóò ò ó ò ò ó ó

+

ô

cosh÷ (

Particular solution:

ñóò ò ó ò ò ó ó

+ [ ô cosh÷ (

Particular solution:

ï ) ñ óò ï

0

=

)+

+ õ [ ô cosh÷ ( −





.

õ 3 ]ñ óò

0 =   −



+

ôƒõ

ï

þ

õ 3 ]ñ

= 0.

cosh÷ (

ï )ñ

= 0.

.

+ ( ô ï + õ ) cosh ÷ ( ï )ñ ó ò – ô cosh÷ ( Particular solution: 0 =  +  .

34.

ñóò ò ó ò ò ó ó

+ õ ) cosh ÷ (

ï

+ (ô

ñóò ò ó ò ò ó ó

+ (ô

ï

38. 39. 40.

0

0

= 0.

ï )ñ óò

– 2 ô cosh÷ (

ï )ñ

= 0.

ï )ñ óò

– 3 ô cosh÷ (

ï )ñ

= 0.

2

= (  +  ) .

+ õ ) cosh ÷ (

Particular solution:

37.

ï )ñ

ñ óò ò ó ò ò ó ó

Particular solution:

36.

= 1, 2, 3.

)–

33.

35.

= 0.

– 2 ô sinh ÷ (

30.

32.

= 0.

ï )ñ ó ò

= (  +  )2 .

+ õ ) sinh ÷ (

ï

+ (ô

Particular solution: 28.

ï )ñ

ñóò ò ó ò ò ó ó

Particular solution: 27.

ï )ñ

3

= (  +  ) .

+ õ cosh÷ ( ï ) ñó ò ò ó + ô [ õ cosh÷ ( ï ) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  cosh  ( Œ  ).

ñóò ò ó ò ò ó ó

+ [ ô + õ cosh÷ ( ï )]ñ ó ò ò ó + ôƒõ cosh÷ ( ï ) ñ = 0. This is a special case of equation 4.1.6.21 with  ( ) =  cosh  ( Œ  ).

ñ óò ò ó ò ò ó ó

( ô ïƒú + õ cosh ï )ñó ò ò ó ò ò ó ó = õ cosh ï ñ , Particular solution: 0 =  + +  cosh  .

þ

= 1, 2, 3.

= ñ + ô (ñó ò cosh ï – ñ sinh ï ).
The substitution ¨ =  cosh  − sinh  leads to a third-order linear equation.

ñóò ò ó ò ò ó ó

= ñ + ô (ñó ò sinh ï – ñ cosh ï ). The substitution ¨ = 
sinh  − cosh  leads to a third-order linear equation.

ñóò ò ó ò ò ó ó

© 2003 by Chapman & Hall/CRC

Page 650

651

4.1. LINEAR EQUATIONS

41.

ñóò ò ó ò ò ó ó

tanh÷ (

ô

+

ï ) ñ óò

Particular solution: 42.

ñóò ò ó ò ò ó ó

+ [ ô tanh÷ (

ï

0

=

)+

+ õ [ ô tanh÷ (





−

.

õ 3 ]ñ óò 

0 =   −

Particular solution:

+

ôƒõ

ï

) – õ 3 ]ñ = 0.

tanh÷ (

+ ( ô ï + õ ) tanh ÷ ( ï )ñó ò – ô tanh÷ ( Particular solution: 0 =  +  .

44.

ñóò ò ó ò ò ó ó

+ õ ) tanh ÷ (

Particular solution: 45.

ñóò ò ó ò ò ó ó

0

0

= 0.

ï )ñ óò

– 2 ô tanh÷ (

ï )ñ

= 0.

ï )ñ óò

– 3 ô tanh÷ (

ï )ñ

= 0.

2

= (  +  ) .

+ õ ) tanh ÷ (

ï

+ (ô

Particular solution:

46.

ï )ñ

ñóò ò ó ò ò ó ó

ï

= 0.

.

43.

+ (ô

ï )ñ

3

= (  +  ) .

+ õ tanh÷ ( ï ) ñó ò ò ó + ô [ õ tanh÷ ( ï ) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  tanh  ( Œ  ).

ñóò ò ó ò ò ó ó

47.

+ [ ô + õ tanh÷ ( ï )]ñó ò ò ó + ôƒõ tanh÷ ( ï ) ñ = 0. This is a special case of equation 4.1.6.21 with  ( ) =  tanh  ( Œ  ).

48.

ñ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

coth÷ (

ô

+

Particular solution: 49.

ñóò ò ó ò ò ó ó

+ [ ô coth÷ (

Particular solution:

50. 51.

+ õ [ ô coth÷ (

ï ) ñ óò ï

0

=

)+ 0

=





−

.

õ 3 ]ñ óò 

−



ï

)–

õ 3 ]ñ

= 0.

+ ôƒõ coth÷ (

ï )ñ

= 0.

.

+ õ coth÷ ( ï ) ñó ò ò ó + ô [ õ coth÷ ( ï ) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  coth  ( Œ  ).

ñóò ò ó ò ò ó ó

+ [ ô + õ coth÷ ( ï )]ñ ó ò ò ó + ôƒõ coth÷ ( ï ) ñ = 0. This is a special case of equation 4.1.6.21 with  ( ) =  coth  ( Œ  ).

ñ óò ò ó ò ò ó ó

4.1.4. Equations Containing Logarithmic Functions 1.

ñ óò ò ó ò ò ó ó

+

ô

ln 

ï ñ óò

Particular solution:

– ( ôƒõ ln  0

=





ï

+

õ 4 )ñ

.

2.

+ ( ô ï + õ ) ln  ( ï )ñó ò – ô ln  ( Particular solution: 0 =  +  .

3.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

ï

+ õ ) ln  (

+ (ô

Particular solution: 4.

ñóò ò ó ò ò ó ó

+ (ô

ï

+ õ ) ln  (

Particular solution: 5.

0

0

= 0.

ï )ñ

= 0.

ï )ñ óò

– 2 ô ln  (

ï )ñ

= 0.

ï )ñ óò

– 3 ô ln  (

ï )ñ

= 0.

2

= (  +  ) . 3

= (  +  ) .

+ 2 ô ï ñó ò – ô [1 + ô ï 2 ln2 ( õ ï )]ñ = 0. ¨ 

 −  ln( ' )¨ = 0. The substitution ¨ = 

 +  ln( ' ) leads to a second-order equation: u

ï 2 ñ óò ò ó ò ò ó ó

© 2003 by Chapman & Hall/CRC

Page 651

652 6. 7. 8.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ï ) ñ óò ò ó

ñóò ò ó ò ò ó ó

+ õ ln  (

ñ óò ò ó ò ò ó ó

+ [ ô + õ ln  (

ñóò ò ó ò ò ó ó

+ ln  (

ï )]ñ óò ò ó

ï )(ï 2 ñ óò ò ó

ñóò ò ó ò ò ó ó

+ ln  (

1

ï )(ï 2 ñ óò ò ó

ñóò ò ó ò ò ó ó

ô ï

+

2

1

– 2ï

ï )ñ

ln  (

ôƒõ

0

ñóò

= ,

2

– 4ï

= 0.

+ 2ñ ) = 0.

=  2.

+ 6ñ ) = 0.

ñóò

2

= ,

ï )ñ óò ò ó

ln  (

Particular solution: 11.

+

This is a special case of equation 4.1.6.21 with  ( ) =  ln  ( Π ).

Particular solutions: 10.

) – ô ]ñ = 0.

This is a special case of equation 4.1.6.20 with  ( ) =  ln  ( Π ).

Particular solutions: 9.

ï

+ ô [ õ ln  (

2

=  3.

ï )ñ

– 2 ô ln  (

2

= .

+ ôñó ò ò ó ò ó + õ ln  ( ï )ñó ò + ôƒõ ln  (  Particular solution: 0 =  −  .

ñóò ò ó ò ò ó ó

= 0.

ï )ñ

= 0.

4.1.5. Equations Containing Trigonometric Functions 4.1.5-1. Equations with sine and cosine. 1.

ñóò ò ó ò ò ó ó

+

ô

sin ÷ (

ï ) ñ óò

Particular solution: 2.

ñóò ò ó ò ò ó ó

+ [ ô sin÷ (

Particular solution:

ï

0

+ õ [ ô sin ÷ ( =

)+ 0





−

=



)–

õ 3 ]ñ

= 0.

sin ÷ (

ï )ñ

= 0.

.

õ 3 ]ñ óò −

ï



+

ôƒõ

.

3.

+ ( ô ï + õ ) sin ÷ ( ï )ñ ó ò – ô sin÷ ( Particular solution: 0 =  +  .

4.

ñóò ò ó ò ò ó ó

ñ óò ò ó ò ò ó ó

ï

+ õ ) sin ÷ (

+ (ô

Particular solution: 5.

ñóò ò ó ò ò ó ó

+ (ô

ï

7. 8.

9.

0

= 0.

ï )ñ óò

– 2 ô sin ÷ (

ï )ñ

= 0.

ï )ñ óò

– 3 ô sin ÷ (

ï )ñ

= 0.

2

= (  +  ) .

+ õ ) sin ÷ (

Particular solution:

6.

0

ï )ñ

3

= (  +  ) .

+ ô sin ÷ ( ï ) ñ ó ò ò ó + õ [ ô sin ÷ ( ï ) – õ ]ñ = 0. ¨ 

 + [  sin ( Œ  ) −  ]¨ = 0. The substitution ¨ = 

 +  leads to a second-order equation: u

ñ óò ò ó ò ò ó ó

+ [ ô + õ sin÷ ( ï )]ñó ò ò ó + ôƒõ sin÷ ( ï ) ñ = 0. ¨ 

 +  sin ( Œ  ) ¨ = 0. The substitution ¨ = 

 +  leads to a second-order linear equation: u

ñóò ò ó ò ò ó ó

= ô sin÷ ( ï ) ñó ò ò ó ò ó + õÿñó ò – ôƒõ sin ÷ ( ï ) ñ .  Particular solutions:  =  :ž¼ ( , = 1, 2, 3), where the ý  are roots of the cubic equation 3 ý −  = 0.

ñóò ò ó ò ò ó ó

+ ô sin ÷ ( ï ) ( ï 3 ñó ò ò ó ò ó – 3 ï 2 ñó ò ò ó + 6 ï ñó ò – 6ñ ) = 0. This is a special case of equation 4.1.6.33 with  ( ) =  sin  ( Œ  ).

ñóò ò ó ò ò ó ó

© 2003 by Chapman & Hall/CRC

Page 652

4.1. LINEAR EQUATIONS

10. 11.

ï 2 ñ óò ò ó ò ò ó ó

ï

sin ÷ (

ô

+

) (ï

( ô sin ï + õ )ñ„ó ò ò ó ò ò ó

ó

ï

ó

=

ï ú

(ô

+ õ sin ï )ñ

óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó ñóò ò ó ò ò ó ó

16.

ñ óò ò ó ò ò ó ó ñóò ò ó ò ò ó ó ñ óò ò ó ò ò ó ó

ï

19.

20. 21.

ñ óò ò ó ò ò ó ó

+ [ ô + õ cos÷ (

ñóò ò ó ò ò ó ó

=

23.

24.

−

+





ñóò ò ó ò ò ó ó

+

cos÷ (

ô

)–

õ 3 ]ñ

= 0.

ôƒõ

cos÷ (

ï )ñ

= 0.

.

ï )ñ ó ò

– ô cos÷ (

ï )ñ óò

– 2 ô cos÷ (

ï )ñ

2

– 3 ô cos÷ (

ï )ñ ó ò

3

+ õ [ ô cos÷ (

ï )]ñ óò ò ó

= 0.

ï )ñ

= 0.

ï )ñ

= 0.

ï

) – õ ]ñ = 0. ¨ 

 + [  cos ( Œ  ) −  ]¨ = 0. leads to a second-order equation: u + ôƒõ cos÷ (

ï )ñ

= 0.

¨ 

 +  cos ( Œ  ) ¨ = 0. leads to a second-order linear equation: u

ï ) ñ óò ò ó ò ó

cos÷ (

ô

ï

= 2, 3.

.

The substitution ¨ = 

 + 

ï

) (ï

3

cos÷ (

ï

) (ï

óò ò ó ò ò ó ó

=

+

– ôƒõ cos÷ (

õÿñóò

ï )ñ

. :ž¼  ( , = 1, 2, 3), where the ý are roots of the cubic equation =    – 3ï

ñóò ò ó ò ó

2

ñóò ò ó

+ 6ï

ñóò

– 6ñ ) = 0.

This is a special case of equation 4.1.6.33 with  ( ) =  cos  ( Π ).

ï 2 ñ óò ò ó ò ò ó ó

ô

– 4ï

ñóò + 6ñ ) = 0. The substitution ‡ =  2 

 −4 
 +6 leads to a second-order equation: ‡

 +  cos ( Œ  )‡ = 0. +

( ô cos ï + õ )ñ

Particular solution: 25.



−

The substitution ¨ = 

 + 

Particular solutions: ý 3 −  = 0. 22.



= (  +  ) .

0

+

þ

,

= (  +  ) .

0

ñóò ò ó ò ò ó ó

ñ

=  +  .

ï ) ñ óò ò ó

cos÷ (

ô

=

0

Particular solution:

.

sin ï

õ 3 ]ñ óò

+ õ ) cos÷ (

ï

ñ

+ õ [ ô cos÷ (

+ õ ) cos÷ (

+ (ô + (ô

õ

+ õ ) cos÷ (

ï

+ (ô

Particular solution: 18.

=

=

0

Particular solution: 17.

sin ï

)+

Particular solution:

.

  + +  sin  . 0 = 

0

ï

+ [ ô cos÷ (

ñ

õ

ï ) ñ óò

cos÷ (

ô

+

Particular solution: 15.

+ 6ñ ) = 0.

=  +  sin  .

0

Particular solution: 14.

ñóò

=  sin  +  .

0

+ õ sin ï )ñ„ó ò ò ó ò ò ó

(ô

sin ï

ô

=

Particular solution: 13.

– 4ï

ñóò ò ó

The substitution ‡ =  

 −4 
 +6 leads to a second-order equation: ‡

 +  sin ( Œ  )‡ = 0. 2

Particular solution: 12.

2

653

(ô

ï

+ õ cos ï )ñ„ó ò ò ó ò ò ó

Particular solution:

ô

ó

0

= 0

2

ñóò ò ó

cos ï

ñ

.

=  cos  +  .

õ

cos ï

ñ

.

=  +  cos  .

© 2003 by Chapman & Hall/CRC

Page 653

654 26. 27.

28.

FOURTH-ORDER DIFFERENTIAL EQUATIONS ( ô ï ú + õ cos ï )ñ ó ò ò ó ò ò ó Particular solution:

ó 0

= õ cos ï ñ , =  + +  cos  .

þ

= 2, 3.

+ 2 ôƒõ cos( õ ï ) ñ„ó ò – ô [ õ 2 sin( õ ï ) + ô sin2 ( õ ï )]ñ = 0. The substitution ¨ = 

 +  sin( ' ) leads to a second-order linear equation of the form 2.1.6.2: ¨ 

 −  sin( ' )¨ = 0.

ñóò ò ó ò ò ó ó

sin4 ï

+ sin2 ï (sin2 ï – 3) ñ ó ò ò ó + sin ï cos ï (2 sin2 ï + 3) ñó ò + ( ô 4 sin4 ï – 3)ñ = 0. Equation of a loaded rigid spherical shell. If  4 = 1 − Œ 2 , the equation can be rewritten as

ñ óò ò ó ò ò ó ó

+ 2 sin3 ï cos ï

ñ óò ò ó ò ó

2

LL( ) − Œ

where L ≡ M

= 0,

This equation falls into two second-order equations: L( ) + Œ

M

2



2

L( ) − Œ

= 0,

+ cot M

M



− cot2  .

= 0,

which differ only in the sign of the parameter Œ . The transformation ˆ = sin 2  , ¨ =  sin  reduces the latter equations to the hypergeometric equations 2.1.2.171:

ˆ (ˆ − 1)¨ ªŠ

Š + ( 25 ˆ − 2)¨ Š
+ 14 (1 TeŒ )¨ = 0.

29. 30.

= ñ + ô (ñó ò sin ï – ñ cos ï ). The substitution ¨ = 
sin  − cos  leads to a third-order linear equation.

ñóò ò ó ò ò ó ó

= ñ + ô (ñó ò cos ï + ñ sin ï ).
The substitution ¨ =  cos  + sin  leads to a third-order linear equation.

ñóò ò ó ò ò ó ó

4.1.5-2. Equations with tangent and cotangent. 31.

+ ôñó ò + ( ô tan ï – 1)ñ = 0. Particular solution: 0 = cos  .

ñóò ò ó ò ò ó ó

32.

+ ( ô ï + õ ) tan÷ ( ï )ñó ò – ô tan÷ (   + . Particular solution: 0 = 

33.

ñ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

+ õ ) tan÷ (

ï

+ (ô

Particular solution: 34.

ñ óò ò ó ò ò ó ó

+ (ô

ï

0

Particular solution: 35.

ñóò ò ó ò ò ó ó

+

ô

tan÷ (

0

ï ) ñ óò

Particular solution: 36.

ñ óò ò ó ò ò ó ó

+ [ ô tan÷ (

Particular solution: 37.

ï

0

ï )ñ

= 0.

ï )ñ ó ò

– 3 ô tan÷ (

ï )ñ

= 0.

)–

õ 3 ]ñ

= 0.

+ ôƒõ tan÷ (

ï )ñ

= 0.

2

3

+ õ [ ô tan÷ (

)+ 0

– 2 ô tan÷ (

= (  +  ) . =





−

.

õ 3 ]ñ ó ò

=

= 0.

ï )ñ ó ò

= (  +  ) .

+ õ ) tan÷ (

ï )ñ

−





ï

.

+ õ tan÷ ( ï ) ñó ò ò ó + ô [ õ tan÷ ( ï ) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  tan  ( Œ  ).

ñóò ò ó ò ò ó ó

© 2003 by Chapman & Hall/CRC

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655

4.1. LINEAR EQUATIONS

38. 39.

40.

+ [ ô + õ tan÷ ( ï )]ñ ó ò ò ó + ôƒõ tan÷ ( ï ) ñ = 0. This is a special case of equation 4.1.6.21 with  ( ) =  tan  ( Œ  ).

ñ óò ò ó ò ò ó ó

= ô tan÷ ( ï ) ñ ó ò ò ó ò ó + õÿñ ó ò – ôƒõ tan÷ ( ï ) ñ .  Particular solutions:  =  :ž¼ ( , = 1, 2, 3), where the ý  are roots of the cubic equation ý 3 −  = 0.

ñ óò ò ó ò ò ó ó

+ ôñó ò – (1 + ô cot ï )ñ = 0. Particular solution: 0 = sin  .

ñóò ò ó ò ò ó ó

41.

+ ( ô ï + õ ) cot÷ ( ï )ñó ò – ô cot÷ ( Particular solution: 0 =  +  .

42.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

ï

+ õ ) cot÷ (

+ (ô

Particular solution: 43.

ñóò ò ó ò ò ó ó

ï

+ (ô

Particular solution: 44.

ñóò ò ó ò ò ó ó

+

ô

cot÷ (

Particular solution: 45.

ñóò ò ó ò ò ó ó

+ [ ô cot÷ (

Particular solution:

46. 47. 48.

ï

– 2 ô cot÷ (

ï )ñ

= 0.

ï )ñ óò

– 3 ô cot÷ (

ï )ñ

= 0.

2

= (  +  ) .

0

= (  +  )3 .

ï ) ñ óò 0

+ õ [ ô cot÷ (

0





−

=

)+

= 0.

ï )ñ óò

0

+ õ ) cot÷ (

ï )ñ

.

õ 3 ]ñ óò −

=





ï

) – õ 3 ]ñ = 0.

+ ôƒõ cot÷ (

ï )ñ

= 0.

.

+ õ cot÷ ( ï ) ñó ò ò ó + ô [ õ cot÷ ( ï ) – ô ]ñ = 0. This is a special case of equation 4.1.6.20 with  ( ) =  cot  ( Œ  ).

ñóò ò ó ò ò ó ó

+ [ ô + õ cot÷ ( ï )]ñó ò ò ó + ôƒõ cot÷ ( ï ) ñ = 0. This is a special case of equation 4.1.6.21 with  ( ) =  cot  ( Œ  ).

ñóò ò ó ò ò ó ó

= ô cot÷ ( ï ) ñó ò ò ó ò ó + õÿñó ò – ôƒõ cot÷ ( ï ) ñ .  Particular solutions:  =  :ž¼ ( , = 1, 2, 3), where the ý  are roots of the cubic equation 3 ý −  = 0.

ñóò ò ó ò ò ó ó

4.1.6. Equations Containing Arbitrary Functions 4.1.6-1. Equations of the form  4 ( )



 +  1 ( )
 +  0 ( ) = ( ). 1.

2.

3.

= ! ( ï )ñ . The transformation  = a

ñóò ò ó ò ò ó ó ñóò ò ó ò ò ó ó

ô ï öï

=I ! O

+õ + ý P (ö

−1

ï

= ‡"a

,

−3

leads to an equation of the same form: ‡ g

g
g
g = a

−8

 (1 a )‡ .

ñ

. + ý )8  +  ,‡ = leads to a simpler equation: ‡ ‘'

‘'

‘'‘ = ¢ The transformation H = (˜ + ( (˜ + )3 M M where ¢ =  − ˜( .

−4

 ( H )‡ ,

M

– ! ó ò ò ó ò ò ó ó ñ = 0, ! = ! ( ï ). Particular solution: 0 =  ( ).

!„ñ óò ò ó ò ò ó ó

© 2003 by Chapman & Hall/CRC

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656

FOURTH-ORDER DIFFERENTIAL EQUATIONS

4.

ñóò ò ó ò ò ó ó

+ !„ñó ò – ô ( ! + ô 3 )ñ = 0,  Particular solution: 0 =   .

5.

ñóò ò ó ò ò ó ó

= ! (ï ).

!

+ ( ! + ô 3 )ñó ò + ô!„ñ = 0,  Particular solution: 0 =  −  .

!

= ! (ï ).

6.

+ ( ô ï + õ ) ! ( ï )ñ„ó ò – ô! ( ï )ñ = 0. Particular solution: 0 =  +  .

7.

ñ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

ï

+ õ ) ! ( ï )ñ

+ (ô

Particular solution: 8.

9.

ñ óò ò ó ò ò ó ó

+ (ô

ï

0

+ õ ) ! ( ï )ñ

óò

– 2 ô! ( ï )ñ = 0.

óò

– 3 ô! ( ï )ñ = 0.

= (  +  )2 .

Particular solution: 0 = (  +  )3 . The substitution H = (  +  )
 −3  linear equation: H 


 + (  +  )  ( )H = 0.

ñ óò ò ó ò ò ó ó

+ ! ( ï )ñ

óò

+

! ó ò ( ï )ñ

= " ( ï ).




 +  ( ) = Xw ( )  + L . M

Integrating yields a third-order linear equation: 10. 11. 12.

13. 14. 15. 16.

leads to a third-order

+ 2 !„óÈò ñó ò + ( !„ó ò ò ó – ! 2 )ñ = 0, ! = ! (ï ). ¨ 

 −  ( )¨ = 0. The substitution ¨ = 

 +  ( ) leads to a second-order linear equation: u

ñóò ò ó ò ò ó ó

ï ñóò ò ó ò ò ó ó

+ ï ! ( ï )ñó ò – [( ï + 1) ! ( ï ) + ï + 4]ñ = 0.  Particular solution: 0 =   .

+ ! ( ï )ñ ó ò + " ( ï )ñ + % ( ï ) = 0. The transformation  = a −1 , = ¨ a −3 leads to an equation of the same form:

ñ óò ò ó ò ò ó ó

¨ g

g
g
g − a −6  (1 a )¨ g
+ V 3a −7  (1 a ) + a −8 (1 a )W ¨ + a −5 % (1 a ) = 0.

= ñ + ! ( ï )(ñ ó ò cosh ï – ñ sinh ï ). The substitution ¨ = 
cosh  − sinh  leads to a third-order linear equation.

ñ óò ò ó ò ò ó ó

= ñ + ! ( ï )(ñó ò sinh ï – ñ cosh ï ). The substitution ¨ = 
sinh  − cosh  leads to a third-order linear equation.

ñóò ò ó ò ò ó ó

= ñ + ! ( ï )(ñó ò sin ï – ñ cos ï ).
The substitution ¨ =  sin  − cos  leads to a third-order linear equation.

ñóò ò ó ò ò ó ó

= ñ + ! ( ï )(ñ ó ò cos ï + ñ sin ï ). The substitution ¨ = 
cos  + sin  leads to a third-order linear equation.

ñ óò ò ó ò ò ó ó

17.

+ !„ñó ò + ( ! tan ï – 1)ñ = 0, Particular solution: 0 = cos  .

18.

ñ óò ò ó ò ò ó ó

+ !„ñ ó ò – (1 + Particular solution:

19.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

=

r óò ò óò ò ó ó r ñ

!

cot ï )ñ = 0, 0 = sin  .

+ ! ( ï ) Oñó ò –

Y
 The substitution ¨ = 
− Y

r óò r ñ P

!

= ! ( ï ).

!

= ! ( ï ).

,

r

=

r (ï

).

leads to a third-order linear equation.

© 2003 by Chapman & Hall/CRC

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4.1. LINEAR EQUATIONS

4.1.6-2. Equations of the form  4 ( )



 +  2 ( )

 +  1 ( )
 +  0 ( ) = ( ). 20.

ñóò ò ó ò ò ó ó

+

!„ñóò ò ó

+ ô ( ! – ô )ñ = 0,

1 b . Particular solutions with  > 0:

1

= cos   £   ,

2 b . Particular solutions with  < 0:

1

= exp  − £ −   ,

ñóò ò ó ò ò ó ó

!

The substitution ¨ = 

 +  21.

23.

2

+ ( ! + ô )ñó ò ò ó + ô!„ñ = 0,

1

= cos  £   ,

2 b . Particular solutions with  < 0:

1

= exp  − £ −   ,

+ ! ( ï )( ï

2

ñóò ò ó ò ò ó ó

+ ! ( ï )( ï

2

2

= ! (ï ).

1 b . Particular solutions with  > 0:

ñóò ò ó ò ò ó ó

= sin   £   . = exp   £ −   .

¨ 

 + (  −  )¨ = 0. leads to a second-order linear equation: u

The substitution ¨ = 

 +  22.

= ! ( ï ).

!

2

= sin  £   . 2

= exp   £ −   .

¨ 

 +  ( )¨ = 0. leads to a second-order linear equation: u

ñóò ò ó

– 2ï

ñóò

+ 2ñ ) = 0.

ñóò ò ó

– 4ï

ñóò

+ 6ñ ) = 0.

Particular solutions: 1 =  , 2 =  2 . The substitution H =  2 

 − 2 
 + 2 leads to a second-order linear equation: "H 

 − 2 H 
+  3  ( ) H = 0. Particular solutions: 1 =  , 2 =  3 . The substitution ¨ =  2 

 − 4 
 + 6 leads to a ¨ 

 +  2  ( )¨ = 0. second-order linear equation: u 2

24.

ñóò ò ó ò ò ó ó

+ (ô

ï

2

+

õï

+ ö ) ! ( ï )ñó ò ò ó – 2 ô! ( ï )ñ = 0. 0

= 

+ 10 !

ó ò ñ óò

Particular solution: 25.

ñ óò ò ó ò ò ó ó

+ 10 !„ñ

óò ò ó

Solution:

2

+ ' + ( .

+ (3 ! =L

1

¨

óò òó 3 1

+ 9!

+L

2

¨

2

2 1

)ñ = 0,

¨

2

+L

3

= ! (ï ).

! ¨ ¨ 1

2 2

+L

4

¨ 3, 2

where ¨ 1 and ¨ 2 are nontrivial linearly independent solutions of the second-order linear equation: ¨3

 +  ¨ = 0. 26.

ñóò ò ó ò ò ó ó

+ ( ! + " )ñó ò ò ó + 2 !„óò ñó ò + ( !„ó ò ò ó +

!" )ñ

= ! ( ï ),

!

= 0,

= " ( ï ).

"

¨ 

 + ¨ = 0. leads to a second-order linear equation: u

The substitution ¨ = 

 + 

4.1.6-3. Other equations. 27.

28.

óò ò ó ò ó

+ï

Particular solution:

0

ñ óò ò ó ò ò ó ó ñóò ò ó ò ò ó ó

+ ! ( ï )ñ

+ ! ( ï )ñó ò ò ó ò

ó

ñ óò ò ó ò ò ó ó

+

!„ñ óò ò ó ò ó

+ "Iñ

– 2" ( ï )ñ = 0.

=  2.

– 2ô

Particular solutions: 29.

" ( ï )ñ ó ò

ñóò ò ó

–ô

+ ô!„ñ

óò

2

−  , 1 = 

óò ò ó

2

! (ï )ñ óò

 . 2 = 

+

ô 4ñ

= 0.

+ ô (" – ô )ñ = 0,

!

1 b . Particular solutions with  > 0:

1

= cos  £   ,

2 b . Particular solutions with  < 0:

1

= exp  − £ −   ,

The substitution ¨ = 

 + 

2

= ! ( ï ),

"

= sin  £   . 2

= " ( ï ).

= exp   £ −   .

¨ 
+( −  )¨ = 0. ¨ 

 +  2 leads to a second-order linear equation: u

© 2003 by Chapman & Hall/CRC

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658 30.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ò ó ó

+ !„ñ ó ò ò ó ò ó + (" + ô )ñ ó ò ò ó + ô!„ñ 1 b . Particular solutions with  > 0:

óò

+ ô"Iñ = 0, £  , 1 = cos  

31.

+ ! ( ï )ñó ò ò ó ò ó + " ( ï )ñó ò ò ó + ï Particular solution: 0 =  .

32.

ñóò ò ó ò ò ó ó

33.

34.

35.

ñóò ò ó ò ò ó ó

+ " ( ï )( ï

2

ñóò ò ó

% (ï )ñ óò

– % ( ï )ñ = 0.

– 2ï

+ 2ñ ) = 0.

ñóò

+ ! ( ï )( ï 3 ñó ò ò ó ò ó – 3 ï 2 ñó ò ò ó + 6 ï ñó ò – 6ñ ) = 0. Particular solutions: 1 =  , 2 =  2 , 3 =  3 . The substitution ¨ =  3 


 −3 2 

 +6 
 −6 leads to a first-order linear equation: ¨u
+  3  ( )¨ = 0.

= ! ( ï )ñó ò ò ó ò ó + ôñó ò – ô! ( ï )ñ .  Particular solutions:  =  ¼ ( , = 1, 2, 3), where the Œ  are roots of the cubic equation Œ 3 −  = 0.

ñóò ò ó ò ò ó ó

= ( ! – ô )ñó ò ò ó ò ó + ( ô! – õ )ñó ò ò ó + ( õ! – ö )ñó ò + ö!„ñ , ! = ! ( ï ).  ¼ ( , = 1, 2, 3), where the Œ  are roots of the cubic equation Particular solutions:  =  Œ 3 + Œ 2 + Œ + ( = 0.

ñóò ò ó ò ò ó ó

37.

ñ óò ò ó ò ò ó ó

40.

ñóò ò ó ò ò ó ó

+ ( ! 3 + ô )ñ ó ò ò ó ò Particular solution:

ó 0

+ (! 2 +  =  − .

ô! 3 )ñ óò ò ó

+ (!

2

1

ï "Iñóò – ô 2I" ñ

+

ô! 2 )ñ óò

!

= 0,

+ ô!

1

ñ

= 0,

= ! ( ï ),

!÷

"

= " ( ï ).

! ÷ (ï

=

).

+ 4ñó ò ò ó ò ó + ô ï ñ = ! ( ï ). The substitution ¨ ( ) =  leads to a constant coefficient nonhomogeneous linear equation: ¨2



 +  ¨ =  ( ).

ï ñóò ò ó ò ò ó ó

+ ô ï ñó ò ò ó ò ó + ( ï 2 ! + õ )ñó ò ò ó + ( ô – 4) ï !„ñó ò + ( õ – 2 ô + 6) !„ñ = 0, ! = ! (ï ). 
¨ 2 

 + (  − 4)  + (  − 2  + 6) leads to a second-order linear equation: The substitution =  ¨2

 +  ¨ = 0.

ï 2 ñ óò ò ó ò ò ó ó ï 4 ñ óò ò ó ò ò ó ó

+

ô ï 3 ñ óò ò ó ò ó

+

Particular solution: 41.

= exp   £ −   .

ñóò ò ó ò ò ó ó

+ ( ! + ô )ñó ò ò ó ò ó + ( ô! + " + ô ï " )ñó ò ò ó + ô  Particular solutions: 1 =  , 2 =  −  .

39.

2

Particular solutions: 1 =  , 2 =  . The substitution H =  2 

 − 2 
 + 2 leads to a second-order linear equation: "H 

 + [  ( ) − 2] H 
+  3 ( ) H = 0. 2

36.

38.

= ! (ï ), " = " ( ï ). = sin   £   .

¨ 
+ ¨ = 0. ¨ 

 +  2 leads to a second-order linear equation: u

The substitution ¨ = 

 + 

ó

2

£ −  , 1 = exp  − 

2 b . Particular solutions with  < 0:

+ ! ( ï )ñó ò ò ó ò

!

ñ óò ò ó ò ò ó ó

+ 6 !„ñ

óò ò ó ò ó

Solution:

0

ï ! (ï )ñ óò

=

3−

.

+ ( ô – 3) ! (ï )ñ = 0.

+ (4 ! ó ò + 11 ! 2 + 10" )ñ + 3(2 !„óò " + 5 !"Ió ò + 6 ! =L

1

¨

3 1

+L

2

óò ò ó

¨

2

"

+ (! ó ò ò ó + 7! ! ó ò + 6! + "Ió ò ò ó + 3" 2 )ñ = 0,

2 1

¨

2

+L

3

¨ ¨ 1

2 2

+L

4

3

+ 30 !" + 10" ! = ! (ï ),

"

ó ò )ñ ó ò

= " ( ï ).

¨ 3, 2

where ¨ 1 and ¨ 2 are nontrivial linearly independent solutions of the second-order linear equation: ¨3

 +  ¨2
+ ¨ = 0. 42.

( !„ñó ò ò ó )ò òó

ó

= 0,

!

= ! ( ï ).

Equation of transverse vibrations of a bar. Solution:

=L

1+L

2 + X 

  −a (L 0  (a )

3

+L

4

a) a. M

© 2003 by Chapman & Hall/CRC

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4.2. NONLINEAR EQUATIONS

4.2. Nonlinear Equations 4.2.1. Equations Containing Power Functions 4.2.1-1. Equations of the form 



 =  ( , ). 1.

ñ óò ò ó ò ò ó ó

=


ñ

)

–5 3

.

Multiply both sides of the equation by 5 J 3 and differentiate the resulting expression with respect to  to obtain  + 5 




  = 0. 3! (5) Integrating this equation three times, we arrive at the chain of equalities:





 + 2 


 − 


 − 2( 


3! 3! 3!





 − ( 

 )2 = 2 L 2 , 

 = 2 L 2  + L 1 , )2 = L 2  2 + L 1  + L 0 ,

(1) (2) (3)

where L 0 , L 1 , and L 2 are arbitrary constants. By eliminating the highest derivatives from (1)–(3) with the help of the original equation, we obtain a first-order equation: (2

.

.

. 
− 3 
)2 = 9( L

2 1

− 4L

0

V 9( L

2.

ñóò ò ó ò ò ó ó

=


ñ ú

2 1

− 4L

0

−2

.

3

. = (  ¨ )3 J

= L 2  2 + L 1  + L 0 . The substitution where the integration of which finally yields:

ù

2

L 2)

L 2 ) + 54 # ¨ − 2¨

J

3 −1 2

W

M¨

.

¨

.

+ 54 # 2

J ,

4 3

leads to a separable equation,

AçX

M

3

.

=L

3.

+ .

This is a special case of equation 4.2.6.1 with  (¨ ) = # 1 b . By integrating, we obtain 2# 2 



 − ( 

 )2 = where L is an arbitrary constant ( ¡ second-order equation:

+1

¡

+

+1

+

4 L , 3

≠ −1). The substitution ¨ ( ) = (
 )3 J

+ ¨

= O 3#  2 +2 ¡

+1

+L

¨ P

2

leads to a

J .

−5 3

The value L = 0 corresponds to the Emden–Fowler equation, whose integrable cases are specified in Section 2.3 for some values of (to those cases there correspond three-parameter ¡ families of particular solutions of the original equation). 2 b . Particular solution: 3.

ñóò ò ó ò ò ó ó = ô ï –3ú

–5

ñ ú

ñ óò ò ó ò ò ó ó

=

ô ï

–

3

ú

+5

Q

8(

¡

.

The transformation  = a 4.

=

−1

,

=a

−3

+ 1)( + 3)(3 ¡ ¡ # ( − 1)4

¡

+ 1) +

S

1 −1

 +L 

4 1−

+ .

¨ (a ) leads to an equation of the form 4.2.1.2: p ¨ g

g
g
g =  ¨2+ .

ñ ú

. This is a special case of equation 4.2.6.5 with  (¨ ) =  ¨p+ . 2

© 2003 by Chapman & Hall/CRC

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660 5.

FOURTH-ORDER DIFFERENTIAL EQUATIONS =ô ï ÷ ñ ú . Generalized homogeneous equation.

ñ óò ò ó ò ò ó ó

1 b . The transformation a =   2 b . The transformation  = H H −  −3 + −5 ¨2+ . 6.

+4

−1

,

+

−1

=H


 leads to a third-order equation.

, ‡ = −3

¨ ( H ) leads to an equation of the same form: ¨u‘'

‘'

‘'‘ =

= ( ôñ + õ ï  )ú , ü = 0, 1, 2, 3. ¨ 



 =  +2¨2+ . The substitution  ¨ =  + '  leads to an equation of the form 4.2.1.2: u

ñ óò ò ó ò ò ó ó

7.

( ô ï + õ )4 ñ ó ò ò ó ò ò ó ó = ö^ñ ú . This is a special case of equation 4.2.6.10 with  (¨ ) = ( ¨p+ .

8.

= (ô ï 2 + õ ï + ö ) 2 ñ ú . This is a special case of equation 4.2.6.12 with  (¨ ) = ¨p+ .

ï 3ú

+1

ñóò ò ó ò ò ó ó

–

3

ú

+5

4.2.1-2. Equations of the form 



 =  ( , , 
 ). 9.

ñóò ò ó ò ò ó ó

=

ôñ ÷ ñóò

+

õ ï

.




 =

By integrating, we find



B +1



+1

+



, +1

 

+1

+ L . For  = 0, the order of this

equation can be reduced by one with the help of the substitution ¨ ( ) =
 . 10.

11.

12.

13.

14.

= ô ï – 4 (ï ñ óò The transformation ¨ g

g
g − 5¨ g

g + 6¨ g
=

ñ óò ò ó ò ò ó ó

– ñ )÷ . a = ln | |, ¨  ¨  .

=  
 −

leads to a third-order autonomous equation:

= ô ïƒ÷ ( ï ñó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =   ¨  . The substitution ¨ =  ¨ 
 ) 

 =   ¨  . leads to a third-order generalized homogeneous equation: (p

ñóò ò ó ò ò ó ó


−

= ô ïƒ÷ ( ï ñó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =   ¨  . The substitution ¨ =  
−2 leads to a third-order generalized homogeneous equation:  ¨p


 − ¨2

 =   +2 ¨  .

ñóò ò ó ò ò ó ó

= ô ïƒ÷ ( ï ñó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =   ¨  . The substitution ¨ =  leads to a third-order generalized homogeneous equation: ¨p


 =   +1 ¨  .

ñóò ò ó ò ò ó ó


−3

= ô 4 ñ + õ ïƒ÷ (ñó ò – ôñ )  . The substitution ‡ = 
−  leads to a third-order equation: ‡


 + ‡

 +  2 ‡
 +  3 ‡ = '  ‡  .

ñóò ò ó ò ò ó ó

4.2.1-3. Equations of the form 



 =  ( , , 
 , 

 ). 15.

+ ôñ ó ò ò ó = õÿñ ÷ + ö . This is a special case of equation 4.2.6.33 with  ( ) =   + ( .

16.

ñ óò ò ó ò ò ó ó

ñ óò ò ó ò ò ó ó

–

5 2

ôñ óò ò ó

+

õÿñ –5 ) 3 . 2  ¨ ˆ = , (ˆ ) = ˆ 3 J 2

9 16

The transformation 4.2.1.1: ¨3Šª

Šª

ŠªŠ =  −2  ¨

ô 2ñ

=

J .

−5 3

leads to an autonomous equation of the form

© 2003 by Chapman & Hall/CRC

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661

4.2. NONLINEAR EQUATIONS

17.

ñóò ò ó ò ò ó ó

+ ôñó ò ò ó + õÿñ = ö^ññó ò ò ó – ö (ñó ò )2 + ü . 1 b . Particular solution: =L where the constants L

sinh( L

1

L 2 , L 3 , and L

1,

4

4 4

L 2 2

( (L

4

 )+L

2

cosh( L

4

 ) + L 3,

are related by two constraints

+ (  − (L 2 1)

−L

2 4

L

2 4

+  = 0,

3

+ , = 0.

L

3)

− L

2 b . Particular solution: =L where the constants L

sin( L

1

L 2 , L 3 , and L

1,

4 4

L 2 1

( (L 18.

ñ óò ò ó ò ò ó ó

+

ôññ óò ò ó

+

õÿñ óò ò ó

Particular solution: 19.

– ô (ñ

=L

1

)2 +

exp( L

2

 )+L

cos( L

4

 ) + L 3,

are related by two constraints

− (  − (L 2 2)

+L

ö^ñ óò

 )−

2

2 4

L

3)

2 4

+  = 0,

3

− , = 0.

L

+ L

= 0.

L

= ôñ 2 ñó ò ò ó – ôñ (ñó ò )2 + õÿñ . 1 b . Particular solution: = L 1 exp( L

3 2

+ L

L

2

+(

2

.

ñóò ò ó ò ò ó ó

where the constants L

L 2 , and L

1,

2 b . Particular solution: where the constants L

L 2 , and L

1,

1

2

3

 )+L

exp(− L

2

3

 ),

are related by the constraint L

2

=L

3 b . There are also solutions 20.

óò

4

4

cos( L

3

 )+L

2

sin( L

3

ü

  +L

and

− 4 L

4 3

+  (L

1

L 2L

−  = 0.

2 3

 ),

are related by the constraint L

= A"

4 3

2 1

+L

2 2)

L

2 3

−  = 0.

= 0.

+ ô (ñó ò )÷ ñó ò ò ó = õÿñ  + ö . This is a special case of equation 4.2.6.34 with  (‡ ) = ‡  and ( ) =   + ( .

ñóò ò ó ò ò ó ó

21.

= ô ï –2 ( ï ñó ò – ñ )÷ ñó ò ò ó . This is a special case of equation 4.2.6.35 with  (¨ ) =  ¨  .

22.

ññ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

– (ñ ó ò ò ó )2 = 0. 1 b . Particular solutions: =L

 + L 2, = L 1 ( + L 2 )−3 J 2 , = L 1 exp( L 3  ) + L 2 exp(− L 3  ), = L 1 cos( L 3  ) + L 2 sin( L 3  ). 1

2 b . Integrating the equation twice, we arrive at a second-order equation:

! 

 − ( 
)2 = L  + L . 1 2 The substitution H = L

1

 +L

2

leads to a generalized homogeneous equation.

© 2003 by Chapman & Hall/CRC

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662 23.

24.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ññóò ò ó ò ò ó ó

– (ñó ò ò ó )2 + ôñ + Particular solutions:

õ

= 0.

=L

1

exp( Π ) + L

=L

1

sin( Π ) + L

2

exp(− Œ  ) −   ,

cos( Œ  ) −   ,

2

Π= ( 2  Π= ( 2 

)1 J 4 ; )1 J 4 .

ññóò ò ó ò ò ó ó

– (ñó ò ò ó )2 + ôñó ò ò ó = 0. 1 b . Integrating the equation two times, we obtain a second-order equation: !

 −(
 )2 +  L 1  + L 2.

=

2 b . Particular solutions:

 ) + L 2 exp(− L 3  ) − L 3−2 , = L 1 sin( L 3  ) + L 2 cos( L 3  ) + L 3−2 , = L 1  + L 2. =L

25.

1

exp( L

3

ññ óò ò ó ò ò ó ó

– (ñ ó ò ò ó )2 = ô [ññ ó ò ò ó – (ñ ó ò )2 ] + õ . 1 b . The substitution ¨ ( ) = !

 − (
 )2 leads to a second-order constant coefficient linear equation of the form 2.1.9.1: ¨ 

 =  ¨ +  . 2 b . Particular solutions:

  −

=L

1

exp ÝL

=L

1

exp  £   −

=L

1

sin  Π  + L

2

4 L

2

 

1

L

4 L 2

2 2 1

exp  − L

2

 

if  ≠ 0,

 − £   + L

cos  Π  ,

Œ

2

=

if  > 0,

2

 (L

2 1



+L

2 2)

if  > 0,

£ − sin   £ −  + L 1  + L 2 if  < 0,  < 0,  = A"   +L 1 if  > 0. ü ññóò ò ó ò ò ó ó – (ñóò ò ó )2 = ô V ññóò ò ó – (ñóò )2 W + õ ï + ö . The substitution ¨ ( ) = ! 

 − (
 )2 leads to a second-order constant coefficient nonhomogeneous linear equation of the form 2.1.9.1: ¨p

 =  ¨ + '  + ( . =

26.

27.

ññóò ò ó ò ò ó ó

– 16 (ñó ò ò ó )2 = Particular solution:

ô ï

2

õï

+

+ö .

L 1  4 + 16 L 2  3 + 12 L 3  2 + L 4  + L 5 , where the constants L 1 , L 2 , L 3 , L 4 , and L 5 are related by three constraints 1 24

=

1 3

L 1 L 3 − 16 L L 1 L 4 − 13 L 2 L L 1 L 5 − 16 L

28.

29.

ñóò ò ó ò ò ó ó

= ô 2 ñ + õ (ñó ò ò ó The substitution ¨ = 2.9.1.1: ¨3

 =  ¨ + 

2 2

= ,

3

=,

2 3

=(.

+ ôñ )  . 

 +  leads to a second-order autonomous equation of the form ¨ .

÷

= ôñ V ññ ó ò ò ó – (ñ ó ò )2 W . This is a special case of equation 4.2.6.42 with  (¨ ) = 0 and (¨ ) =  ¨  .

ñ óò ò ó ò ò ó ó

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4.2. NONLINEAR EQUATIONS

4.2.1-4. Equations of the form 



 =  ( , , 
 , 

 , 


 ). 30.

ñ óò ò ó ò ò ó ó

+ ô 3 ñ óò ò ó ò ó + ô Particular solutions:

2

ñ óò ò ó

+ ô 1 ñ ó ò + ô 0 ñ = õÿññ ó ò ò ó – õ (ñ ó ò )2 + ü . = L exp( Œ   ) + , −1 0 , where L is an arbitrary constant and Œ = Œ  4

are roots of the algebraic equation Π31.

ñóò ò ó ò ò ó ó

+

ôññóò ò ó ò ó

=

õ ïƒ÷

+  3Œ

3

+ O

ñóò ò ó ò ò ó ó

+

ôññóò ò ó ò ó

, Œ  0P

−

2

+  1Π+ 

0

= 0.

.

Integrating, we arrive at a third-order equation: 32.

2

1 


 +  ! 

 −  ( 
)2 = 2



B +1

 

+1

+L .

– ôñó ò ñóò ò ó = 0.

This equation arises in hydrodynamics. 1 b . Particular solutions:

=L

 + L 2, = L 1 exp( L 2  ) −  = 6(  + L 1 )−1 . 1

−1

L 2,

2 b . Integrating, we arrive at a third-order autonomous equation:


 +  !

 −  (
 )2 = L .

^_ 33.

34. 35.

Reference: A. D. Polyanin and V. F. Zaitsev (2002).

ñ óò ò ó ò ò ó ó + ôññ óò ò ó ò ó – ôñ Particular solutions:

ñóò ò ó ò ò ó ó

+

ôññóò ò ó ò ó

óò ñ óò ò ó

=

õÿñ

.

=L

1

exp( Π ) + L

=L

1

sin( Π ) + L

+ õ (ñó ò )÷

ñóò ò ó

=

ö ï

2 2

exp(− Œ  ),

cos( Π ),

+ý .

This is a special case of equation 4.2.6.50 with  (‡ ) = '‡  and ( ) = (˜  + .

M

ñóò ò ó ò ò ó ó = ô ñ ÷ ñ óò ñóò ò ó ò ó

.

This is a special case of equation 4.2.6.51 with  ( ) =  36. 37. 38.

39.

40.

Π=  1J 4, Π=  1J 4.

 .

ô ï –5 ) 3 ñ –5 ) 3 .

ï ñóò ò ó ò ò ó ó

+ 4ñó ò ò ó ò

ó

=

ï ñóò ò ó ò ò ó ó

+ 4ñó ò ò ó ò

ó

= ô (ï

ñ )

.

ï ñóò ò ó ò ò ó ó

+ 2ñó ò ò ó ò

ó

= ô (ï

ñóò

– ñ ) .

ï ñóò ò ó ò ò ó ó

+ ( ô + 3)ñó ò ò ó ò

=

õ ïƒ÷ (ï ñ óò

¨ 



 =  ¨ leads to an equation of the form 4.2.1.1: I

The substitution ¨ ( ) =  The substitution ¨ ( ) = 

J .

−5 3

¨ 



 =  ¨  . leads to an equation of the form 4.2.1.2: I

The substitution ¨ ( ) =  
− leads to a third-order equation: ¨u


 =  ¨  (Section 3.2 presents its solutions for , = − 72 , − 52 , −2, − 43 , − 67 , − 12 , 0, and 1).

ó

The substitution ¨ =  
+  ¨ 


 = '  ¨  .

ï 2 ñ óò ò ó ò ò ó ó

+ 8ï

ñóò ò ó ò ó

2

ôñ )

.

leads to a third-order generalized homogeneous equation:

+ 12ñó ò ò ó =

The substitution ¨ ( ) = 

+

ô ï

) ñ

)

–10 3 –5 3

. ¨ 



 =  ¨ leads to an equation of the form 4.2.1.1: u

J .

−5 3

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664 41.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ï 4 ñ óò ò ó ò ò ó ó

+ 6ï

3

+ 7ï

ñóò ò ó ò ó

2

ñóò ò ó

ï ñóò

+

=

ôñ –5 ) 3 .

The substitution a = ln | | leads to an equation of the form 4.2.1.1:



 = 

J .

−5 3

42.

ññ óò ò ó ò ò ó ó = ôñ óò ñ óò ò ó ò ó . Having integrated this equation, we obtain the third-order equation


 = L solvable cases are specified in Subsection 3.2.2.

43.

ññóò ò ó ò ò ó ó

– ñó ò ñóò ò ó ò

ó

=

ô ƒï ÷ ñ

2

.

44.

ññóò ò ó ò ò ó ó

– ñó ò ñóò ò ó ò

ó

ôñóò

.

+ 3(ñ

óò ò ó

=

   B +1




 = O

Integrating yields a third-order linear equation:

.

P

óò ñ óò ò ó ò ó

ô ï ÷

ññ óò ò ó ò ò ó ó

+ 4ñ

. This is a special case of equation 4.2.6.58 with  ( ) =   .

46.

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

ó

+ 3(ñó ò ò ó )2 =

ôñ

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

ó

+ 3(ñó ò ò ó )2 =

ôñ ÷

ññ óò ò ó ò ò ó ó

+

48.

+L

Integrating yields a third-order constant coefficient nonhomogeneous linear equation: /


 = L − .

45.

47.

+1

 , whose

The substitution ¨ = The substitution ¨ = 2 3

2

ñ óò ñ óò ò ó ò ó

2

)2 =

)

–10 3

.

¨ 



 = 2  ¨ leads to an equation of the form 4.2.1.1: u

J .

−5 3

.

¨ 



 = 2  ¨ ´J 2 . leads to an equation of the form 4.2.1.2: u

– 13 (ñ

óò ò ó

)2 = ô .

This is a special case of equation 4.2.6.59 with  = 23 and  ( ) =  . Integrating the equation twice, we arrive at a second-order equation of the form 2.8.1.54:

  3! 

 − 2( 
)2 = 32  49.

ññóò ò ó ò ò ó ó

+

3 2

ñóò ñóò ò ó ò ó

2

+L

1

 + L 2.

+ 12 (ñó ò ò ó )2 = ô .

Integrating the equation twice, we arrive at a second-order equation of the form 2.8.1.53:

! 

 − 1 ( 
)2 = 1   4 2 50.

ññóò ò ó ò ò ó ó

+

3 2

ñóò ñóò ò ó ò ó

+ 12 (ñó ò ò ó )2 = ( ô

ññóò ò ó ò ò ó ó

– ñó ò ñóò ò ó ò

ó

=

ô ƒï ÷ ñ ñóò ò ó ò ó

+ õ )ñ

.

Integrating yields a third-order linear equation: 52.

ï ññóò ò ó ò ò ó ó

– ï (ñó ò ò ó )2 = ô .

)

–1 2

+L

1

 + L 2.

.

2 = ( g
) leads to a constant coefficient linear equation:

The transformation  =  (a ), 2 (5) g =  +  . 51.

ï

2




 = L exp O



B +1

 

+1

P

.

Integrating the equation twice, we arrive at a second-order equation:

53.

ï ññóò ò ó ò ò ó ó

ï ñóò ñóò ò ó ò ó

! 

 − ( 
)2 =  ln | | + L  + L . 1 2

+ ôññó ò ò ó ò ó . Integrating yields a third-order linear equation of the form 3.1.2.7:


 = L3  . =

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4.2. NONLINEAR EQUATIONS

54.

ñ 3 ñ óò ò ó ò ò ó ó

= 4ñ 2 ñó ò ñó ò ò ó ò ó + 3ñ 2 (ñó ò ò ó )2 – 6(ñó ò )4 . This is a special case of equation 4.2.6.67 with  ≡ 0. Solution in parametric form:

 =A X 55.

56.

2ˆ

ü

+L

ñóò ò óñóò ò ó ò ò ó ó

= ô (ñó ò ò ó ò ó )2 .

Solution:

=

a

+L L 0+L

L

0

 M

4

2

ˆ +L

+L

1

=L

3,

ˆ

exp ”A X

4

2ˆ

ü 3−2

 + ( L 2 + L 3  ) 1−  1  + L 2 exp( L 3  ) 1



4

ˆ M

+L

2

ˆ +L

1

•

.

if  ≠ 1, if  = 1.

– 12 (ñ ó ò ò ó ò ó )2 =  ( ï ñ ó ò – ñ ) + ñ Differentiating with respect to  yields

ñ óò ò ó ñ óò ò ó ò ò ó ó

óò

+ .

 − ƒ… − ý W = 0. 

 V (5)

(1)

Equating the second factor in (1) with zero and integrating it, we find the solution: =ƒ



6

6!



+ý

5

5!

+L

4



4

+L

3

3



+L

2

2



+L

1

 + L 0.

The constants L  and parameters ƒ , ý , and = are related by the constraint 48 L

2

L

4

− 18 L

2 3

= − ƒsL

+ ýhL

0

1

+= ,

obtained by means of substituting the solution into the original equation. The other solution, which corresponds to setting the first factor in (1) to zero, is given by: =L

57.

¶

1

¶  + L 0,

where

ƒ L

¶

0

= ôñ  ñó ò (ñó ò ò ó ò ó )s . This is a special case of equation 4.2.6.72 with  ( ) =  ° = 1, see equation 4.2.1.42. The first integral has the form:À

−ý L

ñóò ò ó ò ò ó ó

For L

 1  +1 = L ( 


 )1− − 1−° , +1   +1 = L ln | 


 | − À , +1 1


(  )1− −  ln | | = L 1−° = 0, equality (1) is changing to the equation 


 = ¬

 (1 − ° ) ® , +1

1

− = = 0.

À

 and (¨ ) = ¨ . For , = −1 and

if , ≠ −1, ° ≠ 1;

(1)

if , ≠ −1, ° = 1;

(2)

if , = −1, ° ≠ 1.

(3)

À 1 1−

¶



À +1 1−

,

which is discussed in Subsection 3.2.2 (the solutions given there generate 3-parametric families of particular solutions of the original equation for , = (1 − ° )ý − 1, where ý = − 72 , − 52 , −2, − 43 , − 67 , − 12 , 0, and 1). 58.

ô 1 (ñ ó ò ò ó ò ò ó ó

)2 + ( ô

ñ ó ò ò ó ò ó + ô 3 ñ ó ò ò ó + 6 ô 4 ñ + ô 5 ï + ô 6 )ñ ó ò ò ó ò ò ó ó + õ 1 (ñó ò ò ó ò ó )2 + ( õ 2 ï + õ 3 )ñó ò ò ó ò ó – ô 4 (ñó ò ò ó )2 + õ 4  ñ óò ò ó 2

+õ There are particular solutions of the form = L 1  4 + L 2  3 + L 3  2 + L five constants L 1 , L 2 , L 3 , L 4 , and L 5 are related by three constraints.

+ õ 6 ï + õ 7 = 0. 4  + L 5 , where the

5

ï

2

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FOURTH-ORDER DIFFERENTIAL EQUATIONS

4.2.2. Equations Containing Exponential Functions 4.2.2-1. Equations of the form 



 =  ( , ). 1.

ñóò ò ó ò ò ó ó

=

ñóò ò ó ò ò ó ó

=

ô]

+õ .

This is a special case of equation 4.2.6.1 with  ( ) =   +  . 2.

3.

ô] +_ ó

+õ .

+ ( ý Œ )

¨ = The substitution ? ¨2



 =   +  .

ñóò ò ó ò ò ó ó

=

ñóò ò ó ò ò ó ó

=

ô ï

–4

]

leads to an autonomous equation of the form 4.2.6.1:

.

This is a special case of equation 4.2.6.2 with  ( ) =    . The substitution a = ln | | leads to an autonomous equation. 4. 5. 6.

7. 8.

ô ï   ]

. This is a special case of equation 4.2.6.15 with  (¨ ) =  ¨ and

ñ óò ò ó ò ò ó ó

ô  ó ñ ÷

=

.

This is a special case of equation 4.2.6.14 with  (¨ ) =  ¨ and

ñ óò ò ó ò ò ó ó

=

ô

= , + 4.

¡

exp( Ė + 

The substitution ? ¨ = ¨2



 =   +  .

ñóò ò ó ò ò ó ó

= ô (ñ +

õ ó

ñóò ò ó ò ò ó ó

= ô (ñ +

õ ó )ú

)–5 )

ï

)+õ . + ( ý Œ )

¡

= B − 1.

2

3

–

õ ó

2

leads to an autonomous equation of the form 4.2.6.1:

.

 ¨ 



 =  ¨ The substitution ¨ = +   leads to an equation of the form 4.2.1.1: u –

õ ó

J .

−5 3

.

 ¨ 



 =  ¨2+ . The substitution ¨ = +   leads to an equation of the form 4.2.1.2: u

4.2.2-2. Other equations. 9.

ñ óò ò ó ò ò ó ó

=

ô ] ñ óò

+

õ _ ó

.

Integrating yields a third-order equation: 10. 11. 12. 13.

ñóò ò ó ò ò ó ó = ô 4 ñ + õ ó (ñ óò




 = Œ



  + ý



  : +L .

– ôñ )  .

 This is a special case of equation 4.2.6.27 with  ( , ¨ ) =   ¨  .

ñóò ò ó ò ò ó ó

=

õ ó (ï ñ óò

– ñ ) .

ñóò ò ó ò ò ó ó

=

õ ó (ï ñ óò

– 2ñ )  .

ñóò ò ó ò ò ó ó

=

õ ó (ï ñ óò

– 3ñ )  .

 This is a special case of equation 4.2.6.23 with  ( , ¨ ) =   ¨  .  This is a special case of equation 4.2.6.24 with  ( , ¨ ) =   ¨  .  This is a special case of equation 4.2.6.25 with  ( , ¨ ) =   ¨  .

© 2003 by Chapman & Hall/CRC

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4.2. NONLINEAR EQUATIONS

14.

ó

ññ óò ò ó ò ò ó ó

– (ñ ó ò ò ó )2 = ô  . 1 b . Integrating the equation twice, we arrive at a second-order equation: ! 

 − ( 
)2 = Œ −2    + L  + L . 1 2 For L 1 = L 2 = 0, it is an equation of the form 2.8.3.47. 2 b . Particular solution:

15.

= L exp( Π ) +

ó

ññóò ò ó ò ò ó ó

 . L Π4 p

– (ñó ò ò ó )2 – ôñó ò + õ = 0. 1 b . Integrating yields a third-order equation: !"


 − 
 

 −  2 b . Particular solutions: LpŒ −  = L exp( Œ  ) + , 4

+ Œ

−1





=L .

LpŒ   = exp( Œ  ) + L exp(− Œ  ) − 3 . 2 Œ Œ

ó

16.

ññóò ò ó ò ò ó ó

– (ñó ò ò ó )2 – ôñó ò ò ó + õ = 0. 1 b . Integrating the equation two times, we obtain a second-order equation: ! 

 − ( 
)2 −   + L 1  + L 2 + Œ −2  = 0. LpŒ 2 −  = L exp( Œ  ) + . 2 b . Particular solution: 4

17.

ññóò ò ó ò ò ó ó

= ô [ññó ò ò ó – ( ñ óò ) ] + õ ó 2

2 b . Particular solution: 18.

= L exp( Π ) +

ó

ññóò ò ó ò ò ó ó

1b .

 
 2b .

19.

LpŒ

– (ñó ò ò ó ) . ¨ !


2 1 b . The substitution ( ) =  − (  ) leads to a second-order constant coefficient linear  equation of the form 2.1.9.1: ¨u

 =  ¨ +   . 2

LpŒ

Œ



2( 2

− )

.

– ôñó ò ò ó ò ó – (ñó ò ò ó )2 + õ = 0. Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  + L 1  + L 2 + Œ −2  = 0. Particular solutions: LpŒ 3 −  = L exp( Œ  ) + , 4

 = 2 Œ

ñ óò ò ó ò ò ó ó

LpŒ

3

exp( Œ  ) + L exp(− Œ  ) −

= ô ] ñ ó ò ñ ó ò ò ó ò ó . This is a special case of equation 4.2.6.51 with  ( ) =   .

 Œ

.

ó

20.

= ( ô]¦ñó ò + õ _ )ñó ò ò ó ò ó .  This is a special case of equation 5.2.6.58 with B = 4,  ( ) =   , and ( ) =   : .

21.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

– 4 ƒñó ò ò ó ò

ó

+6

2

ñóò ò ó

The substitution ¨ ( ) =  − 

–4



3

ñóò

+

4

ñ

=

ô

exp 

ï  ñ –5 ) 3 .

8 3

¨ 



 =  ¨ leads to an equation of the form 4.2.1.1: u

ó

J .

−5 3

22.

– 4 ƒñ ó ò ò ó ò ó + 6 2 ñ ó ò ò ó – 4 3 ñ ó ò + 4 ñ = ô  (1–ú ) ñ ú .  ¨ 



 =  ¨2+ . The substitution ¨ ( ) =  −  leads to an equation of the form 4.2.1.2: u

23.

ññ óò ò ó ò ò ó ó

+ 4ñ ó ò ñ Solution: 2 = L

24.

ññ óò ò ó ò ò ó ó

ñ óò ò ó ò ò ó ó

óò ò ó ò ó

+ 3(ñ 3 3 + L 2 

óò ò ó

ó

)2 = ô  . 2 + L 1  + L 0 + 2 Œ

−4

  .

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ô ] + õ . This is a special case of equation 4.2.6.60 with  ( ) =   +  .

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4.2.3. Equations Containing Hyperbolic Functions 4.2.3-1. Equations with hyperbolic sine. 1. 2.

3.

4. 5.

= ô sinhú ( ƒñ ) + õ . + This is a special case of equation 4.2.6.1 with  ( ) =  sinh ( Œ

ñóò ò ó ò ò ó ó

= ô sinh( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  sinh( Œ ¨ ) +  .

ñ óò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

= ô (ñ + õ sinh ï )2 – õ sinh ï . The substitution ¨ = +  sinh  leads to an autonomous equation of the form 4.2.6.1: ¨ 



 =  ¨ 2 .

ñóò ò ó ò ò ó ó

= ô ï – 4 sinh ú ( ƒñ ). + This is a special case of equation 4.2.6.2 with  ( ) =  sinh ( Œ

ñóò ò ó ò ò ó ó ñ óò ò ó ò ò ó ó

=

ô

sinh( ƒñ )ñ

óò

+ õ sinh(

ï

).

Integrating yields a third-order equation: 6. 7. 8. 9. 10.

Œ



cosh( Œ

)+

ý



cosh(ý± ) + L .

= ô 4 ñ + õ sinh( ï )(ñ ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  sinh( Œ  )¨  . = õ sinh( ï )( ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  sinh( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ sinh( ï )( ï ñ ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  sinh( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ sinh( ï )( ï ñ„ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  sinh( Œ  )¨  .

ñóò ò ó ò ò ó ó

– (ñ ó ò ò ó )2 = ô sinh( ï ). 1 b . Integrating the equation twice, we arrive at a second-order equation: !

 − (
 )2 = Œ −2 sinh( Œ  ) + L 1  + L 2 .

ññ óò ò ó ò ò ó ó

= L sinh( Π ) +

 . LpΠ4

– (ñó ò ò ó )2 – ôñó ò + õ sinh( ï ) = 0. 1 b . Integrating yields a third-order equation: !"


 − 
 

 − 

ññóò ò ó ò ò ó ó

2 b . Particular solution: 12.




 =

).

ñ óò ò ó ò ò ó ó

2 b . Particular solution: 11.

)+ .

=

 Œ ( 2 − L

Œ

2 6)

+ Œ

−1

cosh( Π ) = L .

V LpΠ3 sinh( Π ) +  cosh( Π )W + L .

– (ñó ò ò ó )2 – ôñó ò ò ó + õ sinh( ï ) = 0. 1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  + L 1  + L 2 + Œ −2 sinh( Œ  ) = 0.

ññóò ò ó ò ò ó ó

2 b . Particular solution:

= L sinh( Π ) +

LpŒ 2 −  . LpŒ 4

© 2003 by Chapman & Hall/CRC

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669

4.2. NONLINEAR EQUATIONS

13.

14.

– (ñó ò ò ó )2 = ô [ññó ò ò ó – (ñó ò )2 ] + õ sinh( ï ) + ö . The substitution ¨ ( ) = ! 

 − (
 )2 leads to a second-order constant coefficient nonhomogeneous linear equation of the form 2.1.9.1: ¨p

 =  ¨ +  sinh( Œ  ) + ( .

ññóò ò ó ò ò ó ó

– ôñ ó ò ò ó ò ó – (ñ ó ò ò ó )2 + õ sinh( ï ) = 0. 1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  
 + L 1  + L 2 + Œ −2 sinh( Œ  ) = 0.

ññ óò ò ó ò ò ó ó

2 b . Particular solution: 15.

16.

ñóò ò ó ò ò ó ó

=

 Œ 3 ( 2 − L

Œ

2 2)

V LpΠsinh( Π ) +  cosh( Π )W + L .

sinh  ( ƒñ )ñóÈ ò ñóò ò ó ò ó .

ô

=

This is a special case of equation 4.2.6.51 with  ( ) =  sinh  ( Œ

).

– ñó ò ñóò ò ó ò ó = ô sinh( ï )ñ 2. This is a special case of equation 4.2.6.57 with  ( ) =  sinh( Œ  ). Integrating yields a 


cosh( Π ) + L3S . third-order linear equation:  =

ññóò ò ó ò ò ó ó

Q Œ

17. 18.

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ô sinh( ï ). Solution: 2 = L 3  3 + L 2  2 + L 1  + L 0 + 2 Œ

ññ óò ò ó ò ò ó ó ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

ó

+ 3(ñó ò ò ó )2 =

sinh( Π ).

−4

sinh  ( ƒñ ) + õ .

ô

This is a special case of equation 4.2.6.60 with  ( ) =  sinh  ( Œ

)+ .

4.2.3-2. Equations with hyperbolic cosine. 19. 20.

21.

22. 23.

= ô coshú ( ƒñ ) + õ . + This is a special case of equation 4.2.6.1 with  ( ) =  cosh ( Œ

ñ óò ò ó ò ò ó ó

= ô cosh( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  cosh( Œ ¨ ) +  .

ñ óò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

= ô (ñ + õ cosh ï )2 – õ cosh ï . The substitution ¨ = +  cosh  leads to an autonomous equation of the form 4.2.6.1: ¨2



 =  ¨ 2 .

ñ óò ò ó ò ò ó ó

= ô ï – 4 coshú ( ƒñ ). + This is a special case of equation 4.2.6.2 with  ( ) =  cosh ( Œ

ñóò ò ó ò ò ó ó ñ óò ò ó ò ò ó ó

=

ô

cosh( ƒñ )ñ

óò

+ õ cosh(

ï

).

Integrating yields a third-order equation: 24. 25.

)+ .




 = Œ



sinh( Œ

)+

ý

).



sinh(ý± ) + L .

= ô 4 ñ + õ cosh( ï )(ñ„ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  cosh( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ cosh( ï )( ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  cosh( Œ  )¨  .

ñóò ò ó ò ò ó ó

© 2003 by Chapman & Hall/CRC

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670 26.

27.

28.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ò ó ó

=

õ

cosh(

ï )(ï ñ óò

– 2ñ )  .

ñóò ò ó ò ò ó ó

=

õ

cosh(

ï )(ï „ñ óò

– 3ñ )  .

This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  cosh( Œ  )¨  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  cosh( Œ  )¨  .

ññóò ò ó ò ò ó ó

– (ñó ò ò ó )2 =

ô

ññ óò ò ó ò ò ó ó

óò ò ó

– (ñ

óò

)2 – ôñ

32.

34.

) = 0.

1 b . Integrating yields a third-order equation: !"


 − 
 

 − 

ññóò ò ó ò ò ó ó

 Œ ( 2 − L

=

+ Œ

−1

sinh( Π ) = L .

V LpΠ3 cosh( Π ) +  sinh( Π )W + L .

Œ

2 6)

ï

– (ñó ò ò ó )2 – ôñó ò ò ó + õ cosh(

) = 0.

1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  + L 1  + L 2 + Œ −2 cosh( Œ  ) = 0.

ññ óò ò ó ò ò ó ó

– (ñ

óò ò ó

ññ óò ò ó ò ò ó ó

– ôñ

LpŒ 2 −  . LpŒ 4

= L cosh( Π ) +

óò ò ó

)2 = ô [ññ

óò

– (ñ

)2 ] + õ cosh(

ï

)+ö .

The substitution ¨ ( ) = ! 

 − (
 )2 leads to a second-order constant coefficient nonhomogeneous linear equation of the form 2.1.9.1: ¨p

 =  ¨ +  cosh( Œ  ) + ( .

óò ò ó ò ó

óò ò ó

– (ñ

)2 + õ cosh(

ï

) = 0.

1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  
 + L 1  + L 2 + Œ −2 cosh( Œ  ) = 0. 2 b . Particular solution:

33.

ï

+ õ cosh(

2 b . Particular solution: 31.

 . LpΠ4

= L cosh( Π ) +

2 b . Particular solution: 30.

).

1 b . Integrating the equation twice, we arrive at a second-order equation: !

 − (
 )2 = Œ −2 cosh( Œ  ) + L 1  + L 2 . 2 b . Particular solution:

29.

ï

cosh(

ñóò ò ó ò ò ó ó

ô

=

=

Œ



 −L

3( 2

Œ

2 2)

V LpΠcosh( Π ) +  sinh( Π )W + L .

cosh  ( ƒñ )ñóÈ ò ñóò ò ó ò ó .

This is a special case of equation 4.2.6.51 with  ( ) =  cosh  ( Œ

ññóò ò ó ò ò ó ó

– ñó ò ñóò ò ó ò

ó

=

ô

cosh(

ï )ñ

2

).

.

This is a special case of equation 4.2.6.57 with  ( ) =  cosh( Π ). Integrating yields a  sinh( Π ) + L3S . third-order linear equation: 


 =

Q Œ

35.

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

Solution: 36.

ññóò ò ó ò ò ó ó

2

=L

3

+ 4ñóÈ ò ñóò ò ó ò

ó 

ó

3

+ 3(ñó ò ò ó )2 =

ô

+L

 +L

2



2

+L

1

+ 3(ñó ò ò ó )2 =

ô

cosh( 0

ï

).

+ 2 Œ

−4

cosh( Π ).

cosh  ( ƒñ ) + õ .

This is a special case of equation 4.2.6.60 with  ( ) =  cosh  ( Œ

)+ .

© 2003 by Chapman & Hall/CRC

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4.2. NONLINEAR EQUATIONS

4.2.3-3. Equations with hyperbolic tangent. 37. 38.

39. 40. 41. 42. 43. 44.

= ô tanhú ( ƒñ ) + õ . + This is a special case of equation 4.2.6.1 with  ( ) =  tanh ( Œ

ñóò ò ó ò ò ó ó

= ô tanh( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  tanh( Œ ¨ ) +  .

)+ .

ñóò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

= ô ï – 4 tanhú ( ƒñ ). + This is a special case of equation 4.2.6.2 with  ( ) =  tanh ( Œ

ñóò ò ó ò ò ó ó

= ô tanh( ƒñ )ñ„ó ò + õ tanh( ï ). This is a special case of equation 4.2.6.21 with  ( ) =  tanh( Œ

ñóò ò ó ò ò ó ó

). ) and ( ) =  tanh(ý± ).

= ô 4 ñ + õ tanh( ï )(ñ„ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  tanh( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ tanh( ï )( ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  tanh( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ tanh( ï )( ï ñ ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  tanh( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ tanh( ï )( ï ñ ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  tanh( Œ  )¨  .

ñ óò ò ó ò ò ó ó

45.

= ñ + ô (ñ ó ò – ñ tanh ï )  . This is a special case of equation 5.2.6.32 with  ( , ‡ ) = ‡  and Y ( ) = cosh  .

46.

ñ óò ò ó ò ò ó ó

ñ óò ò ó ò ò ó ó

=

ô

tanh ( ƒñ )ñ

óò ñ óò ò ó ò ó

.

This is a special case of equation 4.2.6.51 with  ( ) =  tanh  ( Œ

).

47.

– ñ ó ò ñ ó ò ò ó ò ó = ô tanh( ï )ñ 2 . This is a special case of equation 4.2.6.57 with  ( ) =  tanh( Œ  ).

48.

ññ óò ò ó ò ò ó ó

+ 4ñ

óò ñ óò ò ó ò ó

+ 3(ñ

óò ò ó

)2 =

ô

tanh  (

ññ óò ò ó ò ò ó ó

+ 4ñ

óò ñ óò ò ó ò ó

+ 3(ñ

óò ò ó

)2 =

ô

tanh  ( ƒñ ) + õ .

49.

ññ óò ò ó ò ò ó ó

ï

)+ õ .

This is a special case of equation 4.2.6.58 with  ( ) =  tanh  ( Œ  ) +  . This is a special case of equation 4.2.6.60 with  ( ) =  tanh  ( Œ

)+ .

4.2.3-4. Equations with hyperbolic cotangent. 50. 51.

= ô cothú ( ƒñ ) + õ . + This is a special case of equation 4.2.6.1 with  ( ) =  coth ( Œ

ñóò ò ó ò ò ó ó

= ô coth( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  coth( Œ ¨ ) +  .

)+ .

ñóò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

© 2003 by Chapman & Hall/CRC

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672 52. 53. 54. 55. 56. 57.

FOURTH-ORDER DIFFERENTIAL EQUATIONS = ô ï – 4 cothú ( ƒñ ). + This is a special case of equation 4.2.6.2 with  ( ) =  coth ( Œ

ñóò ò ó ò ò ó ó

).

= ô coth( ƒñ )ñ ó ò + õ coth( ï ). This is a special case of equation 4.2.6.21 with  ( ) =  coth( Œ

ñ óò ò ó ò ò ó ó

) and ( ) =  coth(ý± ).

= ô 4 ñ + õ coth( ï )(ñ ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  coth( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ coth( ï )( ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  coth( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ coth( ï )( ï ñ„ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  coth( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ coth( ï )( ï ñ„ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  coth( Œ  )¨  .

ñóò ò ó ò ò ó ó

58.

= ñ + ô (ñó ò – ñ coth ï )  . This is a special case of equation 5.2.6.32 with  ( , ‡ ) = ‡  and Y ( ) = sinh  .

59.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

=

ô

coth ( ƒñ )ñó ò ñóò ò ó ò ó .

This is a special case of equation 4.2.6.51 with  ( ) =  coth  ( Œ

).

60.

– ñó ò ñóò ò ó ò ó = ô coth( ï )ñ 2. This is a special case of equation 4.2.6.57 with  ( ) =  coth( Œ  ).

61.

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

ó

+ 3(ñó ò ò ó )2 =

ô

coth  (

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

ó

+ 3(ñó ò ò ó )2 =

ô

coth  ( ƒñ ) + õ .

62.

ññóò ò ó ò ò ó ó

ï

)+ õ .

This is a special case of equation 4.2.6.58 with  ( ) =  coth  ( Œ  ) +  . This is a special case of equation 4.2.6.60 with  ( ) =  coth  ( Œ

)+ .

4.2.4. Equations Containing Logarithmic Functions 4.2.4-1. Equations of the form 



 =  ( , ). 1. 2.

3.

4.

= ô lnú ( ƒñ ) + õ . + This is a special case of equation 4.2.6.1 with  ( ) =  ln ( Œ

ñóò ò ó ò ò ó ó

= ô ln( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨ 



 =  ln( Œ ¨ ) +  .

)+ .

ñóò ò ó ò ò ó ó

= ô ln( ƒñ +  ï The substitution ¨ = ¨2



 =  ln( Œ ¨ ) +  .

ñ óò ò ó ò ò ó ó

2

)+õ . + ( ý Œ )

leads to an autonomous equation of the form 4.2.6.1:

2

leads to an autonomous equation of the form 4.2.6.1:

= ô ï – 4 ln ú ( ƒñ ). + This is a special case of equation 4.2.6.2 with  ( ) =  ln ( Œ

ñóò ò ó ò ò ó ó

).

© 2003 by Chapman & Hall/CRC

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4.2. NONLINEAR EQUATIONS

5. 6. 7. 8. 9. 10.

= ôñ ( ï + þ ln ñ ). This is a special case of equation 4.2.6.14 with  (¨ ) =  ln ¨ .

ñ óò ò ó ò ò ó ó

= ô ï – 4 ( ƒñ + þ ln ï ). This is a special case of equation 4.2.6.15 with  (¨ ) =  ln ¨ .

ñóò ò ó ò ò ó ó

= ô ï –3 (ln ñ – ln ï ). This is a special case of equation 4.2.6.3 with  (¨ ) =  ln ¨ .

ñóò ò ó ò ò ó ó

= ô ï –5 (ln ñ – 3 ln ï ). This is a special case of equation 4.2.6.4 with  (¨ ) =  ln ¨ .

ñóò ò ó ò ò ó ó

= ô ï –5 ) 2 (2 ln ñ – 3 ln ï ). This is a special case of equation 4.2.6.5 with  (¨ ) = 2  ln ¨ .

ñóò ò ó ò ò ó ó

= ô ï ÷ –4 (ln ñ – ù ln ï ). This is a special case of equation 4.2.6.6 with  (¨ ) =  ln ¨ and , = −B .

ñ óò ò ó ò ò ó ó

4.2.4-2. Other equations. 11. 12. 13. 14. 15. 16. 17.

= ô 4 ñ + õ ln( ï )(ñ ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  ln( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ ln( ï )(ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  ln( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ ln( ï )(ï ñ„ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  ln( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ ln( ï )(ï ñ„ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  ln( Œ  )¨  .

ñóò ò ó ò ò ó ó

– (ñó ò ò ó )2 = ô ln( ï ). This is a special case of equation 4.2.6.36 with  ( ) =  ln( Œ  ).

ññóò ò ó ò ò ó ó

ï ñ óò ò ó ò ò ó ó

+ 4ñ ó ò ò ó ò ó = ô (ln ï + ln ñ ). ¨ 



 =  ln ¨ . The substitution ¨ ( ) =  leads to an equation of the form 4.2.6.1: I

ñóò ò ó ò ò ó ó = ô ln( ƒñ )ñóÈò ñóò ò ó ò ó . This is a special case of equation 4.2.6.51 with  ( ) =  ln( Œ

).

18.

– ñó ò ñóò ò ó ò ó = ô ln( ï )ñ 2. Integrating yields a third-order linear equation:

19.

+ 4ñóÈ ò ñóò ò ó ò ó + 3(ñóò ò ó )2 = ô lnú ( ï ) + õ . + This is a special case of equation 4.2.6.58 with  ( ) =  ln ( Œ  ) +  .

20.

ññóò ò ó ò ò ó ó




 = V  ln( Œ  ) −  + L3W .

ññóò ò ó ò ò ó ó

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ô ln ú ( ƒñ ) + õ . + This is a special case of equation 4.2.6.60 with  ( ) =  ln ( Œ

ññ óò ò ó ò ò ó ó

)+  .

© 2003 by Chapman & Hall/CRC

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FOURTH-ORDER DIFFERENTIAL EQUATIONS

4.2.5. Equations Containing Trigonometric Functions 4.2.5-1. Equations with sine. 1. 2.

3.

4. 5.

= ô sinú ( ƒñ ) + õ . This is a special case of equation 4.2.6.1 with  ( ) =  sin + ( Œ

ñóò ò ó ò ò ó ó

= ô sin( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  sin( Œ ¨ ) +  .

ñ óò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

= ô (ñ + õ sin ï )2 – õ sin ï . The substitution ¨ = +  sin  leads to an autonomous equation of the form 4.2.6.1: ¨ 



 =  ¨ 2 .

ñóò ò ó ò ò ó ó

= ô ï – 4 sin ú ( ƒñ ). This is a special case of equation 4.2.6.2 with  ( ) =  sin + ( Œ

ñóò ò ó ò ò ó ó ñ óò ò ó ò ò ó ó

=

ô

sin( ƒñ )ñ

óò

+ õ sin(

ï

).

7. 8. 9. 10.

= ô 4 ñ + õ sin( ï )(ñ ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  sin( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ sin( ï )( ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  sin( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ sin( ï )( ï ñ ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  sin( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ sin( ï )( ï ñ„ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  sin( Œ  )¨  .

ñóò ò ó ò ò ó ó

– (ñ ó ò ò ó )2 = ô sin( ï ). 1 b . Integrating the equation twice, we arrive at a second-order equation: !

 − (
 )2 = − Œ −2 sin( Œ  ) + L 1  + L 2 .

ññ óò ò ó ò ò ó ó

2 b . Particular solution: 11.

 . LpΠ4

= L sin( Π ) +

– (ñó ò ò ó )2 – ôñó ò + õ sin( ï ) = 0. 1 b . Integrating yields a third-order equation: !"


 − 
 

 − 

ññóò ò ó ò ò ó ó

2 b . Particular solution: 12.

).

  


 = − cos( Œ ) − cos(ý± ) + L . Œ ý

Integrating yields a third-order equation: 6.

)+ .

=−

 Π( 2 + L

Œ

2 6)

− Œ

−1

cos( Π ) = L .

V  cos( Π ) + LpΠ3 sin( Π )W + L .

– (ñó ò ò ó )2 – ôñó ò ò ó + õ sin( ï ) = 0. 1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  + L 1  + L 2 − Œ −2 sin( Œ  ) = 0.

ññóò ò ó ò ò ó ó

2 b . Particular solution:

= L sin( Œ  ) −

 + LpΠ2 . LpΠ4

© 2003 by Chapman & Hall/CRC

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4.2. NONLINEAR EQUATIONS

13.

14.

– (ñó ò ò ó )2 = ô [ññó ò ò ó – (ñó ò )2 ] + õ sin( ï ) + ö . The substitution ¨ ( ) = ! 

 − (
 )2 leads to a second-order constant coefficient nonhomogeneous linear equation of the form 2.1.9.1: ¨ 

 =  ¨ +  sin( Œ  ) + ( .

ññóò ò ó ò ò ó ó

– ôñó ò ò ó ò ó – (ñó ò ò ó )2 + õ sin( ï ) = 0. 1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  
 + L 1  + L 2 − Œ −2 sin( Œ  ) = 0.

ññóò ò ó ò ò ó ó

2 b . Particular solution: 15. 16.

ñóò ò ó ò ò ó ó

=

 Π3 ( 2 + L

18.

V  cos( Œ  ) − LpŒ sin( Œ  )W + L .

sin  ( ƒñ )ñóÈ ò ñóò ò ó ò ó .

ô

=

This is a special case of equation 4.2.6.51 with  ( ) =  sin  ( Œ

).

– ñó ò ñóò ò ó ò ó = ô sin( ï )ñ 2. This is a special case of equation 4.2.6.57 with  ( ) =  sin( Œ  ). Integrating yields a third order linear equation: 


 = L − cos( Œ  )S .

ññóò ò ó ò ò ó ó

Q

17.

Œ

2 2)

Œ

+ 4ñóÈ ò ñóò ò ó ò ó + 3(ñóò ò ó )2 = ô sin( ï ). Solution: 2 = L 3  3 + L 2  2 + L 1  + L 0 + 2 Œ

ññóò ò ó ò ò ó ó

sin( Π ).

−4

+ 4ñóÈ ò ñóò ò ó ò ó + 3(ñóò ò ó )2 = ô sin  ( ƒñ ) + õ . This is a special case of equation 4.2.6.60 with  ( ) =  sin  ( Œ

ññóò ò ó ò ò ó ó

)+ .

4.2.5-2. Equations with cosine. 19. 20.

21.

22. 23.

ñóò ò ó ò ò ó ó

cosú ( ƒñ ) + õ .

ô

=

This is a special case of equation 4.2.6.1 with  ( ) =  cos + ( Œ

= ô cos( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  cos( Œ ¨ ) +  .

ñóò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

= ô (ñ + õ cos ï )2 – õ cos ï . The substitution ¨ = +  cos  leads to an autonomous equation of the form 4.2.6.1: ¨2



 =  ¨ 2 .

ñ óò ò ó ò ò ó ó

= ô ï – 4 cosú ( ƒñ ). This is a special case of equation 4.2.6.2 with  ( ) =  cos + ( Œ

ñóò ò ó ò ò ó ó ñ óò ò ó ò ò ó ó

=

ô

cos( ƒñ )ñ

óò

+ õ cos(

ï

).

Integrating yields a third-order equation: 24. 25.

)+ .




 = Œ



sin( Œ

)+

ý

).



sin(ý± ) + L .

= ô 4 ñ + õ cos( ï )(ñ„ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  cos( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ cos( ï )(ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  cos( Œ  )¨  .

ñóò ò ó ò ò ó ó

© 2003 by Chapman & Hall/CRC

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676 26.

27.

28.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ñ óò ò ó ò ò ó ó

=

õ

cos(

ï )(ï ñ óò

– 2ñ )  .

ñóò ò ó ò ò ó ó

=

õ

cos(

ï )(ï „ñ óò

– 3ñ )  .

This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  cos( Œ  )¨  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  cos( Œ  )¨  .

ññóò ò ó ò ò ó ó

– (ñó ò ò ó )2 =

ô

ññ óò ò ó ò ò ó ó

– (ñ

óò ò ó

óò

)2 – ôñ

32.

34.

) = 0.

1 b . Integrating yields a third-order equation: !"


 − 
 

 − 

ññóò ò ó ò ò ó ó

=

 Π( 2 + L

Œ

2 6)

ï

– (ñó ò ò ó )2 – ôñó ò ò ó + õ cos(

+ Œ

−1

sin( Π ) = L .

V  sin( Œ  ) − LpŒ 3 cos( Œ  )W + L .

) = 0.

1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  + L 1  + L 2 − Œ −2 cos( Œ  ) = 0.

LpΠ2 +  . LpΠ4

= L cos( Œ  ) −

ññóò ò ó ò ò ó ó

– (ñó ò ò ó )2 = ô [ññó ò ò ó – (ñó ò )2 ] + õ cos(

ññóò ò ó ò ò ó ó

– ôñó ò ò ó ò

ï

)+ö .

The substitution ¨ ( ) = ! 

 − (
 )2 leads to a second-order constant coefficient nonhomogeneous linear equation of the form 2.1.9.1: ¨p

 =  ¨ +  cos( Œ  ) + ( .

ó

– (ñó ò ò ó )2 + õ cos(

ï

) = 0.

1 b . Integrating the equation two times, we obtain a second-order equation: !

 − (
 )2 −  
 + L 1  + L 2 − Œ −2 cos( Œ  ) = 0. 2 b . Particular solution:

33.

ï

+ õ cos(

2 b . Particular solution: 31.

 . LpΠ4

= L cos( Π ) +

2 b . Particular solution: 30.

).

1 b . Integrating the equation twice, we arrive at a second-order equation: ! 

 − ( 
)2 = − Œ −2 cos( Œ  ) + L 1  + L 2 . 2 b . Particular solution:

29.

ï

cos(

ñ óò ò ó ò ò ó ó

ô

=

cos ( ƒñ )ñ

=−

óò ñ óò ò ó ò ó

 Π3( 2 + L

Œ

2 2)

V LpΠcos( Π ) +  sin( Π )W + L .

.

This is a special case of equation 4.2.6.51 with  ( ) =  cos  ( Œ

ññóò ò ó ò ò ó ó

– ñó ò ñóò ò ó ò

ó

=

ô

cos(

ï )ñ

2

).

.

This is a special case of equation 4.2.6.57 with  ( ) =  cos( Π ). Integrating yields a third sin( Π ) + L3S . order linear equation: 


 =

Q Œ

35.

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

Solution: 36.

ññóò ò ó ò ò ó ó

2

+ 4ñó ò

=L

3

ó 

ñóò ò ó ò ó

3

+ 3(ñó ò ò ó )2 =

ô

+L

 +L

2



2

+L

1

+ 3(ñó ò ò ó )2 =

ô

cos( 0

ï

).

+ 2 Œ

−4

cos( Π ).

cos  ( ƒñ ) + õ .

This is a special case of equation 4.2.6.60 with  ( ) =  cos  ( Œ

)+ .

© 2003 by Chapman & Hall/CRC

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677

4.2. NONLINEAR EQUATIONS

4.2.5-3. Equations with tangent. 37. 38.

39. 40. 41. 42. 43. 44.

= ô tanú ( ƒñ ) + õ . This is a special case of equation 4.2.6.1 with  ( ) =  tan + ( Œ

ñ óò ò ó ò ò ó ó

= ô tan( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  tan( Œ ¨ ) +  .

)+ .

ñ óò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

= ô ï – 4 tanú ( ƒñ ). This is a special case of equation 4.2.6.2 with  ( ) =  tan + ( Œ

ñóò ò ó ò ò ó ó

= ô tan( ƒñ )ñ„ó ò + õ tan( ï ). This is a special case of equation 4.2.6.21 with  ( ) =  tan( Œ

ñóò ò ó ò ò ó ó

). ) and ( ) =  tan(ý± ).

= ô 4 ñ + õ tan( ï )(ñ„ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  tan( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ tan( ï )( ï ñ ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  tan( Œ  )¨  .

ñ óò ò ó ò ò ó ó

= õ tan( ï )( ï ñ„ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  tan( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ tan( ï )( ï ñ„ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  tan( Œ  )¨  .

ñóò ò ó ò ò ó ó

45.

= ñ + ô (ñó ò + ñ tan ï )  . This is a special case of equation 5.2.6.32 with  ( , ‡ ) = ‡  and Y ( ) = cos  .

46.

= ô tan ( ƒñ )ñóÈ ò ñóò ò ó ò ó . This is a special case of equation 4.2.6.51 with  ( ) =  tan  ( Œ

47.

ñóò ò ó ò ò ó ó ñóò ò ó ò ò ó ó

).

– ñó ò ñóò ò ó ò ó = ô tan( ï )ñ 2. This is a special case of equation 4.2.6.57 with  ( ) =  tan( Œ  ).

ññóò ò ó ò ò ó ó

48.

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ô tan  ( ï ) + õ . This is a special case of equation 4.2.6.58 with  ( ) =  tan  ( Œ  ) +  .

49.

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ô tan  ( ƒñ ) + õ . This is a special case of equation 4.2.6.60 with  ( ) =  tan  ( Œ

ññ óò ò ó ò ò ó ó ññ óò ò ó ò ò ó ó

)+ .

4.2.5-4. Equations with cotangent. 50. 51.

= ô cotú ( ƒñ ) + õ . This is a special case of equation 4.2.6.1 with  ( ) =  cot + ( Œ

ñ óò ò ó ò ò ó ó

= ô cot( ƒñ +  ï ) + õ . The substitution ¨ = + (ý Œ ) ¨2



 =  cot( Œ ¨ ) +  .

)+ .

ñóò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.1:

© 2003 by Chapman & Hall/CRC

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678 52. 53. 54. 55. 56. 57.

FOURTH-ORDER DIFFERENTIAL EQUATIONS = ô ï – 4 cotú ( ƒñ ). This is a special case of equation 4.2.6.2 with  ( ) =  cot + ( Œ

ñóò ò ó ò ò ó ó

).

= ô cot( ƒñ )ñó ò + õ cot( ï ). This is a special case of equation 4.2.6.21 with  ( ) =  cot( Œ

ñóò ò ó ò ò ó ó

) and ( ) =  cot(ý± ).

= ô 4 ñ + õ cot( ï )(ñ„ó ò – ôñ )  . This is a special case of equation 4.2.6.27 with  ( , ¨ ) =  cot( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ cot( ï )(ï ñ„ó ò – ñ )  . This is a special case of equation 4.2.6.23 with  ( , ¨ ) =  cot( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ cot( ï )(ï ñ„ó ò – 2ñ )  . This is a special case of equation 4.2.6.24 with  ( , ¨ ) =  cot( Œ  )¨  .

ñóò ò ó ò ò ó ó

= õ cot( ï )(ï ñ ó ò – 3ñ )  . This is a special case of equation 4.2.6.25 with  ( , ¨ ) =  cot( Œ  )¨  .

ñ óò ò ó ò ò ó ó

58.

= ñ + ô (ñó ò – ñ cot ï )  . This is a special case of equation 5.2.6.32 with  ( , ‡ ) = ‡  and Y ( ) = sin  .

59.

= ô cot ( ƒñ )ñóÈ ò ñóò ò ó ò ó . This is a special case of equation 4.2.6.51 with  ( ) =  cot  ( Œ

60.

ñóò ò ó ò ò ó ó ñóò ò ó ò ò ó ó

).

– ñó ò ñóò ò ó ò ó = ô cot( ï )ñ 2. This is a special case of equation 4.2.6.57 with  ( ) =  cot( Œ  ).

ññóò ò ó ò ò ó ó

61.

+ 4ñóÈ ò ñóò ò ó ò ó + 3(ñóò ò ó )2 = ô cot ( ï ) + õ . This is a special case of equation 4.2.6.58 with  ( ) =  cot  ( Œ  ) +  .

62.

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ô cot ( ƒñ ) + õ . This is a special case of equation 4.2.6.60 with  ( ) =  cot  ( Œ

ññóò ò ó ò ò ó ó ññ óò ò ó ò ò ó ó

)+ .

4.2.6. Equations Containing Arbitrary Functions 4.2.6-1. Equations of the form 



 =  ( , ). 1.

ñóò ò ó ò ò ó ó

= ! (ñ ).

Autonomous equation. By integrating, we obtain 2 



 − ( 

 )2 = 2 X

substitution ¨ ( ) = | 
|3 J 2.

3.

2

leads to a second-order equation: ¨

=

= ï – 4 ! (ñ ). This is a special case of equation 4.2.6.55 with  leads to an autonomous equation.



ñ óò ò ó ò ò ó ó

1

=

2

=

= ï –3 ! (ñ ï ). Homogeneous equation. The transformation a = ln  , ¨ equation of the form 4.2.6.79.

ñóò ò ó ò ò ó ó

3

3 2

Q

 ( )

Xv ( ) M

M

+ 2 L . The

+ L2S ¨

J .

−5 3

= 0. The substitution a = ln | |

=  

leads to an autonomous

© 2003 by Chapman & Hall/CRC

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679

4.2. NONLINEAR EQUATIONS

4.

5.

6.

= ï –5 ! (ñ ï –3 ). The transformation  = a ¨2g

g
g
g =  (¨ ).

ñóò ò ó ò ò ó ó

−1

=¨ a

,

−3

= ï –5 ) 2 ! (ñ ï –3 ) 2 ). g The transformation  =  , =  3 J 2 ¨ ¨2g

g
g
g − 5 ¨2g

g = − 9 ¨ +  (¨ ). 2 16

leads to an autonomous equation of the form 4.2.6.1:

ñóò ò ó ò ò ó ó

leads to an autonomous equation of the form 4.2.6.33:

= ï –  –4 ! ( ï  ñ ). Generalized homogeneous equation.

ñ óò ò ó ò ò ó ó

1 b . The transformation a = ln  , H =  

leads to an autonomous equation. 2 b . The transformation H =   , ¨ =  
  leads to a third-order equation.

7.

8.

9. 10.

= ñ ï – 4 ! ( ï ñ ú ). Generalized homogeneous equation. The transformation H =   + , ¨ third-order equation.

ñóò ò ó ò ò ó ó

= ! (ñ + ô 3 ï 3 + ô 2 ï 2 + ô 1 ï + ô 0 ). The substitution ¨ = +  3  3 +  2  2 +  1  +  ¨2



 =  (¨ ).

ñ óò ò ó ò ò ó ó

= ! (ñ + ô ï 4 ). The substitution ¨ = + 

ñóò ò ó ò ò ó ó ï (ô ï

+ õ )4 ñó ò ò ó ò ò ó

ó

= ! (ñ

ï

4 –3

0

leads to an equation of the form 4.2.6.1:

).

leads to an autonomous equation of the form

3

ô ï  ï

11.

ñóò ò ó ò ò ó ó

= (ô

ï

+ õÿñ + ö . + ñ + • This is a special case of equation 5.2.6.19 with B = 4.

12.

ñ óò ò ó ò ò ó ó

= (ô

ï

2

õÿñ

+

+ ö )–3 !e”

õï


 leads to a

leads to an equation of the form 4.2.6.1: ¨u



 =  (¨ ) + 24 .

 +  ¨ The transformation ˆ = ln û û, =  û û  4.2.6.79. û û +

=

+ ö )–5 ) 2 !e”

ñ

(ô ï

2



õï

+

+ ö )3 )

2

•

.

, ¨ = M + ' + ( (  2 + ' + ( )3 J equation of the form 4.2.6.33 with respect to ¨ = ¨ (ˆ ): 1 b . The transformation ˆ = X



2

¨ Šª

ªŠ

ŠªŠ − 5 ¢ ¨ ªŠ

Š + 2

9 16

¢ 2 ¨ =  (¨ ),

where

2

leads to an autonomous

¢ =  2 − 4 !( .

Therefore, having integrated the latter equation, we obtain

¨ Š
¨ Šª

ªŠ
Š − 1 (¨ Šª

Š )2 − 5 ¢ (¨ Š
)2 = − 9 ¢ 2 ¨ 2 4 32 3J The substitution H (¨ ) = |¨3Š
|

H ?

? =

15 8

2

3 2

Q

9 − 32 ¢

2

¨

2

where

.

3 2

= 

.


)


 − 2

1 2

.

(

 )2 +

+ ' + ( , ¨ =

.

 (¨ ) ¨ + L . M

+ Xv (¨ ) ¨ + L2S´H

2 b . The first integral of the original equation has the form:

. ( 
 −

+X

leads to a second-order equation:

J +

−1 3

¢çH

2

J .

1 2

.

M

J .

−5 3



 

 + 3  !

 − 2  (
 )2 = X| (¨ ) ¨ + L , M

−3 2

© 2003 by Chapman & Hall/CRC

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680 13.

14.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ó

ñ óò ò ó ò ò ó ó

ó

=   ! (ñ –  ).  This is a special case of equation 4.2.6.47 with  =  = ( = 0. The substitution ¨ ( ) =  −  leads to an autonomous equation.

ó

= ñ! (  ñ ú ).  The transformation H = ž  + , ¨ ( H ) = 
  leads to a third-order equation.

ñóò ò ó ò ò ó ó

15.

= ï – 4 ! ( ïƒú ] ). The transformation H =  +  , ¨ ( H ) =  
 leads to a third-order equation.

16.

ñ óò ò ó ò ò ó ó

17.

18.

19.

20.

ñóò ò ó ò ò ó ó

ó

ó

= ! (ñ + ô ) – ô .  The substitution ¨ = +   leads to an equation of the form 4.2.6.1: ¨ 



 =  (¨ ).

= ! (ñ + ô cosh ï ) – ô cosh ï . The substitution ¨ = +  cosh  leads to an autonomous equation of the form 4.2.6.1: ¨2



 =  (¨ ).

ñóò ò ó ò ò ó ó

= ! (ñ + ô sinh ï ) – ô sinh ï . The substitution ¨ = +  sinh  leads to an autonomous equation of the form 4.2.6.1: ¨2



 =  (¨ ).

ñ óò ò ó ò ò ó ó

= ! (ñ + ô cos ï ) – ô cos ï . The substitution ¨ = +  cos  leads to an autonomous equation of the form 4.2.6.1: ¨2



 =  (¨ ).

ñóò ò ó ò ò ó ó

= ! (ñ + ô sin ï ) – ô sin ï . The substitution ¨ = +  sin  leads to an autonomous equation of the form 4.2.6.1: ¨2



 =  (¨ ).

ñóò ò ó ò ò ó ó

4.2.6-2. Equations of the form 



 =  ( , , 
 ). 21.

22.

23. 24.

ñóò ò ó ò ò ó ó

= ! (ñ )ñó ò + " ( ï ).

By integrating, we find 


 = X  ( ) + X ( )  + L . For ( ) ≡ 0, the order of this M M equation can reduced by one with the help of the substitution ¨ ( ) =
 . = ï – 4 ! ( ï ñó ò – ñ ). The transformation a = ln | |, ¨ ¨2g

g
g − 5¨2g

g + 6¨2g
=  (¨ ).

ñóò ò ó ò ò ó ó

= ! ( ï , ï ñó ò – ñ ). The substitution ¨ =  
−

ñóò ò ó ò ò ó ó

=  
 −

leads to a third-order autonomous equation:

¨ 
 )

 =  ( , ¨ ). leads to a third-order equation: (u

= ! ( ï , ï ñó ò – 2ñ ). ¨ 


 − ¨2

 =  2  ( , ¨ ). The substitution ¨ =  
− 2 leads to a third-order equation:  u

ñóò ò ó ò ò ó ó

25.

= ! ( ï , ï ñó ò – 3ñ ). ¨ 


 =   ( , ¨ ). The substitution ¨ =  
− 3 leads to a third-order equation: u

26.

ñ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

=

ñ ï

–4

! (ï ñ óò ¦ ñ

The transformation H = 

).


 , ¨ = 

2



  leads to a second-order equation.

© 2003 by Chapman & Hall/CRC

Page 680

681

4.2. NONLINEAR EQUATIONS

27.

28. 29. 30.

= ô 4 ñ + ! ( ï , ñó ò – ôñ ). ¨ 

 +  2 2 ¨ 


 +  2 ¨ 
+  3 ¨ = The substitution ¨ = 
−  leads to a third-order equation: u ¨  ( , ).

ñóò ò ó ò ò ó ó

= ! ( ï , ñó ò sinh ï – ñ cosh ï ) + ñ . The substitution ¨ = 
sinh  − cosh  leads to a third-order equation.

ñóò ò ó ò ò ó ó

= ! ( ï , ñó ò cosh ï – ñ sinh ï ) + ñ . The substitution ¨ = 
cosh  − sinh  leads to a third-order equation.

ñóò ò ó ò ò ó ó

= ! ( ï , ñó ò sin ï – ñ cos ï ) + ñ . The substitution ¨ = 
sin  − cos  leads to a third-order equation.

ñóò ò ó ò ò ó ó

31.

= ! ( ï , ñó ò cos ï + ñ sin ï ) + ñ . The substitution ¨ = 
cos  + sin  leads to a third-order equation.

32.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

=

r óò ò óò ò ó ó r ñ

+I ! O

ï , ñ óò



Y  The substitution ¨ =  − Y

–

r óò r ñ P

r

,

r (ï

=

).

leads to a third-order equation.

4.2.6-3. Equations of the form 



 =  ( , , 
 , 

 ). 33.

ñóò ò ó ò ò ó ó

+

ôñóò ò ó

= ! (ñ ).

Having integrated this equation, we obtain 2 



 − ( 

 )2 +  ( 
)2 = 2 XZ ( )

where L is an arbitrary constant. The substitution ¨ ( ) = |"
 |3 J equation:

34.

¨

= −3 ¨ 4 

J +

−1 3

3 2

Q

X  ( ) M

+ L2S ¨

M

+ 2L ,

leads to a second-order

J .

−5 3

ñóò ò ó ò ò ó ó + ! (ñóò )ñóò ò ó = " (ñ ). Having integrated this equation, we obtain a third-order autonomous equation: 2 



 − ( 

 )2 + 2  ( 
) = 2 X¥ ( )

M

+ 2L ,

where

The substitution ¨ ( ) = 
leads to a second-order equation. 35.

2

= ï –2 ! ( ï ñ ó ò – ñ )ñ ó ò ò ó . The transformation a = ln | |, ¨ =  
 −

 (‡ ) = Xȇ/ (‡ ) ‡ . M

ñ óò ò ó ò ò ó ó

leads to a third-order equation:

¨ g

g
g − 5¨ g

g + 6¨ g
=  (¨ )¨ g
. Integrating it, we obtain a second-order autonomous equation:

¨ g

g − 5¨ g
+ 6¨ = X  (¨ ) ¨ + L . M ¨ 1 ¨2
The substitution H ( ) = 5 g leads to an Abel equation of the second kind: HH ?
− H =

1 25

Q

−6¨ + X

 (¨ ) ¨ + L2S M

(see Subsection 1.3.1).

© 2003 by Chapman & Hall/CRC

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682 36.

FOURTH-ORDER DIFFERENTIAL EQUATIONS – (ñó ò ò ó )2 = ! ( ï ). Integrating the equation twice, we arrive at a second-order equation:

ññóò ò ó ò ò ó ó

! 

 − ( 
)2 = X 37.

38.

39.



( − a )  (a ) a + L

M

0

ñóò ò ó ò ò ó ó

= ô 2 ñ + ! (ñó ò ò ó The substitution ¨ = 2.9.1.1: ¨ 

 =  ¨ + 

+ ôñ ). 

 +  ¨( ).

leads to a second-order autonomous equation of the form

ñóò ò ó ò ò ó ó = ! (ñ , ñóò ò ó ). The substitution ¨ ( ) = A (
 )2 leads to a third-order equation: 2   , A 12 ¨2
 . 

= ô 2 ñ + ! ( ï , ñó ò ò ó + The substitution ¨ = 

 + 

41.

ñóò ò ó ò ò ó ó

= ñ!  ññó ò ò ó – ñó ò 2  . This is a special case of equation 4.2.6.42.

42.

ñ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

= ñ ó ò ò ó !u ññ ó ò ò ó – ñ 1 b . Particular solution: where the constants L

1,

ôñ

óò 2 

4 3

+ ñ"*

ññ óò ò ó

–ñ

=L

exp( L

3

L

45.

46.

1, 4 3

1 2

2 3

−L

2 b . Particular solution: where the constants L

L 2 , and L

+L

 (− L

2

L

2

exp(− L

3

 ),

are related by the constraint

1

2 1

óò 2  .

 )+L 2 3)

 (4 L 1 L 2 L

=L 2 3

¨*¨u


+ 1 2 ¨
2 ¨

= 2    

). ¨ 

 =  ¨ +  ( , ¨ ). leads to a second-order equation: u

L 2 , and L L

44.

 + L 2.

– (ñó ò ò ó )2 = ! ( ï )[ññó ò ò ó – (ñó ò )2 ] + " ( ï ). This is a special case of equation 4.2.6.93. The substitution ¨ ( ) = ! 

 − ( 
)2 leads to a second-order linear equation: ¨u

 =  ( )¨ + ( ).

ññóò ò ó ò ò ó ó

40.

43.

1

cos( L

3

− (4 L

 )+L

2

1

L 2 L 32 ) = 0.

sin( L

3

 ),

are related by the constraint 2 3

−L

2 2

L

2 3)

− (− L

2 1

L

2 3

−L

2 2

L 32 ) = 0.

= ïƒú ! ( ï 2 ñó ò ò ó – 2 ï ñó ò + 2ñ ). The substitution ¨ =  2 

 − 2 
+ 2 leads to a second-order equation:  ¨ 

 − 2¨ 
=  + +3  (¨ ). For = −4, the substitution H (¨ ) = 13  ¨2
leads to an Abel equation of the ¡?
second kind: HH − H = 19  (¨ ) (see Subsection 1.3.1).

ñóò ò ó ò ò ó ó

= ! ( ï , ï ñó ò – ñ , ñó ò ò ó ). The substitution ¨ ( ) =  
−

ñóò ò ó ò ò ó ó

leads to a third-order equation.

= ! ( ï , ï 2 ñó ò ò ó – 2 ï ñó ò + 2ñ ). ¨ 

 − 2¨2
= The substitution ¨ =  2 

 − 2 
 + 2 leads to a second-order equation:  u ¨ 3   ( , ).

ñóò ò ó ò ò ó ó ñóò ò ó ò ò ó ó

=

ñóò !IO

ñ óò ò ó , ñ óò ñ óò

–ñ

ñ óò ò ó ñ óò

P

.

Particular solution: = L 1 exp( L 2  )+ L 3 , where L 1 is an arbitrary constant and the constants L 2 and L 2 are related by the constraint L 23 =  ( L 2 , − L 2 L 3 ).

© 2003 by Chapman & Hall/CRC

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4.2. NONLINEAR EQUATIONS

4.2.6-4. Equations of the form 



 =  ( , , 
 , 

 , 


 ). 47.

ñóò ò ó ò ò ó ó

+

ôñóò ò ó ò ó

õÿñóò ò ó

+

+

ö^ñóò

=

ó

 ! (ñ  –

ó

).

 The substitution ¨ ( ) =  −  leads to an autonomous equation: ¨ 



 + (4 Œ +  )¨ 


 + (6 Œ 2 + 3  Œ +  )¨ 

 + (4 Œ 3 + 3 Œ 2 + 2 Œ + ( )¨ 
+ ( Œ 4 + Œ 3 + Œ

2

+ (Œ )¨ =  (¨ ),

which can be reduced to a third-order equation by means of the substitution H (¨ ) = ¨I
. For  = −4 Œ and ( = 8 Œ 3 − 2 Œ , the above equation coincides, up to notation, with equation 4.2.6.33 and can be reduced to a second-order equation. 48.

ñ óò ò ó ò ò ó ó

+

ôññ óò ò ó ò ó

= ! ( ï ).

Integrating, we arrive at a third-order equation: 49.

ñóò ò ó ò ò ó ó

+

ôññóò ò ó ò ó

– ôñó ò ñóò ò ó = ! (ï ).

Integrating, we arrive at a third-order equation: 50.

ñóò ò ó ò ò ó ó

+

ôññóò ò ó ò ó

+ ! (ñó ò )ñó ò ò ó = " ( ï ).




 +  ! 

 − 12  ( 
)2 = X  ( )  + L . M 


 +  ! 

 −  ( 
)2 = Xv ( )  + L . M

Integrating, we arrive at a third-order equation:




 +  ! 

 − 12  ( 
)2 +  ( 
) = X ( )  + L , M 51.

ñ óò ò ó ò ò ó ó

óò ñ óò ò ó ò ó

= ! (ñ )ñ

where

 (‡ ) = X  (‡ ) ‡ . M

.

Integrating, we arrive at a third-order autonomous equation of the form 3.5.1.1: L exp X¯ ( ) S .

Q

52. 53. 54.

55.

M

ï ñóò ò ó ò ò ó ó

+ 4ñó ò ò ó ò

ï ñóò ò ó ò ò ó ó

+ ( ô + 3)ñó ò ò ó ò

ó

= ! (ï

The substitution ¨ ( ) = 

ñ

).

¨ 



 =  (¨ ). leads to an equation of the form 4.2.6.1: I

= ! (ï , ï

ó

The substitution ¨ =  
+ 

ï 2 ñ óò ò ó ò ò ó ó

+ 8ï

ñóò ò ó ò ó

ï 4 ñ óò ò ó ò ò ó ó

+

ñóò

ô 3 ï 3 ñ óò ò ó ò ó

+

+ ôñ ).

¨ 


 =  ( , ¨ ). leads to a third-order equation: u

+ 12ñó ò ò ó = ! ( ï

The substitution ¨ ( ) =  ¨2



 =  (¨ ).




 =

2

2

ñ

).

leads to an autonomous equation of the form 4.2.6.1:

ô 2 ï 2 ñ óò ò ó

+ô

1

ï ñóò

= ! (ñ ).

The substitution a = ln | | leads to an autonomous equation:











(1) g g g g + (  3 − 6) g g g + (11 − 3 3 +  2 ) g g + (2  3 −  2 +  1 − 6) g =  ( ), the order of which can be lowered with the help of the substitution ¨ ( ) =
g . For  3 = 6 and  1 =  2 − 6, equation (1) coincides, up to notation, with equation 4.2.6.33 and can be reduced to a second-order equation.

56.

ï 4 ñ óò ò ó ò ò ó ó

+

ô ï 3 ñ óò ò ó ò ó

+

õ ï 2 ñ óò ò ó

+ö

ï ñóò

+ sñ =

ï – ! (ñ ï

). The transformation a = ln  , ¨ =   leads to an autonomous equation of the form 4.2.6.79.

© 2003 by Chapman & Hall/CRC

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684 57.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

ññóò ò ó ò ò ó ó

– ñó ò

= ! ( ï )ñ 2 .

ñóò ò ó ò ó




 = X  ( )  + L2S . Q M

Integrating yields a third-order linear equation: 58.

ññóò ò ó ò ò ó ó

+ 4ñóÈ ò ñóò ò ó ò

Solution: 59.

2

=L

3

+ 3(ñó ò ò ó )2 = ! ( ï ).

ó 

3

+L

2



2

+L

1

 +L



0

+ 13 X  ( − a )3  (a ) a . M 0

+ ôñ ó ò ñ ó ò ò ó ò ó + ( ô – 1)(ñ ó ò ò ó )2 = ! ( ï ). Integrating the equation two times, we obtain a second-order equation:

ññ óò ò ó ò ò ó ó



! 

 +  − 2 ( 
)2 = L  + L 1

0

2

+ X  ( − a )  (a ) a . M 0

60.

ññ óò ò ó ò ò ó ó

+ 4ñ ó ò ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 = ! (ñ ). ¨ 



 = 2   A £ ¨  . The substitution ¨ = 2 leads to an equation of the form 4.2.6.1: u

61.

ññóò ò ó ò ò ó ó

– ñó ò ñóò ò ó ò

= ! ( ï )ññó ò ò ó ò ó .

ó




 = L exp X  ( )  S . Q M

Integrating yields a third-order linear equation: 62.

ññóò ò ó ò ò ó ó

+ ñó ò ñóò ò ó ò

= ! ( ï )ññó ò ò ó ò ó .

ó




Integrating yields a third-order equation of the form 3.5.1.2: !   = L exp X¯ ( )  S .

! = ! (ï

ññóò ò ó ò ò ó ó

" = " (ï

Q

M

63.

+ ( ! – 1)ñóÈ ò ñóò ò ó ò ó + !"Iññóò + "Ióò ñ 2 = 0, ), ). The functions that solve the third-order linear equation "


 + ( ) = 0 are solutions of the given equation.

64.

ññóò ò ó ò ò ó ó

+ (4ñó ò +

+ 3(ñó ò ò ó )2 + 3 !„ñó ò ñóò ò ó + " (ï ) = 0,

!„ñ )ñ óò ò ó ò ó

!

= ! ( ï ).



The substitution ¨ = (! 
) leads to a first-order linear equation: ¨u
+  ¨ + = 0. Solution:  2 = L 2  2 + L 1  + L 0 + X  (  − a )2 ¨ ( a ) a , M 0 ¨ Ž −Ž ( ) ( ) L −X  ( )  S ,  ( ) = X¯ ( )  ;  0 is an arbitrary number. where ( ) =  Q 3 M M

65.

ññ óò ò ó ò ò ó ó

+ (4ñ ó ò + !„ñ )ñ ó ò ò ó ò ó + 3(ñ ó ò ò ó )2 + (3 !„ñ ó ò + "Iñ )ñ ó ò ò ó + " (ñ ó ò )2 + %„ññ ó ò + s = 0. Here,  =  ( ), = ( ), % = % ( ), ° = ° ( ). The substitution ¨ = !"
 leads to a third-order nonhomogeneous linear equation: ¨u


 +  ¨2

 + ¨2
+ % ¨ + ° = 0.

66.

(ñ +

ô ï

+ õ )ñ

óò ò ó ò ò ó ó

óò

+ 4(ñ

Solution: ( +  +  )2 = L 67.

ññ óò ò ó ò ò ó ó

= 4ñ

óò ñ óò ò ó ò ó

3

óò ò ó

+ 3(ñ

The transformation ˆ =

+ ô )ñ




3

óò ò ó ò ó

+L

)2 – 6

2

(ñ

óò ò ó

+ 3(ñ +L

ñ

2

óò

)4 2



 , ¨ = −O

)2 = ! ( ï ).

+ V ññ




2



1 3 0 + 3 X  ( − a )  (a ) a . M 0

1 + L

óò ò ó

– (ñ

ñó óò )2 W+!IO ò ñ P

.

leads to a second-order linear equation with

P respect to ¨ 2 : (¨ 2 )Šª

Š = 24ˆ 2 + 2  (ˆ ). Integrating it, we obtain Š ¨ 2 = L ˆ + L + 2ˆ 4 + 2 X (ˆ − a )  (a ) a . 2 1 Š M 0

© 2003 by Chapman & Hall/CRC

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685

4.2. NONLINEAR EQUATIONS

Taking into account that ˆ 
= ¨ , 
 = ˆ , 
Š = ˆ  ¨ , we find the solution in parametric form:

 =X where ¨ = A 68.

Q

L 2ˆ + L

1

M¨

4

+ 2ˆ

ˆ

+L

=L

3,

Š

+ 2 X Š (ˆ − a )  (a ) aÝS M 0

4

exp O X

J

1 2

ˆ

¨M

ˆ

P

,

.

– 2ññó ò ñó ò ò ó ò ó + ! ( ï )ñ 2 ñó ò ò ó ò ó + 2(ñó ò )2 ñó ò ò ó – ! ( ï )ññó ò ñó ò ò ó + 2 !„ó ò ( ï )ñ 2 ñó ò ò ó + 2 ! ( ï )(ñó ò )3 + [ ! 2 ( ï ) – 2 !„ó ò ( ï )]ñ (ñó ò )2 +

ñ 2 ñ óò ò ó ò ò ó ó

!„ó ò ò ó (ï )ñ 2 ñ óò

= 0.

The solution satisfies the second-order linear equation "

 +  ( )
 − H ( , L 1 , L 2 ) = 0, where H = H ( , L 1 , L 2 ) is the Weierstrass elliptic function determined by the second-order autonomous equation H 

 + H 2 = 0. 69.

ñ 2 ñ óò ò ó ò ò ó ó

– 2ññó ò ñóò ò ó ò ó + ! (ï )ñ 2 ñóò ò ó ò ó + 2(ñóò )2 ñóò ò ó – ! (ï )ññóò ñóò ò ó + 2 !„ó ò ( ï )ñ 2 ñó ò ò ó + 2 ! ( ï )(ñó ò )3 + [ ! 2 ( ï ) – 2 !„ó ò ( ï )]ñ (ñó ò )2 +

!„ó ò ò ó (ï )ñ 2 ñ óò

=


ï ñ

3

.

The solution satisfies the second-order linear equation "

 +  ( )
 − H ( , L 1 , L 2 ) = 0, where H = H ( , L 1 , L 2 ) is the solution of the first Painlev´e transcendent H 

 + H 2 = #* . 70.

ñ 2 ñ óò ò ó ò ò ó ó – 2ññóò ñóò ò ó ò ó + [ô + ! (ï )]ñ 2 ñóò ò ó ò ó – ñ (ñóò ò ó )2 – [3ô + ! (ï )]ñ ñ„óÈò ñóò ò ó + 2(ñó ò )2ñó ò ò ó + [ ô! ( ï ) + " (ï )]ñ 2ñó ò ò ó + 2 ô (ñó ò )3 – ô! ( ï )ñ (ñó ò )2 +  ô " (ï )ñ 2 ñóò

= % ( ï )ñ 3.

The solution satisfies the second-order linear equation "

 +  
 − H ( , L 1 , L 2 ) = 0, where H = H ( , L 1 , L 2 ) is the solution of the second-order linear equation H 

 +  ( ) H 
+ ( ) H = % ( ). 71.

72.

ñóò ò óñóò ò ó ò ò ó ó

– 3(ñó ò ò ó ò ó )2 = ! ( ï

ñóò

– ñ )(ñó ò ò ó )5 .

The Legendre transformation  = ‡ g
, ‡ g

g
g
g = −  (‡ ).

ñóò ò ó ò ò ó ó

= aև g
− ‡ leads to an equation of the form 4.2.6.1:

= ! (ñ )ñó ò " (ñóò ò ó ò ó ).

Integrating yields a third-order autonomous equation:

¨ M ¨ = Xv ( ) ( ) M X

+L ,

where

¨ = 


 ,

the order of which can be lowered by means of the substitution H ( ) =
 . 73.

ï ñóò ò ó ò ò ó ó

+ 2ñó ò ò ó ò

ó

= (ï

ñóò ò ó

)–5 !IO

75.

ï 2 ñ óò ò ó ò ò ó ó

+ 2ï

ñóò ò ó ò ó

= ! (ï

−5 2

2

P

.

leads to a third-order equation of the form 3.5.2.27:

¨2
J O  , £ ¨ P

¨ 


 = ¨ 74.

ü


−

The substitution ¨ ( ) = 

ï ñ óò ò ó ï ñ óò – ñ

ñóò ò ó

– 2ï

ñóò

where

+ 2 ñ )" ( ï

2

ñóò ò ó ò ó

 (ˆ ) = ˆ

−5

 (ˆ ).

).

The substitution ¨ ( ) =  2 

 − 2 
 + 2 leads to a second-order equation of the form ¨ 
). 2.9.4.36: ¨3

 =  (¨ ) (2

ñóò ò ó ò ò ó ó

= ! ( ï )" ( ï

ñóò ò ó ò ó

– 3ï

2

ñóò ò ó

+ 6ï

ñóò – 6ñ ). The substitution ¨ ( ) =  3 


 − 3 2 

 + 6 
 − 6 leads to a first-order separable equation: ¨2
=  3  ( ) (¨ ). 3

© 2003 by Chapman & Hall/CRC

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FOURTH-ORDER DIFFERENTIAL EQUATIONS

4.2.6-5. Other equations. 76.

ññóò ò ó ò ò ó ó

– 16 (ñó ò ò ó )2 = Particular solution:

ï 2 ! 1 (ñ óò ò ó ò ò ó ó

)+ï

! 2 (ñ óò ò ó ò ò ó ó

)+

! 3 (ñ óò ò ó ò ò ó ó

).

L 1  4 + 16 L 2  3 + 12 L 3  2 + L 4  + L 5 , where the constants L 1 , L 2 , L 3 , L 4 , and L 5 are related by three constraints =

1 24

1 3

L 1 L 3 − 16 L L 1 L 4 − 13 L 2 L L 1 L 5 − 16 L 77.

ññóò ò ó ò ò ó ó

– 16 (ñó ò ò ó )2 =

ñóò ò ó ò ó! 1 (ñ óò ò ó ò ò ó ó

Particular solution:

2 2

=  1(L

1 ),

3

=  2(L

1 ),

2 3

=  3(L

1 ).

) + ñó ò ò ó! 2 (ñó ò ò ó ò ò ó

ó) ï 2 + ! 3 (ñ ó ò ò ó ò ò ó ó

)+

ï ! 4 (ñ ó ò ò ó ò ò ó ó

)+

! 5 (ñ ó ò ò ó ò ò ó ó

).

L 1  4 + 16 L 2  3 + 12 L 3  2 + L 4  + L 5 , where the constants L 1 , L 2 , L 3 , L 4 , and L 5 are related by three constraints =

1 3

1 24

L 1 L 3 − 16 L L 1 L 4 − 13 L 2 L L 1 L 5 − 16 L

2 2

1 2

L 1  2 ( L 1 ) +  3 ( L 1 ), 3 = L 1  1 ( L 1 ) + L 2  2 ( L 1 ) +  4 ( L 1 ), 2 3 = L 2  1 ( L 1 ) + L 3  2 ( L 1 ) +  5 ( L 1 ). =

78.

= H ( ï , ñó ò , ñó ò ò ó , ñó ò ò ó ò ó ). ¨ 


 =  ( , ¨ , 2 ¨ 
, 2 ¨ 

 ). The substitution ¨ ( ) = 
leads to a third-order equation: u

79.

ñóò ò ó ò ò ó ó

80.

81.

82.

83.

ñóò ò ó ò ò ó ó

= H (ñ , ñó ò , ñó ò ò ó , ñó ò ò ó ò ó ). Autonomous equation. The substitution ¨ ( ) = ("
 )2 leads to a third-order equation: = ï –3 H (ñ ï , ñó ò , ï ñó ò ò ó , ï 2 ñó ò ò ó ò ó ). Homogeneous equation. The transformation a = ln  , ¨ equation of the form 4.2.6.79.

ñóò ò ó ò ò ó ó

1 2

£ ¨ ¨

). 

=  

leads to an autonomous

= ï –  –4 H ( ï ñ , ï +1 ñó ò , ï +2 ñó ò ò ó , ï +3 ñó ò ò ó ò ó ). Generalized homogeneous equation. The transformation a = ln  , ¨ autonomous equation of the form 4.2.6.79.

ñóò ò ó ò ò ó ó

=  

leads to an

= H ( ï , ï ñ ó ò – ñ , ñ ó ò ò ó , ñ ó ò ò ó ò ó ). This is a special case of equation 5.2.6.78 with B = 4. The substitution ¨ = 
 − a third-order equation.

ñ óò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó = H (ï , ï ñóò  The substitution ¨ = ¨2

  .

84.

¨*¨


+ 1 ¨
¨

= 2  ( , A £ ¨ , 1 ¨
, A 2   2   

– 2ñ , ñó ò ò ó ò ó ). =  
− 2 leads to a third-order equation:






=  ( , ¨ , ), where

= H ( ï , ï 2 ñó ò ò ó – 2 ï ñó ò + 2ñ , ñó ò ò ó ò ó ). The substitution ¨ ( ) =  2 

 − 2 
 + 2 leads to a second-order equation: (  ( , ¨ ,  −2 ¨2
).

ñóò ò ó ò ò ó ó

leads to

−2

¨2
)
 =

© 2003 by Chapman & Hall/CRC

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687

4.2. NONLINEAR EQUATIONS

85.

ñóò ò ó ò ò ó ó

= ñHwO

ñ óò ñ

ñ óò ò ó ñ

,

,




The transformation ˆ =

¨ 86.

ñóò ò ó ò ò ó ó

2

=

ñ óò ò ó ò ó ñ

.

P


 

 , ¨ = −O

¨ Šª

Š + ¨ (¨ Š
)2 + 4ˆ ¨*¨ Š
+ 3¨

ñ ï

–4

HwO ï  ñ ú

ï ñ óò ñ

,

2

P

+ 6ˆ 2 ¨ + ˆ

ï 2 ñ óò ò ó ñ

,

2

leads to a second-order equation: 4

ï 3 ñ óò ò ó ò ó ñ

,

=  (ˆ , ¨ + ˆ 2 , ¨*¨ Š
+ 3ˆ ¨ + ˆ 3 ).

P

.

 
 Generalized homogeneous equation. The transformation a =   + , H = third-order equation. 87.

ñóò ò ó ò ò ó ó

=

ñ ï

–4

HwO

ï ñ óò ñ

,

The transformation H = 88.

ñ óò ò ó ò ò ó ó

=

ñ óò ò ó ñ óò HwO ó , ñ óò ñ ò

ï 2 ñ óò ò ó ñ

 


ï 3 ñ óò ò ó ò ó ñ

,

ñ óò ò ó ñ óò

–ñ

,

ñ óò ò ó ò ó ñ óò

.

P

 2 

 , ¨ =

leads to a

leads to a second-order equation. .

P

Autonomous equation. This is a special case of equation 4.2.6.79. Particular solution: = L 1 exp( L 2  ) + L 3 , where L 1 is an arbitrary constant and the constants L L 23 =  ( L 2 , − L 2 L 3 , L 22 ). 89.

ó

ó

ó

ó

2

and L

2

are related by the constraint

ó

=  – ^ HÊm ^ ñ ,  ^ ñ ó ò ,  ^ ñ ó ò ò ó ,  ^ ñ ó ò ò ó ò ó  .  Equation invariant under “translation–dilatation” transformation. The substitution ‡ =  9 leads to an autonomous equation of the form 4.2.6.79.

ñ óò ò ó ò ò ó ó

90.

= ï – 4 H ( ïƒú  ^ ] , ï ñó ò , ï 2 ñó ò ò ó , ï 3 ñó ò ò ó ò ó ). The transformation H =  +  9  , ¨ =  
 leads to a third-order equation.

91.

ñóò ò ó ò ò ó ó

ñóò ò ó ò ò ó ó

= ñHwO+

^ ó ñ ú

,

ñ óò ñ óò ò ó , ñ ñ

,

ñ óò ò ó ò ó ñ

P

.

The transformation H =  9 + , ¨ = 
  leads to a third-order equation. 92.

H (ñ óò ò ó¦ñ , ñ ñóò ò ó

– ñó ò 2 , ñó ò ò ó ò ó¦ñó ò , ñó ò ò ó ò ò ó ó¦ñ ) = 0. Autonomous equation. This is a special case of equation 4.2.6.79. 1 b . Particular solution:

 ), where the constants L 1 , L 2 , L 2 are related by the constraint   L 32 , 4 L 1 L 2 L 32 , L 32 , L 34  = 0. 2 b . Particular solution: = L 1 cos( L 3  ) + L 2 sin( L 3  ), where L 1 , L 2 , and L 2 are related by the constraint   − L 32 , −( L 12 + L 22 ) L 32 , − L 32 , L 34  = 0. 93.

=L

1

exp( L

3

 )+L

2

exp(− L

3

H  ï , ñ ñóò ò ó

– (ñó ò )2 , ññó ò ò ó ò ó – ñó ò ñóò ò ó , ññóò ò ó ò ò ó ó – (ñóò ò ó )2  = 0. ¨ !


¨ 

 ) = 0. The substitution ( ) =  − (  )2 leads to a second-order equation:  ( , ¨ , ¨u
, 2

© 2003 by Chapman & Hall/CRC

Page 687

688 94.

FOURTH-ORDER DIFFERENTIAL EQUATIONS

HwO

ñ óò ò ó ò ò ó ó ñ óò ò ó ò ò ó ó , ñ ñ óò ñ óò

– ñó ò ò ó ò

ó

= 0.

P

A solution of this equation is any function that solves the third-order linear equation:




 = L where the constants L 95.

H ( ï , ñ óò ò ó

1

and L

+ ôñ , ñó ò ò ó ò ò ó ó – ô The substitution ¨ = 

 + 

2 2

1

+L

2,

are related by the constraint  ( L

1, −

L 2 ) = 0.

ñ , ñ óò ò ó ò ò ó ó

+ ôñó ò ò ó ) = 0. leads to a second-order equation:  ( , ¨ , ¨u

 −  ¨ , ¨2

 ) = 0.

© 2003 by Chapman & Hall/CRC

Page 688

Chapter 5

Higher-Order Differential Equations 5.1. Linear Equations 5.1.1. Preliminary Remarks In this Chapter, we denote higher derivatives by ( 

)



to mean M

.   M 1 b . The general solution of a homogeneous linear equation of the B th-order   ( ) (  ) +   −1 ( ) (  −1) + ))) +  1 ( ) 
+  0 ( ) = 0 has the form:

=L

1 1(

 )+L

2 2(

(1)

 ) + ))) + L   ( ).

(2)

Here, 1 ( ), 2 ( ),  ,  ( ) make up a fundamental set of solutions (the  are linearly independent solutions;  1 0); L 1 , L 2 ,  , L  are arbitrary constants. 2 b . Let

0

=

0(

 ) be a nontrivial particular solution of equation (1). Then the substitution =

0(

H ( )  M

 )ù

leads to a linear equation of the (B − 1)st-order for H ( ). Let 1 = 1 ( ) and 2 = 2 ( ) be two nontrivial linearly independent particular solutions of equation (1) with ≡ 0. Then the substitution =

1

X

2

¨

 − M

2

X

1

¨

leads to a linear equation of the (B − 2)nd-order for ¨ = ¨ ( ).

M



3 b . Some additional information about higher-order linear equations can be found in Subsections 0.4.1–0.4.3.

5.1.2. Equations Containing Power Functions 5.1.2-1. Equations of the form   ( ) (  ) +  0 ( ) = ( ). 1.

ñ ó(6) + ôñ

= 0. 1 b . Solution for  = 0: =L 2 b . Solution for  = , =L

1

cos ,! + L

2

6

1

+L

2

2

 + L 3

+L

4



3

+L

5



4

+L

6

 5.

> 0:

sin ,! + cos 

1 2

,!   L + sin 

3 1 2

cosh ˆ + L

,!   L

5

4

sinh ˆ 

cosh ˆ + L

6

sinh ˆ  ,

where

ˆ =

2

3 2

,! .

© 2003 by Chapman & Hall/CRC

Page 689

690

HIGHER-ORDER DIFFERENTIAL EQUATIONS 3 b . Solution for  = − , =L

2.

ñ ó(2÷

1

cosh ,! + L

= ô 2÷ Solution: )

6

< 0:

sinh ,! + cosh 

2

1 2

,!   L

+ sinh 

ñ

1

  ½   + L 2  − +

9

,

=ô ï Solution: )

ñ

+õ , =

where 4.

ñ ó( ÷

½ Á

,!   L

,

−1



=1

where Y  =  cos  ,  =  sin  ; L B B constants.

ñ ó( ÷

cos ˆ + L 5

4

sin ˆ 

cos ˆ + L

6

sin ˆ  ,

where

ˆ =

2

3 2

,! .

. =L

3.

3

1 2

ô ½

LlÁ = =0

Á 

=0



1,

9 9  t ¼ ( #  cos  + $  sin  ), L 2 , #  , $  ( , = 1, 2,  , B − 1) are arbitrary

> 0.

a  +1 S a, LlÁ ´ÁtX ÷ exp ´Áa − Q 0  (B + 1) M

´Á = exp O

2 @!™  , B +1P

and ™ 2 = −1.

=ô ï _ ñ . For specific ý , see equations 5.1.2.2, 5.1.2.3 (with  = 0), 5.1.2.5–5.1.2.9, and 5.1.2.10 (with  = 0). )

1 b . Let B ≥ 2, ý > −B , and (B + ý )( ° + 1) ≠ 1, 2,  , B − 1, where ° = 0, 1,  Then the ¾ ¾ ¾ that can be represented as: equation has B solutions ( ) = 

−1

« 

,1+

: J ,(:

+ −1)

: +  , J  

 = 1, 2,  , B .

(1)

Here, «  , + , ( H ) is a Mittag-Leffler type special function defined by: -

½ «  , + ,- ( H ) = 1 + ÷   H  ,  =1 (2) » À −1 »  À −1  B ( ° + 0 ) + 1 1 w ¡ ,  = = [B ( ° + 0 ) + 1]  [B ( ° + 0 ) + B ]  B ( ° + 0 + 1) + 1  =0 w =0 ¡ ¡ ¡ where (ˆ ) is the gamma function, 0 is an arbitrary number, and > 0. w ¡ If ý ≥ 0, solutions (1) are linearly independent. Series expansions of (1) are convenient for small  . 2 b . Let B ≥ 2, ý < −B , and (B + ý )( ° + 1) ≠ −1, −2,  , −(B − 1), where ° = 0, 1,  Then ¾ ¾ question the equation in has B solutions ¾ that can be represented as: ( ) =  −1 «  ,−1−: J ,−1−(: + ) J   (−1)   : +   ,  = 1, 2,  , B , (3) where «  , + , ( H ) is the Mittag-Leffler type special function defined by (2). If ý ≤ −2B , solutions (3) are linearly independent. Series expansions of (3) are convenient for large  . 3 b . The transformation  = a −1 , = ¨ a 1−  leads to an equation of similar form: ¨ (g  ) =  (−1)  +1 a −2  −: ¨ . ^_ Reference: M. Saigo and A. A. Kilbas (2000).

© 2003 by Chapman & Hall/CRC

Page 690

691

5.1. LINEAR EQUATIONS

5.

6.

ï 2 ÷ ñ ó( ÷

)

ïƒ÷ ñ ó(2÷

)

= ôñ . The transformation  = a ¨ (g  ) = (−1)   ¨ .

ôñ

= Solution:

−1

= ¨ a

,

1−

 leads to a constant coefficient linear equation:

.

½ V ´  J 2 = L   =1

1

®

 2ý  £   + L  2 | ;   2ý  £   W ,

where ®  ( H ) and | ;  (H ) are the modified Bessel functions; ý 1 , ý 2 ,  , ý  are roots of the equation ý  = £  . 7.

ï 3÷ ñ ó(2÷

)

=

ôñ

.

The transformation  = a 8.

ïƒ÷ +1 ) 2 ñ ó(2÷

+1)

=

Solution:

ôñ

−1

=¨ a

,

1−2

 leads to an equation of the form 5.1.2.6: a  ¨ (2 g  )= ¨ .

.

2 J ½ L  Vø £   + ™ ø  +1 J 2  2ý  £   W , −  −1 J 2  2 ý   =0 where ø + ( H ) are the Bessel functions; ý 0 , ý 1 ,   , ý 2  are roots of the equation ý 2  ™ 2 = −1.



(2 +1) 4

=

9.

10.

11.

ï 3÷ +3 ) 2 ñ ó(2÷

= ôñ . The transformation  = a a  +1 J 2 ¨ (2 g  +1) = −  ¨ .

ï

,

= ¨ a

−2

 leads to a linear equation of the form 5.1.2.8:

−1

,

= ¨ a

1−

 leads to a linear equation of the form 5.1.2.3:

ñ ó÷

12.

13.

(ô

+1 ( )

= (ö

ï

+ ý )ñ . (˜ + The transformation ˆ = M , ¨ =  +  (  +  )  ¨ (Š  ) = ¢ −  ˆ ¨ , where ¢ = ˜( −  .

(ô ï

14.

−1

)

+ õ )2 ÷

(ô

= − ™ ;

+1)

= ôñ + õ ïƒ÷ . The transformation  = a ¨ (g  ) = (−1)  ( a ¨ +  ).

ï 2÷ +1 ñ ó(÷

+1

+ õ )÷ (ö ï

+ ý ) ÷ ñ ó( ÷

õï

ñ ó( ÷

)

leads to an equation of the form 5.1.2.3:

M

üIñ

=

−1

.

 +  ¨ , = leads to a constant coefficient linear 1 b . The transformation ˆ = ln (˜ + ( (˜ + )  −1 M M equation.   +  ¨ , = leads to the Euler equation 5.1.2.39: 2 b . The transformation = (˜ + ( (˜ + )  −1 M ¨ (û  ) = , ¢ −  ¨ , where ¢ =  −M ˜( . M

ï

2

+

+ ö )÷

)

=

üIñ

.

The transformation ˆ = X linear equation.



(ô

)

ï

+ õ )÷ (ö

ï

+ ý )3 ÷

ñ ó(2÷

=

2

M



+ ' + (

üIñ

, ¨ = ( 

2

+ ' + ( )



1− 2

leads to a constant coefficient

.

 +  ¨ , = The transformation ˆ = (˜ + ( (˜ + )2  M M  ) ¨ ˆ ¨ (2 −2  Š =, ¢ , where ¢ =  − ˜( . M

−1

leads to an equation of the form 5.1.2.6:

© 2003 by Chapman & Hall/CRC

Page 691

692 15.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ï

(ô

) (ö ï

+ õ )÷

+ ý )3 ÷

+1 2

) ñ ó ÷

+3 2 (2 +1)

=

üIñ

.

 +  ¨ , = (˜ + ( (˜ + )2  M M  +1) ¨ ˆ +1 J 2 ¨ (2 −2  −1 Š =, ¢ , where ¢ =  − ˜( . M

The transformation ˆ =

leads to an equation of the form 5.1.2.8:

5.1.2-2. Equations of the form   ( ) (  ) +  1 ( )
 +  0 ( ) = ( ). 16. 17. 18.

19. 20.

+ ô ï ñó ò + ôƒü ï –1 ñ = 0. Integrating yields an (B − 1)st-order linear equation:

ñ ó( ÷

)

)

ñ ó( ÷

) + ô ï +1 ñó ò

23.

+  

=L .

+ ô ( ü + ù ) ï ñ

−1)

+  

+1

H = 0.

= 0. = ¨ a 1−  leads to an equation of the form 5.1.2.16: The transformation  = a −1 , ¨ (g  ) + 'a Á ¨2g
+ @!a Á −1 ¨ = 0, where  =  (−1)  +1, @ = 1 − , − 2B . + ( ô ï + õ ) ï ñó ò – ô ï ñ = 0. Particular solution: 0 =  +  .

ñ ó( ÷

)

ñ ó( ÷

)

ï

+ õ ) ï

+ (ô

ñóò

ñ ó( ÷

)

ï

+ (ô

+ õ ) ï

ñ ó÷

+ (ô ï

– 2ô

ñóò

= 0.

– 3ô

ï ñ

= 0.

= (  +  )3 .

0

– ô (ù – 1) ï ñ = 0. Particular solution: 0 = (  +  )  −1 . The substitution H = (  +  )
 −  (B − 1) leads to an (B − 1)st-order linear equation: H (  −1) + (  +  )  H = 0. ( )

+ õ ) ï  ñ óò

ï ñ

= (  +  )2 .

0

Particular solution:

22.

−1)

+ ô ï +1 ñó ò – ô (ù – 1) ï ñ = 0. The substitution H =  
− (B − 1) leads to an (B − 1)st-order equation: H ( 

ñ ó( ÷

Particular solution: 21.

( 

+ ô ï +1 ñó ò – ôþ Particular solution: linear equation:

ñ ó( ÷

)

ï ñ

0

= 0, þ = 1, 2, uuu , ù – 1. =  + . The substitution H =  
−

c  −+

−1

O

H ( + ) +   H = 0, öP

where

¡

leads to an (B − 1)st-order

c = M .  M

5.1.2-3. Other equations. 24.

25.

= ô ÷ ñ + õ ï  (ñ ó ò ò ó – ôñ ). This is a special case of equation 5.1.6.18 with  ( ) = '  . The substitution ¨ = 

 −    −2) +  ¨ (2  −4) + ) )) +   −1 ¨ = '  ¨ . leads to an (2B − 2)nd-order linear equation: ¨ (2

ñ ó(2÷ ñ ó( ÷

)

+ ô ÷ –1 ñ ó(÷ –1) + ­­­ + ô 1 ñó ò + ô 0 ñ = 0. Constant coefficient homogeneous linear equation. To solve this equation, determine the B roots of the characteristic equation: Œ  +   −1 Œ  −1 + ))) +  1 Œ +  0 = 0. If the roots Œ 1 , Œ 2 ,  , Œ  are all real and different, then the general solution of the original equation is: = L 1 exp( Œ 1  ) + L 2 exp( Œ 2  ) + ))) + L  exp( Œ   ). The general case, which involves the cases of multiple and/or complex roots of the characteristic equation, is discussed in Subsection 0.4.1. )

© 2003 by Chapman & Hall/CRC

Page 692

693

5.1. LINEAR EQUATIONS

ñ ó( ÷

26.

)

ô ï  ñ ó(ú

+

Particular solution:

ñ ó( ÷

27.

)

ï 

+ (ô

0

÷ – ú ) ñ ó( ú

–õ

Particular solution:

ñ ó( ÷

28.

)

ôñ ó(÷

+

–1)

ï ñ ó( ÷

)

– ùþxñ

ó( ÷

õ ÷ )ñ

.

ú ï ñ

– ôƒõ





= 0.

ïƒú ñ

+ ôƒõ

ï ñ

= 0.

. = 0.

−  . 0 = 

+ô

–1)



)

õ ïƒú ñóò

+

+



= =

0

Particular solution: 29.

ú¹ï

– ( ôƒõ

)

ù

= 0,

Solution:

=  (+

= 2, 3, 4,



+1) −1

1−

O

uuu

 M 3P M

þ

,

+

(

1−

= 1, 2, 3,

uuu

 ¨ ),

where ¨ is the general solution of the constant coefficient linear equation: ¨ (  ) +  ¨ = 0. 30. 31.

ï ñ ó( ÷

)

ï ñ ó( ÷

)

ï ñ ó( ÷

)

+ ùñ

ó( ÷

–1)

+ ùñ

ó( ÷

–1)

=

ô ï ñ

+õ .

ô ï 2ñ

+õ . leads to an equation of the form 5.1.2.3: ¨ ( 

The substitution ¨ =  =

The substitution ¨ = 

32.

+ (ù – þ

– 1)ñ

ï ñ ó( ÷

)

+

ô ï  ñ ó(ú

ï ñ ó( ÷

)

=

÷ 7 –1 

=0

[( ô


Here, #  = 1, #

 0

–1)

ï

– (ô

)

Particular solution: 34.

ó( ÷

0

+1

+

ô ï  ñ óò

–


+



ï –1 ñ

– ôþ

+ . 0 = 

Particular solution: 33.

leads to a constant coefficient linear equation: ¨ ( 

ôþ ï –1 + ï

)ï +




+1 ]

ñ ó( 

= ¨ + .

=  ¨ +  .

= 0.

+ ù )ñ = 0.

=  .



)

)

)

.

= 0;  and #3Á are arbitrary numbers ( @ = 1, 2,  , B − 1).

 7 −1 #3Á Á =0

Œ Á . Let the roots Œ 1 , Œ 2 ,  , Œ  −1 of the algebraic equation  ( Œ ) = 0 be all different, and  ( ) ≠ 0. Then the solution is as follows:     
( ) S . = L 1   1 + L 2   2 + ))) + L  −1   n −1 + L     −   ( ) Q Denote  ( Œ ) =

35.

7÷



=0

(ô



ï

+

õ  ) ñ ó( 

)

+1

= 0.

The Laplace equation. Particular solutions:

where

.

(a ) =

:ž¼  =X ¼ 9 7

Á

7 =0

Áa Á , Æ (a ) =

Á

=0

.

1 exp a + X (a ) Q

.Æ

(a ) aÝS a , (a ) M M

Áa Á ; ƒ  and ý  are found from the condition

exp O a + X

.Æ

:ž¼ (a ) a û = 0. (a ) M P û 9 ¼ û

In many cases, the path of integration has to be chosen on the complex plane.

© 2003 by Chapman & Hall/CRC

Page 693

694 36. 37. 38. 39.

HIGHER-ORDER DIFFERENTIAL EQUATIONS + 2ù ï ñ ó(÷ –1) + ù (ù – 1)ñ ó(÷ –2) = ô ï 2 ñ + õ . The substitution ¨ =  2 leads to a constant coefficient linear equation: ¨ ( 

ï 2 ñ ó( ÷

)

ï 2 ñ ó( ÷

)

+ 2ù ï ñ ó(÷ –1) + ù (ù – 1)ñ ó(÷ –2) = ô ï 3 ñ + õ . The substitution ¨ =  2 leads to an equation of the form 5.1.2.3: ¨ ( 

ï (ï

+ þ ) ñ ó( ÷ ) + ï ( ô Particular solution:

ô ÷ ƒï ÷ ñ ó(÷

ï

0

– ï – ù )ñ  =  .

ó( ú

)

– ô (ï + þ

) ï

ñ

)

= ¨ + .

)

=  ¨ +  .

= 0.

+ ô ÷ –1 ïƒ÷ –1 ñ ó(÷ –1) + ­­­ + ô 1 ï ñó ò + ô 0 ñ = 0. Euler equation. If all roots Œ  ( , = 1, 2,   , B ) of the algebraic equation )

½  Á

ÁžŒ ( Œ − 1)  ( Œ − @ + 1) = − 

0

=1

are different, the general solution of the original differential equation is given by: =L

1|

 |  1 + L 2 | |  2 + ))) + L  | | Dn .

In the general case, the substitution a = ln | | leads to a constant coefficient linear equation of the form 5.1.2.25:

½

40. 41. 42. 43. 44. 45. 46.

Á

Ážc ( c − 1)  ( c − @ + 1) = − 

0

,

where

=1

c = M .  M

+ ù ï 2÷ ñ ó(÷ –1) = ô ï ñ . The substitution ¨ =  leads to an equation of the form 5.1.2.5:  2  ¨ ( 

ï 2÷ +1 ñ ó(÷

)

ï 2÷ +1 ñ ó(÷

)

+ ù ï 2÷ ñ ó(÷ –1) = ôñ . The substitution ¨ =  leads to an equation of the form 5.1.2.10:  2 

+ 2ù ïƒ÷ –1 ñ ó(2÷ The substitution ¨ = 

ïƒ÷ ñ ó(2÷

)

+1

¨ (  ) =  ¨ .

= ôñ .   leads to an equation of the form 5.1.2.6:   ¨ (2

–1)

+ 2ù ï 3÷ –1 ñ ó(2÷ –1) = ôñ .   The substitution ¨ =  leads to an equation of the form 5.1.2.7:  3  ¨ (2

ï 3÷ ñ ó(2÷

ï ÷ +1 ñ ó(2÷

7

+1)

ï 3÷ +3 ) 2 ñ ó(2÷

)

= ¨ .

J ¨ (2 

+1 2

+ (2ù + 1)ï 3÷ +1 ) 2 ñ ó(2÷ ) = ôñ . The substitution ¨ =  leads to an equation of the form 5.1.2.9:  3 

+1)

= ¨ .

+1)

J ¨ (2 

+3 2

ï )ñ ó(÷ ) + g ÷ –2 (ï )ñ ó(÷ –1) + ­­­ + g 1 (ï )ñ óò ò ó . Here, the Á ( ) are polynomials of degree ≤ @ , as:

= ¨ .

)

)

ï ñ . + (2ù + 1)ï ÷ ñ ó(2÷ ) = ô The substitution ¨ =  leads to an equation of the form 5.1.2.8:  

g ÷

= ¨ .

)

+1)

= ¨ .

+ ( ô 1 ï + õ 1 )ñ ó ò – þ ô 1 ñ = 0. is a positive integer,  1 ≠ 0. ¡ A particular solution of this equation is the polynomial of degree that can be written

–1 (

½+

. 1  + O− [ ®´ − + −1 (  0 =  1P  =0  Á +1 Á with @ ≠ −1. where c = M , ®´ =  @ +1 ^_ M Reference: E. Kamke (1977).

−1

. c  + ))) + 1 c

¡

2

+  1 c )]   + ,

© 2003 by Chapman & Hall/CRC

Page 694

695

5.1. LINEAR EQUATIONS

47.

[ ô ÷ ïƒ÷ + g ÷ –1 ( ï )]ñ ó(÷ ) + ­­­ + [ ô 1 ï + g 0 ( ï )]ñó ò + . Here, the Á ( ) are polynomials of degree ≤ @ . Assume that for some integer ≥ 0:

= 0.

¡

½ Á

ô 0ñ

L + Á @ ! Á = 0,

=0

where

! , L +Á = ¡ @ ! ( − @ )! ¡

and is the least of the numbers satisfying this condition. Then there exists a solution in the ¡ form of a polynomial of degree such that no polynomial of a smaller degree satisfies the ¡ original equation.

^_ 48.

Reference: E. Kamke (1977).

V ï g ( S ) – i ( S )WUñ = 0,

. Here,

S

. =

≡

ï ý ý ï

.

( H ) and Æ = Æ ( H ) are arbitrary polynomials of degree  and  , respectively. . Suppose Æ ( H + 1) = ( H + 1) Æ 1( H + 1), where the polynomial Æ 1 ( H + 1) is such that ( H ) and Æ 1 ( H + 1) do not have common factors. Then the original equation admits a formal solution in the power series form: 0

^_

½

= ÷



=0

#    ,

where

.

#  +1 (B ) . = #  Æ (B + 1)

Reference: H. Bateman and A. Erd´elyi (1953, Vol. 1).

5.1.3. Equations Containing Exponential and Hyperbolic Functions 5.1.3-1. Equations with exponential functions.

ó ñóò

1.

ñ ó( ÷

)

+ ( ô ï + õ )  Particular solution:

2.

ñ ó( ÷

)

ï

+ (ô

ó

 ñ óò

+ õ )

Particular solution: 3.

ñ ó( ÷

)

ï

+ (ô

ó ñóò

+ õ ) 

Particular solution: 4.

5. 6.

7.

ñ ó( ÷

ó

– ô ñ = 0. 0 =  +  . – 2 ô 0

= 0. 2

= (  +  ) . – 3 ô

0

ó

ó

 ñ

óñ

= 0. 3

= (  +  ) .

ó

+ ( ô ï + õ )   ñ ó ò – (ù – 1) ô  ñ = 0. Particular solution: 0 = (  +  )  −1 . The substitution H = (  +  )
 −  (B − 1) leads to  an (B − 1)st-order linear equation: H (  −1) + (  +  )  H = 0. )

ó

ó

+ ô ï  ñó ò – ôþ ñ = 0, Particular solution: 0 =  + .

ñ ó( ÷

)

þ

= 1, 2,

uuu , ù

– 1.

ó

= ô ÷ ñ + õ (ñó ò ò ó – ôñ ).  This is a special case of equation 5.1.6.18 with  ( ) =    . The substitution ¨ = 

 −     −2) +  ¨ (2  −4) + ) )) +   −1 ¨ =   ¨ . leads to an (2B − 2)nd-order linear equation: ¨ (2

ñ ó(2÷ ñ ó( ÷

)

)

+ ( ô



ó

–õ

÷ – ú ) ñ ó( ú

Particular solution:

0

=

)





– ôƒõ

ú óñ

= 0.

.

© 2003 by Chapman & Hall/CRC

Page 695

696 8. 9.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó( ÷

ó ñóò

+ ôñ ó(÷ –1) + õ Particular solution: )

ñ ó(÷ ) + ô  ó ñ ó(ú

)

0

– ( ôƒõ

Particular solution: 10.

ñ ó( ÷

)

÷ 7 –1

=



=0

ó

 +1  

(


0

+ ôƒõ  =  − .

ú ó 



=

+

óñ

= 0.

õ ÷ )ñ

= 0.

.

õ
 +1 –
 )ñ ó( ) .

+

Here, #  = 1, # 0 = 0;  and #  are arbitrary numbers ( , = 1, 2,  , B − 1). Ù , where the “ + are roots of the polynomial equation Particular solutions: + =  

 7 −1 

11. 12. 13. 14.

=0

# 

ï ñ ó( ÷

)

ï ñ ó( ÷

)

+1

“  = 0.

ó

+ ô ï   ñ ó( ú Particular solution:

– [ô (ï + þ  0 =   .

)

ó( ÷

+ (ù – þ – 1)ñ Particular solution:

–1)

+

ó

+ þ )ñ ó(÷ ) + ï ( ô – ï – ù )ñ  Particular solution: 0 =   .

ó

( ô ïƒú + õ + ö )ñ ó(÷ Particular solution:

)

ó

+ ï + ù ]ñ = 0.

ô ï  ó ñóò

+ . 0 = 

ï (ï

ó 

)

ó( ú

)

– ô (ï + þ

ó

= õ ñ , þ = 0, 1,  + +  +( . 0 = 

15.

þ = 0, 1, ( ô ïƒú  + õ )ñ ó(÷ ) = (–1)÷ õÿñ , + −   +  . Particular solution: 0 = 

16.

ODô

ó

+

÷ 7 –1 ï  (÷ õ Pñó 

)

=

=0

Particular solution:

óñ

– ôþ

0

ô ó ñ 

= 

= 0. ) 

uuu , ù uuu , ù

óñ

= 0.

– 1. – 1.

.

+

7

−1



=0

  .

5.1.3-2. Equations with hyperbolic functions. 17. 18.

ñ ó(2÷

)

=

ô ÷ ñ

+ õ sinh  (

ñ ó( ÷

)

+

ô

sinh 

ï ñ ó( ú

ñ ó( ÷

)

+ ( ô sinh 

ï

–

Particular solution: 20. 21.

– ôñ ).

ú

ï

This is a special case of equation 5.1.6.18 with  ( ) =  sinh  ( Π ). Particular solution:

19.

ï )(ñ óò ò ó

– ( ôƒõ

)

0

=





õ ÷ – ú ) ñ ó( ú 0

=





sinh  . )

– ôƒõ

ú

+

õ ÷ )ñ

.

)

+ ô ï sinh  ï Particular solution:

ï ñ ó( ÷

)

ñ ó( ú

0

– [ô (ï + þ  =  . )

ï ñ

sinh 

+ ( ô ï + õ ) sinh ú ( ï )ñ ó ò – ô sinhú ( Particular solution: 0 =  +  .

ñ ó( ÷

= 0.

) sinh 

ï )ñ ï

= 0. = 0.

+ ï + ù ]ñ = 0.

© 2003 by Chapman & Hall/CRC

Page 696

697

5.1. LINEAR EQUATIONS

22.

23.

ñ ó(2÷

)

ô ÷ ñ

=

ñ ó( ÷

)

cosh 

ô

+

ï ñ ó( ú

ñ ó( ÷

)

ï

+ ( ô cosh 

–

ñ ó( ÷

)

26.

ï ñ ó( ÷

)

ô ï

+

cosh 

27.

28.

29.

ñ ó(2÷

)

ñ ó(2÷

)

ï ñ ó( ú

33.

ï )ñ óò

– ô coshú (

– [ô (ï + þ

)

ï )ñ

= 0.

ï

) cosh 



=

ñ

+ ô (ñó ò sinh ï – ñ cosh ï ).

+ ï + ù ]ñ = 0.

The substitution ¨ = 
sinh  − cosh  leads to an (2B − 1)st-order linear equation.

ñ ó( ÷

)

tanh 

ô

+

ï ñ ó( ú

ñ ó( ÷

)

ï

+ ( ô tanh

ñ ó( ÷

)

ï ñ ó( ÷

)

+

ô ï

0

ñ ó( ÷

)

)

+

ô

coth

ï ñ ó( ú

ñ ó( ÷

)

+ ( ô coth

ï

–õ

ñ ó( ÷

)

+ (ô

ï

ï ñ ó( ÷

)

+

ô ï

0

0

ï ñ ó( ú

Particular solution:

0

ú

– ôƒõ

)

= 0.

ï ñ

tanh

= 0.

. – ô tanhú (

ï )ñ óò )

– [ô (ï + þ

ï )ñ

= 0.

ï

) tanh 



+

ï

+ ù ]ñ = 0.

=  .

ú

– ( ôƒõ





=

=

+ õ ) coth ú ( coth





ï

coth

+

õ ÷ )ñ

= 0.

.

÷ – ú ) ñ ó( ú 0

õ ÷ )ñ

=  +  .

0

Particular solution:

+

.



=

0

ï

tanh 

õ ÷ – ú ) ñ ó( ú

ï ñ ó( ú

tanh





=

+ õ ) tanh ú (

ï

+ (ô

ú

– ( ôƒõ

) 0

–

Particular solution: 36.

= 0.

The substitution ¨ = 
cosh  − sinh  leads to an (2B − 1)st-order linear equation.

Particular solution: 35.

ï ñ

cosh 

.

+ ô (ñó ò cosh ï – ñ sinh ï ).

Particular solution: 34.

ú

– ôƒõ

ñ

Particular solution: 32.



)

=

Particular solution: 31.

= 0.

=  .

0

Particular solution: 30.

õ ÷ )ñ

=  +  .

0

Particular solution:

+

.



=

ï

cosh

õ ÷ – ú ) ñ ó( ú 0

Particular solution:





=

+ õ ) cosh ú (

ï

+ (ô

ú

– ( ôƒõ

) 0

Particular solution: 25.

– ôñ ).

This is a special case of equation 5.1.6.18 with  ( ) =  cosh  ( Π ).

Particular solution: 24.

ï )(ñ óò ò ó

+ õ cosh  (



)

– ôƒõ

ú

ï ñ

= 0.

ï )ñ

= 0.

coth

.

ï ) ñ óò

– ô cothú (

=  +  . )

– [ô (ï + þ



) coth 

ï

+ ï + ù ]ñ = 0.

=  .

© 2003 by Chapman & Hall/CRC

Page 697

698

HIGHER-ORDER DIFFERENTIAL EQUATIONS

5.1.4. Equations Containing Logarithmic Functions 1. 2.

= ô ÷ ñ + õ ln ï (ñ ó ò ò ó – ôñ ). This is a special case of equation 5.1.6.18 with  ( ) =  ln  .

ñ ó(2÷ ñ ó( ÷

)

)

ï ñ ó( ú

ln 

ô

+

Particular solution: 3.

ñ ó( ÷

)

+ ( ô ln 

ï

–õ

– ( ôƒõ

)

0

5. 6.

0

+ ( ô ï + õ ) ln  ( Particular solution:

ñ ó( ÷ ñ ó( ÷

)

)

ï

+ õ ) ln  (

+ (ô

10.

.

ú

– ôƒõ

)

õ ÷ )ñ

= 0.

ï ñ

ln 

.

ñ ó( ÷

)

ï

+ õ ) ln  (

+ (ô

ñ ó( ÷

)

ï

0

= 0.

– 3 ô ln  (

ï )ñ

= 0.

3

= (  +  ) .

– ô (ù – 1) ln  (

 0 = (  +  )

+ ô ï ln  ( ï )ñó ò – ôþ ln  ( Particular solution: 0 =  + .

ñ ó( ÷

)

+ ô ï ln  ( ï )ñ Particular solution:

ï ñ ó( ÷

)

ó( ú

0

= 0.

ï )ñ

= (  +  ) .

ï )ñ ó ò

= 0.

– 2 ô ln  ( 2

ï )ñ óò

= 0.

ï )ñ

ï )ñ

– ô ln  ( =   + .

ï )ñ óò

0

+ õ ) ln  (

+ (ô

ï )ñ óò

0

Particular solution:

9.

=

+

)

Particular solution: 8.





ï

+ ôñ ó(÷ –1) + õ ln  ( ï )ñó ò + ôƒõ ln  (  Particular solution: 0 =  −  .

ñ ó( ÷

Particular solution: 7.



ln 



=

÷ – ú ) ñ ó( ú

Particular solution:

4.

ú

−1

þ

= 0,

ï

) ln  (

)

= 0.

.

ï )ñ

– [ô (ï + þ  =  .

ï )ñ

= 1, 2,

uuu , ù

– 1.

) + ï + ù ]ñ = 0.

5.1.5. Equations Containing Trigonometric Functions 5.1.5-1. Equations with sine and cosine. 1.

ñ ó( ÷

)

+

ô

sin ,ï

ñ ó( ú

Particular solution: 2.

4. 5.

ú

– ( ôƒõ 0





=



sin ,ï +



ú

+ ôñ ó(÷ –1) + õ sin ú ( ï )ñ ó ò +  Particular solution: 0 =  −  .

ôƒõ

)

+ ( ô sin 

ï

–

ñ ó( ÷

õ ÷ – ú ) ñ ó( ú

. – ôƒõ

ñ ó( ÷

Particular solution:

3.

)

0

=

)

.

)

õ ÷ )ñ sin 

)

ñ ó( ÷

)

+ (ô

ï

+ õ ) sin ú (

Particular solution:

0

ï )ñ óò

– 2 ô sin ú ( 2

= (  +  ) .

ï ñ

sinú (

+ ( ô ï + õ ) sin ú ( ï )ñó ò – ô sinú ( Particular solution: 0 =  +  .

ñ ó( ÷

= 0.

ï )ñ ï )ñ

= 0.

ï )ñ

= 0.

= 0. = 0.

© 2003 by Chapman & Hall/CRC

Page 698

699

5.1. LINEAR EQUATIONS

6.

ñ ó( ÷

)

ï

Particular solution: 7. 8.

ñ ó(2÷

)

=

ô ÷ ñ

+

ô ï

0

ï ñ ó( ÷

)

ïƒú

(ô

ó( ÷

0 )



ñ ó( ÷

)

+

cos ,ï

ô

ñ ó( ú

ñ ó( ÷

)

ï

+ ( ô cos 

–

ñ ó( ÷

)

+

)

ôñ ó(÷ ï

–1)

0

ñ ó( ÷

)

ï

0

Particular solution: 16.

ñ ó( ÷

)

+ (ô

ï

0

+ õ ) cosú (

Particular solution: 17. 18.

ñ ó(2÷

)

=

ô ÷ ñ

+

ô ï

0

ú



=

ï

sin 

7

−1



=0

® ù ñ

= 0, 1,

uuu , ù

– 1.

.

  .

cos )ï +



1 2

+

þ

+ ù ]ñ = 0.

õ ÷ )ñ

= 0.

. )



– ôƒõ

ú

+

ôƒõ

ï ñ

cos

= 0.

.

ï )ñ óò

cosú (

ï )ñ óò

– ô cosú (

ï )ñ

ï )ñ óò

– 2 ô cosú (

=  +  .

2

= (  +  ) .

ï )ñ óò

– 3 ô cosú ( 3

ï )ñ

= 0.

= 0.

ï )ñ

= 0.

ï )ñ

= 0.

= (  +  ) .

ï )(ñ óò ò ó

– ôñ ).

This is a special case of equation 5.1.6.18 with  ( ) =  cos  ( Π ).

ï ñ ó( ÷

)

ï ) ñ ó( ú

cos (

ï ú

(ô

+ õ cos ï )ñ

ó( ÷

Particular solution: 20.

ô

=  sin  +

+ õ cos  (

Particular solution: 19.

=

−  . 0 = 

+ õ ) cosú (

+ (ô

)

+ õ cosú (

Particular solution: 15.

ï

sin 

– ( ôƒõ

+ õ ) cosú (

+ (ô

õ

õ ÷ – ú ) ñ ó( ú

Particular solution:

ñ ó( ÷

=  .

0 =  

Particular solution:

ï

)+

+ 12 ® ù  ñ , +   +  sin  . 0 = 

=

0 )

ï

) sin  (



=0

Particular solution:

14.

= 0.

– ôñ ).

– [ô (ï + þ

)

÷ 7 –1 ï  (÷ õ Pñó

Particular solution:

13.

ï )(ñ óò ò ó

ï ) ñ ó( ú

sin  (

+ õ sin ï )ñ

ODô sin ï +

10.

12.

3

This is a special case of equation 5.1.6.18 with  ( ) =  sin  ( Π ).

Particular solution:

11.

ï )ñ

– 3 ô sin ú (

= (  +  ) .

+ õ sin  (

Particular solution: 9.

ï )ñ óò

+ õ ) sin ú (

+ (ô

O

ô

cos ï +

0 )

)

) cos  (

ï

+ 12 ® ù  ñ , +   +  cos  . 0 = 

=

=0

Particular solution:



=  .

õ

÷ 7 –1 ï  (÷ õ Pñó 

– [ô (ï + þ

0

cos 

)

=

ô

cos 

=  sin  +

ï

7

−1



=0

+

1 2

® ù ñ

ï

) + ï + ù ]ñ = 0.

þ

= 0, 1,

uuu , ù

– 1.

.

  .

© 2003 by Chapman & Hall/CRC

Page 699

700 21. 22.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó(2÷

)

= (–1)÷ ñ + ô (ñó ò sin ï – ñ cos ï ). The substitution ¨ = 
sin  − cos  leads to an (2B − 1)st-order linear equation.

ñ ó(2÷

)

= (–1)÷ ñ + ô (ñó ò cos ï + ñ sin ï ). The substitution ¨ = 
cos  + sin  leads to an (2B − 1)st-order linear equation.

5.1.5-2. Equations with tangent and cotangent. 23.

ñ ó( ÷

)

tan 

ô

+

ï ñ ó( ú

Particular solution: 24.

)

0

26. 27.

30.

ôƒõ

tanú (

33. 34.

)

ñ ó( ÷

)

ï

+ õ ) tan ú (

+ (ô

ñ ó( ÷

)

0

+ õ ) tan ú (

ï

+ (ô

ñ ó( ÷

)

+

ô

cot)ï

ñ ó( ú

ñ ó( ÷

)

+ ( ô cot,ï –

)

ï )ñ

ï )ñ

= 0. = 0.

ï )ñ ó ò

– 3 ô tanú (

ï )ñ

= 0.

2

= (  +  ) . 3

= (  +  ) .

ú

cot,ï +

=





õ ÷ – ú ) ñ ó( ú 0 =  

.



)

– ôƒõ

ú

õ ÷ )ñ

cot)ï

ñ

.

+ ( ô ï + õ ) cotú ( ï )ñó ò – ô cotú ( Particular solution: 0 =  +  . )

ñ ó( ÷

)

+ õ ) cotú (

ï

+ (ô

ñ ó( ÷

)

+ (ô

ï

ï ñ ó( ÷

0

+ õ ) cotú (

0

= 0.

= 0.

ï )ñ

= 0.

ï )ñ óò

– 3 ô cotú (

ï )ñ

= 0.

3

= (  +  ) .

+ ô ï cot ( ï )ñ ó(ú ) – [ ô ( ï + þ  Particular solution: 0 =   . )

ï )ñ

ï )ñ

– 2 ô cotú (

= (  +  ) .

+ ù ]ñ = 0.

= 0.

ï )ñ ó ò

2

ï

)+

= 0.

)

ñ ó( ÷

ï

) tan  (

+ ôñ ó(÷ –1) + õ cotú ( ï )ñó ò + ôƒõ cotú (  Particular solution: 0 =  −  .

ñ ó( ÷

= 0.

ï )ñ

– ( ôƒõ 0

= 0.

– 2 ô tanú (

)

Particular solution:

36.

0

ñ

ï )ñ óò

+ ô ï tan ( ï )ñ ó(ú ) – [ ô ( ï + þ  Particular solution: 0 =   .

ï ñ ó( ÷

Particular solution: 35.

.

+ ( ô ï + õ ) tan ú ( ï )ñó ò – ô tanú ( Particular solution: 0 =  +  .

ñ ó( ÷

Particular solution:

32.

=

)

)

Particular solution: 31.

= 0.

+ ôñ ó(÷ –1) + õ tanú ( ï )ñó ò +  Particular solution: 0 =  −  .

ñ ó( ÷

0

Particular solution:

29.





õ ÷ )ñ

tan ,ï

+ ( ô tan)ï – õ

÷ – ú ) ñ ó( ú

.

+

ú

)

Particular solution: 28.



=

ï

tan



– ôƒõ

ñ ó( ÷

Particular solution:

25.

ú

– ( ôƒõ

) cot  (

ï

)+

ï

+ ù ]ñ = 0.

© 2003 by Chapman & Hall/CRC

Page 700

701

5.1. LINEAR EQUATIONS

5.1.6. Equations Containing Arbitrary Functions 5.1.6-1. Equations of the form   ( ) (  ) +  1 ( ) 
+  0 ( ) = ( ). 1.

ñ ó( ÷

)

= ! ( ï ).

=

Solution: 2.

ñ ó( ÷

)

= ! ( ï )ñ .

½

−1

Á

=0

The transformation  = a (−1)  a −2   (1 a )¨ . 3.

4. 5.

7.

)

!„ñ ó ÷

= (ö

( )

ï

–! ó÷ ñ

= 0, Particular solution:

„! ñ ó ÷

(2 +1)

+! ó ÷

(2 +1)

½ 2 

ñ

10.

0

= " ( ï ),

12.

!

(−1)    (2  − 

=0

=

is an arbitrary number.

0

 leads to an equation of similar form: ¨ (g 

)

=

leads to a simpler equation: ¨ (Š 

)

=

( (˜ + ) 

M

−1

= ! (ï ).

  = X ( )  + L . M

) ( )

+ ( ô ï + õ ) ! ( ï )ñ„ó ò – ô! ( ï )ñ = 0.   + . Particular solution: 0 = 

ñ ó( ÷

)

ñ ó( ÷

)

ï

+ õ ) ! ( ï )ñ„ó ò – 2 ô! ( ï )ñ = 0.

+ (ô

ñ ó( ÷

)

+ (ô

ï

0

= (  +  )2 .

0

= (  +  )3 .

+ õ ) ! ( ï )ñ„ó ò – 3 ô! ( ï )ñ = 0.

+ ( ô ï + õ ) ! ( ï )ñ„ó ò – (ù – 1) ô! (ï )ñ = 0. Particular solution: 0 = (  +  )  −1 . The substitution H = (  +  )
 −  (B − 1) leads to an (B − 1)st-order linear equation: H (  −1) + (  +  )  ( )H = 0.

ñ ó( ÷

)

ñ ó( ÷

)

+ ï ! ( ï )ñó ò – þ¯! ( ï )ñ = 0, þ = 1, 2, uuu , + Particular solution: 0 =  . The substitution H =  equation:

c  −+

11.

1−

M

= ! ( ï ). =  ( ).

!

( )

Particular solution:

9.

= ¨ a

,

ô ï öï

+ ý )–2÷ I ! O

Particular solution: 8.

−1

+õ ñ . +ý P  +  ¨ , The transformation ˆ = (˜ + ¨ − M ¢  (ˆ ) , where ¢ =  − ˜( .

ñ ó( ÷

First integral: 6.

 ( − a )  −1  (a ) a , where  (B − 1)! M 0

LlÁ Á + X 

ñ ó( ÷

−1

O

H ( + ) +  ( ) H = 0, öP

ù

– 1.


−

where

¡

leads to an (B − 1)st-order

c = M .  M

+ ! ( ï )ñó ò + " ( ï )ñ + % ( ï ) = 0. The transformation  = a −1 , = ¨ a 1−  leads to an equation of similar form:

ñ ó( ÷

)

¨ (g  ) + (−1)  a −2  ¦ −a 2  (1 a )¨ g
+ V (B − 1)aª (1 a ) + (1 a )W ¨ + a 

)

+ ! ( ï )ñó ò +

!„ó ò (ï )ñ

= " ( ï ).

Integrating yields an (B − 1)st-order linear equation:



( −1)

−1

% (1 a ) § = 0.

+  ( ) = X ( )  + L .

M

© 2003 by Chapman & Hall/CRC

Page 701

702 13.

14.

15.

16.

17.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó(2÷

)

ñ ó(2÷

)

ñ ó(2÷

)

ñ ó(2÷

)

=

ñ

+ ! ( ï )(ñó ò cosh ï – ñ sinh ï ).

=

ñ

+ ! ( ï )(ñó ò sinh ï – ñ cosh ï ).


The substitution ¨ =  cosh  − sinh  leads to an (2B − 1)st-order linear equation. The substitution ¨ = 
sinh  − cosh  leads to an (2B − 1)st-order linear equation. = (–1)÷

ñ

+ ! ( ï )(ñ

= (–1)÷

ñ

+ ! ( ï )(ñó ò cos ï + ñ sin ï ).

sin ï – ñ cos ï ).

óò

The substitution ¨ = 
sin  − cos  leads to an (2B − 1)st-order linear equation. The substitution ¨ = 
cos  + sin  leads to an (2B − 1)st-order linear equation.

ñ ó( ÷

)

r (ó ÷ ) r ñ

=

r óò r ñ P

+ ! ( ï ) OUñó ò –



Y  The substitution ¨ =  −

r

,

=

r (ï

).

leads to an (B − 1)st-order linear equation.

Y

5.1.6-2. Other equations. 18.

ñ ó(2÷

)

=

ô ÷ ñ

+ ! ( ï )(ñó ò ò ó – ôñ ).

The substitution ¨ = 

 − 

leads to an (2B − 2)nd-order linear equation:

¨ (2  19.

ñ ó( ÷

)

+ ! ( ï )( ï

2

– 2ï

ñ óò ò ó

ñ óò

  +  ¨ (2

−2)

−4)

+ ))) +  

−1

¨ =  ( )¨ .

+ 2ñ ) = 0.

Particular solutions: 1 =  , 2 =  2 . The substitution H =  linear equation of the (B − 2)nd-order. 20.

21.

22.

ñ ó( ÷

+2)

ñ ó(2÷

)

+ ! ( ï )[ ï

2

ñ óò ò ó

24.

=

ô 2ñ

+ ! ( ï )[ñ

ó( ÷

)

ôñ

+

The substitution ¨ = (  ) + 

ñ ó( ÷

)

+ ! ( ï )ñ

ó( ú

)

– [ô

ñ ó( ÷

)

+ (! – ô

÷ – ú ) ñ ó( ú

+ ù (ù + 1)ñ ] = 0.

÷

].

leads to an B th-order linear equation: ¨ ( 

+ô

ú ! (ï )]ñ

–ô

ú !„ñ

 =  .

0 )

Particular solution:

ñ ó( ÷

+ ô!„ñ = 0,

)

+

ôñ ó(÷

–1)

+

!„ñóò

ñ ó( ÷

)

+ ! ( ï )ñ

ó( ÷

–1)

−  . 0 = 

+ " ( ï )ñ

ó( ÷

The substitution ¨ ( ) = (  ( )¨ + % ( ) = 0.

–2) −2)

)

= [  (  ) +  ]¨ .

= 0.

!

= 0,

 . 0 = 

Particular solution: 25.



 − 2 
+ 2 leads to a

The substitution ¨ ( ) =  2 

 − 2B/ 
 + B (B + 1) leads to an B th-order linear equation: ¨ (  ) +  2  ( )¨ = 0.

Particular solution: 23.

ï ñ óò

– 2ù

2

!

= ! ( ï ).

= ! (ï ).

+ % ( ï ) = 0.

¨ 

 +  ( )2 ¨ 
+ leads to a second-order linear equation: u

© 2003 by Chapman & Hall/CRC

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703

5.1. LINEAR EQUATIONS

26.

ñ ó( ÷

)

ô ÷ –1 ñ ó(÷

+

–1)

+

+ô

­­­

ñóò

1

+

= ! ( ï ).

ô 0ñ

Constant coefficient nonhomogeneous linear equation. The general solution of this equation is the sum of the general solution of the corresponding homogeneous equation (see 5.1.2.25) and any particular solution of the nonhomogeneous equation. If the roots Π1 , Π2 ,  , Π of the characteristic equation

Π + 

−1

Π

−1

+ ))) +  1 Π+ 

0

=0

are all different, the original equation has the general solution:

  ½ LlÁ   ¡ + . 
 (¡ Œ"Á ) X  ( ) −  ¡  M  Á =1 

½ 

=

Á

=1

(with complex roots, the real part should be taken). Paragraph 0.4.1-2 lists the forms of particular solutions corresponding to some special forms of the right-hand side function of the nonhomogeneous linear equation. 27.

ñ ó( ÷

+ ! (ï )

)

Here, L +  =

÷ 7 –1 

=0

(–1) 

ü ! = ÷  –1 ï ÷ – –1 ñ ó(÷ – –1) = 0.

!

are binomial coefficients. ¡ , ! ( − , )! ¡ Particular solutions: + =  + , where = 1, 2,  , B − 1. ¡  7 −1 The substitution H = (−1)  , ! L   −1   −  −1 (  −  −1) leads to a first-order linear equation:  =0 H 
+   −1  ( ) H = 0. Having solved this equation, we obtain an (B − 1)st-order linear equation of the form 5.1.6.34 for ( ). 28.

ñ ó( ÷

)

÷ 7 –1

=



=0

(ô

–ô

 +1 !

 ) ñ ó( 

Here,  =  ( );   = 1,  Particular solutions:

)

.

= 0;   are arbitrary numbers ( , = 1, 2,  , B − 1). ¼  ( , = 1, 2,  , B − 1), where the Œ are roots of the   =    7 −1    +1 Œ = 0. polynomial equation  =0 29. 30.

ï ñ ó( ÷

)

ï ñ ó( ÷

)

+ ( ô + ù – 1)ñ

–1)

+

ï !„ñ ó(ú

– [( ï + þ

)

ï ñ ó( ÷

)

+ ùñ

ó( ÷

–1)

=

0

ï

1–2

÷ !

33.

ï 2 ñ ó( ÷

+2)

+

 ï ñ ó( ÷

)ñ

ó( ÷

+1)

+ ñ

ñ óò

+

ôñ

).

leads to an (B − 1)st-order linear equation: ¨ ( 

) ! + ï + ù ]ñ = 0,

!



=  .

The transformation a =  (−1)  [  (a )¨ + (a )]. 32.

= ! ( ï )( ï

The substitution ¨ =  
+  Particular solution:

31.

ó( ÷

0

(1 

ï )ñ

−1

, ¨

ó( ÷

)

÷ " (1  ï

+ï =

– –1



+ ! ( ï )[ ï

2−

2

= ! ( ï ).

−1)

=  ( )¨ .

).

 leads to an B th-order linear equation: ¨ (g 

ñóò ò ó

+ (  – 2ù ) ï

+ ( – ù + ù

ñóò

2

)

=

+ ù )ñ ] = 0.

The substitution ¨ ( ) =  2 

 + ( ƒ − 2B ) 
 + (ý − ƒ…B + B 2 + B ) leads to an B th-order linear equation: ¨ (  ) +  ( )¨ = 0.

ï (ï

+þ

)

+ ï ( ! – ï – ù )ñ

Particular solution:

0



=  .

ó( ú

)

– (ï + þ

) !„ñ = 0,

!

= ! ( ï ).

© 2003 by Chapman & Hall/CRC

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704 34.

35. 36.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ïƒ÷ ñ ó(÷

)

+ õ ÷ –1 ïƒ÷ –1 ñ ó(÷ –1) + ­­­ + õ 1 ï ñó ò + õ 0 ñ = ! ( ï ). g Nonhomogeneous Euler equation. The substitution  =   (  ≠ 0) leads to a constant coefficient nonhomogeneous linear equation of the form 5.1.6.26.

ïƒ÷ ñ ó(÷

)

+ (ù – þ – 1)ïw÷ –1 ñ ó(÷ Particular solution: 0 =  + .

ïƒú ñ ó(÷

)

+

)

ïƒú !„ñ ó(ú

=

÷ 7 –1 ïƒú 

=0

)

(ô

[

polynomial equation

= ! (ï ).

!

–ô

 +1 ! 7

−1



=0

D (



)+ô



 

+1



=2

!  ( ï ) ñ ó( 

)

= " ( ï )( ï

Particular solution: equation.

7÷

 =ú

+1

!  ( ï ) ñ ó( 

)

ï ï

7÷



=3

!  ( ï ) ñ ó( 

)

0

= " ( ï )( ï

+

1 2

® ù 

+ ! ( ï ) sin 

+

1 2

® ù 

+ ! ( ï ) cos 

ï

+

ï

+

® þ  WUñ

= 0.

® þ  Wñ

= 0.

1 2

ñóò

– þxñ ),

þ

= 1, 2,

leads to an (B − 1)st-order linear

uuu , ù

=  + . The substitution ¨ ( ) =  
 − 2

1 2

– ñ ).

=  . The substitution ¨ ( ) =  
 −

= " ( ï )( ï

Particular solution: linear equation. 42.

0

ñóò

ñ ó(  ) .

Π = 0.

cos ï ñ ó(÷ ) + cos ï ! ( ï )ñ ó(ú ) – V cos  Particular solution: 0 = cos  .

7÷

+1 ]

= 0; and   are arbitrary numbers ( , = 1, 2,  , B − 1). ¡ ¼  ( , = 1, 2,  , B − 1), where the Œ are roots of the   = 

0

sin ï ñ ó(÷ ) + sin ï ! ( ï )ñ ó(ú ) – V sin  Particular solution: 0 = sin  .

40.

41.

– þ¯!„ñ = 0,

÷

– (ù ! =

Here,  =  ( );   = 1,  Particular solutions:

39.

ï !„ñóò

+

+ þ ! = ú ! )ñ = 0. D + 1) w are binomial coefficients, and (  ) is the gamma Here,  =  ( ), L   = w B ! (  − B + 1) w function. Particular solution: 0 =   .

ïƒ÷ ñ ó(÷

37.

38.

–1)

ñ óò ò ó

– 2ï

ñ óò

¡

– 1. leads to an (B − 1)st-order

+ 2ñ ).

Particular solutions: 1 =  , 2 =  2 . The substitution ¨ ( ) =  2 

 − 2 
 + 2 leads to an (B − 2)nd-order linear equation. 43.



7÷

=4

!  ( ï ) ñ ó( 

)

= " ( ï )( ï

3

ñóò ò ó ò ó

– 3ï

2

ñóò ò ó

+ 6ï

ñóò

– 6ñ ).

Particular solutions: 1 =  , 2 =  2 , 3 =  3 . The substitution ¨ ( ) =  3 


 − 3 2 

 + 6 
− 6 leads to an (B − 3)rd-order linear equation. 44.

7÷

 =ú

+1

!  ( ï ) ñ ó( 

)

+ " (ï )



7ú

=0

(–1) 

ü ! = ú  ïƒú – ñ ó(ú – ) = 0.

À ! À binomial are coefficients. ¡ , ! ( − , )! ¡ =  , where ° = 1, 2,  , Particular solutions:

Here, L +  =

¡

.

© 2003 by Chapman & Hall/CRC

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705

5.2. NONLINEAR EQUATIONS

The substitution H =

7  =+ 7÷

45.



=0

+1

(!



  ( ) c  − + – ô!



+1 )

−1

7+ 

=0

(−1)  , ! L +   +

−



)

leads to an (B −

  − + H 
 + ( ) H = 0, where c =  . M M

¡

)th-order linear equation:

ñ ó( ) = 0.

Here,   =   ( ) ( , = 1, 2,  , B );    Particular solution: 0 =   . 46.

 ( + − 

7 ÷ ï [!  + (ü – þ )! 

+1 ]

=0

+1

≡

0

≡ 0.

+1

≡

0

≡ 0.

ñ ó( ) = 0.

Here,   =   ( ) ( , = 1, 2,  , B );   Particular solution: 0 =  + .

5.2. Nonlinear Equations 5.2.1. Equations Containing Power Functions 5.2.1-1. Fifth- and sixth-order equations. 1.

ñ ó(5) = ôññ óò ò ó ò ò ó ó

– ô (ñ

óò ò ó

)2 +

õï

+ö .

This is a special case of equation 5.2.6.1 with  ( ) =  +  . 2.

ññ ó(5) = ô ï

+õ .

This is a special case of equation 5.2.6.17 with B = 2 and  ( ) =  +  . 3.

ññ ó(5) = ôñóÈò ñóò ò ó ò ò ó ó

.

1 b . For  ≠ −1, integrating the equation two times, we arrive at a third-order autonomous equation: 



 − 12 (

 )2 = L 1  +1 + L 2 . The substitution ¨ ( ) = 12 (
 )2 leads to a second-order equation: ¨*¨

− 1 (¨
)2 = 1 L  +1 + 1 L . 4 2 1 2 2





For  = 1, this is a solvable equation of the form 2.8.1.53. 2 b . For  = −1, integrating the equation two times, we arrive at a third-order autonomous equation: 



 − 12 ( 

 )2 = L 1 ln | | + L 2 . 3 b . Particular solution: 4.

=L

1

3ññ

ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

= 0.

2ññ

ó(5) + 5ñóÈò ñóò ò ó ò ò ó ó

+ 5ñó ò ò óÈñó ò ò ó ò



3

+L

2



2

+L

3

 + L 4.

This is a special case of equation 5.2.1.3 with  = − 35 . Integrating the equation three times, we arrive at a second-order equation: 3!

 − 2(
 )2 = L 1  2 + L 2  + L 3 . The substitution = ¨ 3 leads to a solvable equation of the form 2.8.1.5: ¨p

 = 19 ( L 1  2 + L 2  + L 3 )¨ −5 .

5.

ó

= 0.

This is a special case of equation 5.2.6.4 with  = 52 and  ( ) = 0. Integrating the equation three times, we arrive at a second-order equation of the form 2.8.1.53: !

 − 14 (
 )2 = L 1  2 + L 2  + L 3.

© 2003 by Chapman & Hall/CRC

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706

HIGHER-ORDER DIFFERENTIAL EQUATIONS

6.

+ (3 ô – 5)ñ„ó ò ò óÈñó ò ò ó ò ó = 0. Integrating the equation three times, we obtain a second-order equation: !

 + 12 (  −3)(
 )2 = L 1  2 + L 2  + L 3.

7.

ññ ó(5) + ôñóò ñóò ò ó ò ò ó ó

8. 9. 10.

ññ ó(5) + ôñóò ñóò ò ó ò ò ó ó

+ õÿñó ò ò óñó ò ò ó ò ó = 0. Integrating yields a fourth-order equation: !"



 + (  − 1)
 


 + 12 (1 −  +  )( 

 )2 = L .

+ 10ñ ó ò ò ó ñ ó ò ò ó ò ó = ô ï ÷ . This is a special case of equation 5.2.6.2 with  ( ) =   .

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò ó = ôñ ÷ . This is a special case of equation 5.2.6.3 with  ( ) = 

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

 .

ñ ó(6) =
ñ –7 ) 5 .

 + 7
 (6)  = 0. Multiplying by 7 J 5 and differentiating with respect to  , we obtain 5! (7) Having integrated this equation three times, we arrive at the chain of equations: 5! 5! 5!

 + 2 
(5)  − 2 

 



 + ( 


 )2 = 2 L 2 , (5)  − 3 




 + 

 


 = 2 L 2  + L 1 , 



 − 8 



 + 92 ( 

 )2 = L 2  2 + L 1  + L 0 , (6)

(1) (2) (3)

where L 0 , L 1 , and L 2 are arbitrary constants. Eliminating the highest derivatives from (1)–(3), with the aid of the original equation, we obtain a third-order equation that can be reduced to a second-order equation (see equation 5.2.1.12 with B = 3). 11.

+ 10(ñó ò ò ó ò ó )2 = ô ïƒ÷ . This is a special case of equation 5.2.6.6 with  ( ) =   .

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó

  =  ( , ). ( )

5.2.1-2. Equations of the form 12.

ñ ó(2÷

)

=


ñ

1+2 1–2

÷

÷

.



2 +1 2 −1



Multiply both sides by

and differentiate with respect to  . As a result, we obtain

  (2B − 1)! (2

+1)


  + (2B + 1)  (2

)

= 0.

Three integrals containing arbitrary constants L 0 , L 1 , and L 2 are presented in 5.2.6.62, where one should let  ≡ 0. Eliminating the highest derivatives from those integrals and the original equation, one can always obtain an (2B − 3)rd-order equation. With the aid of the transformation

 M. ,

a =X

¨ =

.

1−2 2



,

where

.

=L

2



2

+L

1

 + L 0,

this equation can be reduced to the autonomous form 5.2.6.77. Therefore, the substitution H (¨ ) = ¨2g
finally leads to an (2B − 4)th-order equation with respect to H = H (¨ ). 13.

ñ ó(2÷

= ôñ  + õ , ü ≠ –1. This is a special case of equation 5.2.6.8. Integrating yields an (2B − 1)st-order equation: )

½

+

−1

(−1) =1

+

+ (

  +

) (2 −

)

+

1  2  (−1) ÑV (  ) W = − 2 , +1

+1

−

+L ,

where L is an arbitrary constant. Furthermore, the order of the obtained autonomous equation can be reduced by one by the substitution ¨ ( ) = 
 .

© 2003 by Chapman & Hall/CRC

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5.2. NONLINEAR EQUATIONS

14. 15.

= ô ï –÷ ñ ú . This is a special case of equation 5.2.6.11 with  ( ) =  "+ .

ñ ó( ÷

)

=ô ï  ñ ú . 1 b . The transformation  = a −1 , ¨ (g  ) = (−1)  #*a −  −(  −1) + −  −1 ¨2+ .

ñ ó( ÷

)

2 b . The transformation ˆ =   +  16.

+

= a −1

1−

 ¨ (a ) leads to an equation of the same form:

, H =


 leads to an (B − 1)st-order equation.

= ô ï + õ . This is a special case of equation 5.2.6.17 with  ( ) =   +  .

ññ ó(2÷

+1)

17.

= ïƒú –÷¿ú –÷ –1 ( ôñ + õ ïƒ÷ –1 )ú . This is a special case of equation 5.2.6.13 with  (¨ ) = (  ¨ +  ) + .

18.

ñ ó(2÷

19.

20. 21.

ñ ó( ÷

)

)

=

ï ú

–2

÷¿ú

2

÷

–2 –1

Oiôñ + õ

24. 25.

26.

.

= ( ôñ + õ ï )ú ; ü = 1, 2, uuu , ù – 1. ¨ =  + '  leads to an autonomous equation of the form 5.2.6.8: The substitution  ¨ (  ) =  +2¨2+ (see also 5.2.1.12 and 5.2.1.13 with  = 0).

ñ ó( ÷

)

ú –¿÷ ú –÷

2 = (ô ï 2 + õ ï + ö ) ñ ú . This is a special case of equation 5.2.6.22 with  (¨ ) = ¨ + .

ñ ó( ÷

)

–1

= ( ô ï + õ )–÷ ( ö ï + ý )ú –÷¿ú –1 ñ ú . This is a special case of equation 5.2.6.21 with  (¨ ) = ¨p+ .

ñ ó( ÷

)

(  ) =  ( , , 
 , 

 ).

= ôñ  ñ ó ò + õ ï ú . This is a special case of equation 5.2.6.34 with  ( ) = 

ñ ó( ÷

)

an (B − 1)st-order equation: 23.

P

This is a special case of equation 5.2.6.14 with  (¨ ) = (  ¨ +  ) + .

5.2.1-3. Equations of the form 22.

ï 2÷ 2–1 ú

( 

−1)

=

  , +1

+1

+

¡



 and ( ) = ' + . Integrating yields  + +1 + L .

+1

= ô ÷ ñ + õ (ñó ò – ôñ )  . This is a special case of equation 5.2.6.38 with  ( , ¨ ) =  ¨  . The substitution ¨ = 
−  leads to an (B − 1)st-order autonomous equation: ¨ (  −1) +  ¨ (  −2) + ))) +   −1 ¨ =  ¨  .

ñ ó( ÷

)

= ô ïƒú –÷ ñ 1–ú (ñó ò )ú . This is a special case of equation 5.2.6.37 with  (¨ ) =  ¨p+ .

ñ ó( ÷

)

= ô ïƒú ( ï ñó ò – ñ )  . This is a special case of equation 5.2.6.39 with  ( , ¨ ) =  +2¨  . The substitution ¨ =  leads to an (B − 1)st-order equation.

ñ ó( ÷

)

= ô ï ( ï ñó ò – þxñ )A .   is a positive integer and B ≥ + 1. The substitution ¨ Here, ¡ ¡ ¨ + (B − 1)st-order equation:  (  − −1) =   - , where = ¨ ( + )  .

ñ ó( ÷


−

)

=  "
 −

¡

leads to an

© 2003 by Chapman & Hall/CRC

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708 27.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó(2÷

+ ôñó ò ò ó + õÿñ = ö^ññó ò ò ó – ö (ñó ò )2 + ü . 1 b . Particular solution: )

=L where the constants L

1,

1

sinh( L

4

 )+L

2

cosh( L

4

 ) + L 3,

L 2 , L 3 , and L 4 are related by two constraints L 42  + (  − (L 3 ) L 42 +  = 0, ( ( L 22 − L 12 ) L 42 − L 3 + , = 0.

2 b . Particular solution: =L where the constants L

28.

ñ ó( ÷

)

+

ôññ óò ò ó

– ô (ñ

Particular solution: 29.

30.

31.

1,

óò

1

sin( L

4

 )+L

2

cos( L

4

 ) + L 3,

L 2 , L 3 , and L 4 are related by two constraints L 42  − (  − (L 3 ) L 42 +  = 0, ( ( L 12 + L 22 ) L 42 + L 3 − , = 0.

)2 +

õÿñ óò ò ó

=L

1

ö^ñ óò

+

exp( L

2

 )−

= 0.

L 2 −1 + L 2 + ( . L 2

= ô ÷ ñ + õ (ñ ó ò ò ó – ôñ )  . This is a special case of equation 5.2.6.50 with  ( , ¨ ) =  ¨  . The substitution ¨ = 

 −    −2) +  ¨ (2  −4) + ))) +   −1 ¨ =  ¨  . leads to an (2B − 2)nd-order autonomous equation: ¨ (2

ñ ó(2÷

)

= ô ïƒú ( ï ñó ò – ñ )  (ñó ò ò ó )A . The substitution ¨ ( ) =  
− leads to an (B − 1)st-order equation:

ñ ó( ÷

)

ñó ÷

(2 )

M

 = ôñ  ññó ò ò ó – ñó ò  2

¨ 
 −2 2 =  + −- ¨  (¨ 
)- . M  −2 O   P

. This is a special case of equation 5.2.6.52 with  (¨ ) = 0 and (¨ ) =  ¨  .

5.2.1-4. Other equations. 32.

ñ ó( ÷

= ôññó ò ò ó ò ò ó ó – ô (ñó ò ò ó )2 . 1 b . Integrating the equation two times, we obtain an (B − 2)nd-order equation: )



( −2)




=  !   −  (  )2 + L

1

 + L 2.

2 b . Particular solutions:

 ) + L 2 exp(− L 3  ) +  −1 L 3 −4 = L 1 sin( L 3  ) + L 2 cos( L 3  ) + (−1) ´J 2  −1 L 3 −4 = L 1 exp( L 2  ) +  −1 L 2 −4 = L 1 + L 2 =L

33.

1

exp( L

3

if B is even number, if B is even number, if B is odd number, if B ≥ 2 is any number.

= ôññó ò ò ó ò ò ó ó – ô (ñó ò ò ó )2 + õ ï + ü . This is a special case of equation 5.2.6.54 with  ( ) = ' + , . Integrating the equation two times, we obtain an (B −2)nd-order equation: (  −2) =  !

 −  (
 )2 + 16 ' 3 + 12 ,! 2 + L 1  + L 2 .

ñ ó( ÷

)

© 2003 by Chapman & Hall/CRC

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5.2. NONLINEAR EQUATIONS

34.

35.

ñ ó(2÷

)

ô 2ñ

+ õ [ñ

ô ï ñ ó( ÷

–1)

ôññ ó(÷

–1)

ó( ÷

–1)

=

ó( ÷

)

+

ôñ ]

.

The substitution ¨ = (  ) + 

ñ ó( ÷

)

ñ ó( ÷

)

+

leads to an B th-order autonomous equation: ¨ (  ) =  ¨ +  ¨  .

+ 2 õÿññó ò + ôƒõ

ï ñ

2

+

ö^ñ

+

öï

= 0.

The functions that solve the (B − 1)st-order autonomous equation solutions of the original equation. 36.

+

õÿñóò

+

+ ôƒõÿñ

2



( −1)

= −

2

− (  are

= 0.

The functions that solve the (B − 1)st-order constant coefficient nonhomogeneous linear equation (  −1) +  = − (  are solutions of the original equation. 37.

38.

39.

40.

ï ñ ó( ÷

)

ï ñ ó( ÷

)

+ ùñ

ô ïƒú ñ ú

=

.

This is a special case of equation 5.2.6.59 with  (¨ ) =  ¨p+ .

ó( ÷

= õ (ï

ñóò

+ ù (ù – 1)ñ

ó( ÷

+ ( ô + ù – 1)ñ

–1)

+ ôñ )  .

This is a special case of equation 5.2.6.60 with  ( , ¨ ) =  ¨  . The substitution ¨ =  leads to an (B − 1)st-order autonomous equation: ¨ (  −1) =  ¨  .

ï 2 ñ ó( ÷

)

ññ ó(2÷

+1)

+ 2ù

ï ñ ó( ÷

–1)

–2)

=

ô ï 2ú ñ ú


+ 

.

This is a special case of equation 5.2.6.61 with  (¨ ) =  ¨ + . =

ôñóÈò ñ ó(2÷

)

.

The equation admits two different (with  ≠ −1) first integrals:

¶   =L (2 )

,

1

 −1 ! (2  ) + (  + 1) ½ (−1) + ( + ) (2  − + ) + 1 (−1)  (  + 1) V (  ) W 2 = L ¶ , 2 2 + =1 ¶ ¶ where L 1 and L 2 are arbitrary constants. Eliminating the highest derivative from the first integrals, we arrive at an (2B − 1)st-order autonomous equation: ½

where L

1

=−

L

+ ¶ 1

(−1) +

=1

+

, L

=

2

L

  +

+ 12 (−1) ÑV (  ) W

2



(

) (2 −

2

. The order of the obtained equation next can be lowered by

)

=L

1

+1

+L

2,

¶

 +1 the standard substitution ¨ ( ) = 
 .

ó(2÷

+1)

ó( ÷

–1)

ñóò ñ ó(÷

–1)

41.

(2ù – 1)ññ

42.

ññ ó(÷

)

ññ ó(÷

)

43.

 +1

−1

+ (2ù + 1)ñ„óÈ òñ

ó(2÷

)

=

ô ïƒú

.

This is a special case of equation 5.2.6.62 with  ( ) =  + . – ñó òñ

=

ôñ

2

.

Integrating yields an (B − 1)st-order linear equation: (  −1) = (  + L ) . The transformation H =  + L  brings it to an equation of the form 5.1.2.3 with  = 0. =

+ ôñó ò .

Integrating yields an (B − 1)st-order constant coefficient nonhomogeneous linear equation: (  −1) = L −  .

© 2003 by Chapman & Hall/CRC

Page 709

710 44.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

÷ ½



ô  ñ ó( ) = õÿññóò ò ó

=0

– õ (ñó ò )2 + ü . = L ´

Particular solutions: the algebraic equation  45. 46.

ï ññ ó(÷

0

7



=0



+ ,

−1 0 ,

where L

is an arbitrary constant and Πare roots of

  Œ  = , Œ 2 .

= ( ï ñó ò + ôñ )ñ ó(÷ –1) . Integrating yields an (B − 1)st-order linear equation of the form 5.1.2.4: )

ù > (ñ + ô ïƒú –1 )ñ ó(÷ ) – ñ ó(ú ) ñ ó(÷ –ú ) + õ ïƒú –1 ñ ó(ú ) = 0, The functions that solve the (B − )th-order linear equation ¡

 + ( −

)

=L

+ ( L +  )

+

þ



( −1)

= L3  .

.

−1

are solutions of the original equation. 47.

ñ ó( ÷

ñ ó÷

–2) ( )

=

Solution: 48. 49.

ñ ó( ÷

ô V a ñ ó( ÷

–1) 2

W .

+L L 0+L

L

=

 + ))) + L  1  + ))) + L 

0

1

ú

 

−3

= ôñ  ñó ò V ñ ó(÷ –1) W . This is a special case of equation 5.2.6.73 with  ( ) =  , (¨ ) =  ¨ + . )

= ô ïƒú 1 ñ ú 2 (ñó ò )ú 3 ­­­ (ñ ó(÷ –1) )ú±° +1 . Generalized homogeneous equation. The transformation ˆ =  

ñ ó( ÷

1

¡

−

¡

3

−2

¡

4

− ))) − (B − 1) 

¡

leads to an (B − 1)st-order equation.

O

÷ ñ ýýï P

7

–1

(ñ

óò

)=

ô ï

½÷

2

ú

–1

(–1)ú

=1

ñ ó(ú ) ñ ó(2÷ –ú

“ =

+1 ,

¡

2

+

,¨ ¡

3

=

+ ))) +


 , where ¡ 

+1

− 1,

+õ . 2 = ( g
) leads to a constant coefficient linear equation:

The transformation  =  (a ), 2 (g  +1) =  +  . 51.

if  ≠ 1, if  = 1.

)

Π=B +

50.

1

+ ( L  −2 + L  −1  )  −2+ 1−   −3 + L  −2 exp( L  −1  ) −3  −3

)

+ (–1)÷ V ñ

ó( ÷

W + (ñó ò )2 =

) 2

ôñ

2

+

õÿñ

+ö .

Differentiating both sides with respect to  and dividing by
 , we arrive at a constant   ) − 2 Œ 

 + 2  +  = 0. coefficient linear equation: 2 (2 52.

2

ú

½÷

–1 =1

(–1)ú

ñ ó(ú ) ñ ó(2÷ –ú

)

+ (–1)÷ V ñ

ó( ÷

W =  (ï

) 2

ñóò

– ñ ) + ñó ò + .

Differentiating both sides of the equation with respect to  , we have

  

 V 2 (2

−1)

− ƒ… − ý/W = 0.

(1)

Equate the second factor to zero to obtain: =

2½  −2 ƒ… 2  ý± 2  −1 + + L   . 2(2B )! 2(2B − 1)!  =0

© 2003 by Chapman & Hall/CRC

Page 710

711

5.2. NONLINEAR EQUATIONS

The integration constants L  and parameters ƒ , ý , and = are related by 2

½

−1

+

(−1) +

=2

! (2B −

¡

)! L + L

¡

 + + (−1)  (B !)2 L  2 = ýhL

2 −

1

− ƒsL

0

+= ,

which is obtained as a result of substituting the above solution into the original equation. In there is¶ a solution corresponding to equating the first factor in (1) to zero: ¶ ¶ addition, ¶ = L 1  + L 0 , where ý L 1 − ƒ L 0 + = = 0. 53.

½÷

2

–1

ú

(–1)ú

ñ ó(ú ) ñ ó(2÷ –ú

=1

)

+ (–1)÷ V ñ

ó( ÷

W + s(ñó ò ò ó )2 =  ( ï

– ñ ) + ñó ò + ,

ñóò

) 2

ù

≥ 3.

For the case ° = 0, see equation 5.2.1.52. Let now ° ≠ 0. Differentiating the equation with respect to  , we have   −1) + 2 ° 


 − ƒ… − ý/W = 0. 

 V 2 (2 Equate the second factor to zero and integrate to obtain:

ƒ…

4

ý±

3

+ L 2  2 + L 1  + L 0 + XXX ¨    , 48 ° 12 ° M M M where ¨ = ¨ ( ) is the general solution of a linear constant coefficient linear equation of the   −4) + ° ¨ = 0. The constants of integration are related by the constraint that form 5.1.2.2: ¨ (2 results from substituting the obtained solution ¶ into¶ the original equation. = L In addition, there is the solution 1  + L 0 , where the constants of integration are ¶ ¶ related by ý L 1 − ƒ L 0 + = = 0. =

54.

÷

½

ú

=1

ô ú

¦ 2

ú½

–1



=1

)

+ƒ

(–1)¦ñ

+

ó( ) ñ ó(2ú –

+ (–1)ú

)

Vñ

ó( ú

W § = ñ

) 2

2

+ 2ñ + .

Differentiating with respect to  , we arrive at a constant coefficient linear equation:

7

+

=1

 +

 +

(2

+ ý = 0.

5.2.2. Equations Containing Exponential Functions 5.2.2-1. Fifth- and sixth-order equations. 1.

ó

ñ ó(5) = ôññóò ò ó ò ò ó ó

– ô (ñó ò ò ó )2 + õ .  1 b . This is a special case of equation 5.2.6.1 with  ( ) =   . Integrating the equation two  times, we obtain a third-order equation: 


 =  !

 −  (
 )2 + L 1  + L 2 + Œ −2  . 2 b . Particular solutions: = L exp( Œ  ) + =

2.

ñ ó(5) = ô]¦ñóò ñóò ò ó ò ò ó ó

5

exp( Œ  ) + L exp(− Œ  ) −

Π

.

. Integrating yields a fourth-order autonomous equation of the form 5.2.6.8 with B = 4:  L exp O   .

Œ

3.

2Œ



LpŒ 5 −  , LpŒ 4

ññ ó(5) = ô ó





 =

P

+õ .  This is a special case of equation 5.2.6.17 with B = 2 and  ( ) =   +  .

© 2003 by Chapman & Hall/CRC

Page 711

712 4.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó 2

Solution: 5.

=L

4

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó



+ 10ñó ò ò óÈñó ò ò ó ò +L

4

3



3

ó

+L

=

+ 10ñó ò ò óÈñó ò ò ó ò

2

ó



2

ô ó +L

1

ô]

=

.

 +L

+ 2 Œ

0

  .

−5

.

This is a special case of equation 5.2.6.3 with  ( ) =   . 6.

ñ ó(6) = ô]

+õ .

This is a special case of equation 5.2.6.8 with B = 6 and  ( ) =   +  . 7.

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó

+ 10(ñó ò ò ó ò ó )2 =

ô ó

.



This is a special case of equation 5.2.6.6 with  ( ) =   . 5.2.2-2. Equations of the form 8.

ñ ó( ÷

)

ñ ó( ÷

)

=

ô] +_ ó

ñ ó( ÷

)

=

ô ï –÷  ]

ñ ó( ÷

)

ô ]

=

  =  ( , ). ( )

+õ .

This is a special case of equation 5.2.6.8 with  ( ) =   +  . 9.

10.

+õ . ¨ = + (ý Œ ) leads This is a special case of equation 5.2.6.9 with = 1. The substitution ? ¡ ¨ to an autonomous equation of the form 5.2.6.8: (  ) =   +  . .

This is a special case of equation 5.2.6.11 with  ( ) =   . 11.

12.

ô ï   ^ ]

=

.

This is a special case of equation 5.2.6.26 with  (¨ ) =  ¨ and

ñ ó( ÷

)

ó
B ^ ñ ú

=

.

This is a special case of equation 5.2.2.23 with 13.

ññ ó(2÷

+1)

=

ô ó

+õ .

=

¡

¡

1

and

This is a special case of equation 5.2.6.17 with  ( ) =   14.

ñ ó( ÷

)

=

ô

= , +B .

¡

exp( Ė + 

ïƒú

)+ õ ,

+ ( ý Œ ) +

¨ = The substitution ? ¨ (  ) =   +  .

þ

= 1, 2,

uuu , ù



2

¡

=

¡

3

= ))) =

= 0. ¡ 

+ .

– 1.

leads to an autonomous equation of the form 5.2.6.8:

5.2.2-3. Other equations. 15.

ñ ó( ÷

)

=

ô ] ñ óò

+

õ _ ó

.

Integrating yields an (B − 1)st-order equation: 16.

ñ ó( ÷

)

=

ôññ óò ò ó ò ò ó ó

– ô (ñ

óò ò ó

)2 +

õ  ó

.



( −1)

=

Œ



  + ý



  : +L .



1 b . This is a special case of equation 5.2.6.54 with  ( ) =   . Integrating the equation two  times, we obtain an (B − 2)nd-order equation: (  −2) =  !

 −  (
 )2 + L 1  + L 2 + Œ −2  .

© 2003 by Chapman & Hall/CRC

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713

5.2. NONLINEAR EQUATIONS

2 b . Particular solutions:

LpŒ  −  LpŒ 4

= L exp( Π ) +

Œ  exp( Œ  ) + L exp(− Œ  ) −  2Œ  

= 17.

18. 19.

õ ó (ñ óò

ñ ó( ÷

)

=

ô ÷ ñ

ñ ó( ÷

)

=

õ ó (ï ñ óò

+

(B is any number), −4

(B is odd number).

– ôñ )  .

 This is a special case of equation 5.2.6.38 with  ( , ¨ ) =    ¨  . – ñ ) .

 This is a special case of equation 5.2.6.39 with  ( , ¨ ) =   ¨  . (2 ù – 1)ññ

ó(2÷

ó(2÷

+ (2ù + 1)ñ„óÈ òñ

+1)

)

ô ó

=

.



This is a special case of equation 5.2.6.62 with  ( ) =   .

20.

ñ ó( ÷

)

ñ ó( ÷

)

= ( ô

ñ ó( ÷

)

=

ô]Èñóò V ñ ó(÷

ñ ó( ÷

)

=

ó
B ^ ñ ú

=

ô]Èñóò ñ ó(÷

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =   . 21. 22.

23.

] ñ óò

õ _ ó )ñ ó(÷

+

–1)

.

 This is a special case of equation 5.2.6.58 with  ( ) =   and ( ) =   : . –1)

W

ú

.

This is a special case of equation 5.2.6.73 with  ( ) =   and (¨ ) = ¨2+ . 1

(ñó ò )ú

2

uuu  ñ ó(÷



–1)

ú±°

.

 The substitution ¨ ( ) =  : , where ý = equation of the form 5.2.6.77.

24.

ñ ó( ÷

)

=


B ^ ] ïƒú

(ñó ò )ú

(ñó ò ò ó )ú

uuu  ñ ó(÷

1

¡

–1)



+

ƒ

2

¡

+ ))) +

2

3

, leads to an autonomous

ú±°

. The transformation H =  ‡  9  , ¨ =  
 , where . = B + ¡ leads to an (B − 1)st-order equation. 1

−1 ¡ 

1−

¡

2 −2

¡

3 −3

¡

4−

))) −(B −1)  , ¡

5.2.3. Equations Containing Hyperbolic Functions 5.2.3-1. Equations with hyperbolic sine. 1.

ñ ó(5) = ôññóò ò ó ò ò ó ó

– ô (ñó ò ò ó )2 + õ sinh(

ï

).

1 b . This is a special case of equation 5.2.6.1 with  ( ) =  sinh( Π ). Integrating the equation two times, we obtain a third-order equation: "


 =  !

 −  (
 )2 + L 1  + L 2 + Œ −2 sinh( Œ  ). 2 b . Particular solution: 2.

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó Solution:

3.

2

=L

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

4



=

Œ 4(Œ

+ 10ñó ò ò óÈñó ò ò ó ò 4

+L

+ 10ñ

3



3

ó

+L

óò ò ó ñ óò ò ó ò ó



−  2L

2

ô

= 2



+L

ô

1

V L sinh( Π ) + Πcosh( Π )W + L .

ï

).

 +L

0

sinh(

2

=

2)

sinh ú (

ï

+ 2 Œ )+õ .

−5

cosh( Π ).

+

This is a special case of equation 5.2.6.2 with  ( ) =  sinh ( Π ) +  .

© 2003 by Chapman & Hall/CRC

Page 713

714 4. 5. 6. 7. 8. 9.

HIGHER-ORDER DIFFERENTIAL EQUATIONS + 10ñó ò ò óÈñó ò ò ó ò ó = ô sinh ú ( ƒñ ) + õ . + This is a special case of equation 5.2.6.3 with  ( ) =  sinh ( Œ

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

)+ .

+ 10(ñó ò ò ó ò ó )2 = ô sinhú ( ï ). + This is a special case of equation 5.2.6.6 with  ( ) =  sinh ( Œ  ).

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó

= ô sinh ú ( ƒñ ) + õ . + This is a special case of equation 5.2.6.8 with  ( ) =  sinh ( Œ

ñ ó( ÷

)

= ô ï –÷ sinhú ( ƒñ ). + This is a special case of equation 5.2.6.11 with  ( ) =  sinh ( Œ

ñ ó( ÷

)+ .

)

).

= ô sinh ú ( ï ) + õ . + This is a special case of equation 5.2.6.17 with  ( ) =  sinh ( Œ  ) +  .

ññ ó(2÷

+1)

= ôññó ò ò ó ò ò ó ó – ô (ñó ò ò ó )2 + õ sinh( ï ). 1 b . This is a special case of equation 5.2.6.54 with  ( ) =  sinh( Œ  ). Integrating the equation two times, we obtain an (B − 2)nd-order equation: (  −2) =  !

 −  (
 )2 + L 1  + L 2 + Œ −2 sinh( Œ  ).

ñ ó( ÷

)

2 b . Particular solutions:

LpŒ  −  LpŒ 4  V L sinh( Œ  ) + Œ  = 2  −4 Œ −  2L 2Œ 4 = L sinh( Œ  ) +

10. 11. 12.

if B is even number, −4

cosh( Π )W + L

if B is odd number.

ò ñ ó(2÷ ) = ô sinhú ( ï ) + õ . (2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ + This is a special case of equation 5.2.6.62 with  ( ) =  sinh ( Œ  ) +  . ñ ó( ÷

)

=

ô

sinh  ( ƒñ )ñóÈ òñ

ó( ÷

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =  sinh  ( Œ

).

– ñó ò ñ ó(÷ –1) = ô sinh( ï )ñ 2. This is a special case of equation 5.2.6.64 with  ( ) =  sinh( Œ  ). Integrating yields an  cosh( Œ  ) + L3S . (B − 1)st-order linear equation: (  −1) =

ññ ó(÷

)

Q Œ

5.2.3-2. Equations with hyperbolic cosine. 13.

– ô (ñó ò ò ó )2 + õ cosh( ï ). 1 b . This is a special case of equation 5.2.6.1 with  ( ) =  cosh( Œ  ). Integrating the equation twice, we obtain a third-order equation: 


 =  !

 −  (
 )2 + L 1  + L 2 + Œ −2 cosh( Œ  ).

ñ ó(5) = ôññóò ò ó ò ò ó ó

2 b . Particular solution: 14.

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó Solution:

15.

2

+ 10ñ = L 4 4 + L

=

Œ 4(Œ

2



−  2L

2)

V L cosh( Π ) + Πsinh( Π )W + L .

cosh( ï ). 3 + L 2  2 + L 1  + L 0 + 2 Œ 3

óò ò ó ñ óò ò ó ò ó

=

ô

−5

sinh( Π ).

+ 10ñ ó ò ò ó ñ ó ò ò ó ò ó = ô coshú ( ï ) + õ . + This is a special case of equation 5.2.6.2 with  ( ) =  cosh ( Œ  ) +  .

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

© 2003 by Chapman & Hall/CRC

Page 714

715

5.2. NONLINEAR EQUATIONS

16.

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò

ó

coshú ( ƒñ ) + õ .

ô

=

+

This is a special case of equation 5.2.6.3 with  ( ) =  cosh ( Π17.

ññ ó(6) + 6ñ óò ñ ó(5) + 15ñ óò ò ó ñ óò ò ó ò ò ó ó

óò ò ó ò ó

+ 10(ñ

coshú (

ô

)2 =

+

ï

)+ .

).

This is a special case of equation 5.2.6.6 with  ( ) =  cosh ( Π ). 18.

ñ ó( ÷

)

ñ ó( ÷

)

coshú ( ƒñ ) + õ .

ô

=

+

This is a special case of equation 5.2.6.8 with  ( ) =  cosh ( Π19.

coshú ( ƒñ ).

ô ï –÷

=

+

This is a special case of equation 5.2.6.11 with  ( ) =  cosh ( Π20.

ññ ó(2÷

+1)

=

coshú (

ô

ï

)+õ .

)+ . ).

+

This is a special case of equation 5.2.6.17 with  ( ) =  cosh ( Π ) +  . 21.

ñ ó( ÷

)

ôññóò ò ó ò ò ó ó

=

ï

– ô (ñó ò ò ó )2 + õ cosh(

).

1 b . This is a special case of equation 5.2.6.54 with  ( ) =  cosh( Œ  ). Integrating the equation twice, we obtain an (B −2)nd-order equation: (  −2) =  !

 −  (
 )2 + L 1  + L 2 + Œ −2 cosh( Œ  ). 2 b . Particular solutions: = L cosh( Œ  ) + = 22.

(2 ù – 1)ññ

Π

2 −4

ó(2÷



−  2L

2

Œ

LpŒ  −  LpŒ 4 V L cosh( Œ  ) + Œ  4

óò ñ ó(2÷

+ (2ù + 1)ñ

+1)

)

ô

=

if B is even number, −4

sinh( Π )W + L

coshú (

ï

)+õ .

if B is odd number.

+

This is a special case of equation 5.2.6.62 with  ( ) =  cosh ( Π ) +  .

23.

24.

ñ ó( ÷

)

=

ô

cosh  ( ƒñ )ñóÈ òñ

ó( ÷

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =  cosh  ( Œ

ññ ó(÷

)

–ñ

ó ò ñ ó( ÷

–1)

=

ô

cosh(

ï )ñ

2

).

.

This is a special case of equation 5.2.6.64 with  ( ) =  cosh( Œ  ). Integrating yields an  sinh( Œ  ) + L3S . (B − 1)st-order linear equation: (  −1) =

Q Œ

5.2.3-3. Equations with hyperbolic tangent. 25.

ñ ó(5) = ôññóò ò ó ò ò ó ó

– ô (ñó ò ò ó )2 + õ tanh(

ï

)+ö .

This is a special case of equation 5.2.6.1 with  ( ) =  tanh( Π ) + ( . 26.

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò

ó

=

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò

ó

=

ô

tanhú (

ô

tanhú ( ƒñ ) + õ .

ï

)+ õ .

+

This is a special case of equation 5.2.6.2 with  ( ) =  tanh ( Π ) +  . 27.

+

This is a special case of equation 5.2.6.3 with  ( ) =  tanh ( Π28.

ññ ó(6) + 6ñ óò ñ ó(5) + 15ñ óò ò ó ñ óò ò ó ò ò ó ó

+ 10(ñ

óò ò ó ò ó

)2 =

ô

tanhú (

+

ï

)+ .

).

This is a special case of equation 5.2.6.6 with  ( ) =  tanh ( Π ).

© 2003 by Chapman & Hall/CRC

Page 715

716 29. 30. 31. 32. 33. 34. 35. 36.

HIGHER-ORDER DIFFERENTIAL EQUATIONS = ô tanhú ( ƒñ ) + õ . + This is a special case of equation 5.2.6.8 with  ( ) =  tanh ( Œ

ñ ó( ÷

)

= ô ï –÷ tanhú ( ƒñ ). + This is a special case of equation 5.2.6.11 with  ( ) =  tanh ( Œ

ñ ó( ÷

)+ .

)

).

= ô tanhú ( ï ) + õ . + This is a special case of equation 5.2.6.17 with  ( ) =  tanh ( Œ  ) +  .

ññ ó(2÷

+1)

ñ ó(2÷

)

= ñ + ô (ñó ò – ñ tanh ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = cosh  .

ñ ó(2÷

+1)

= ñ tanh ï + ô (ñ„ó ò – ñ tanh ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = cosh  .

(2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„ó ò ñ ó(2÷ ) = ô tanhú ( ï ) + õ . + This is a special case of equation 5.2.6.62 with  ( ) =  tanh ( Œ  ) +  .

ñ ó( ÷

)

=

ô

tanh ( ƒñ )ñ

ó ò ñ ó( ÷

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =  tanh  ( Œ

ññ ó(÷

).

– ñ ó ò ñ ó(÷ –1) = ô tanh( ï )ñ 2 . This is a special case of equation 5.2.6.64 with  ( ) =  tanh( Œ  ). )

5.2.3-4. Equations with hyperbolic cotangent. 37. 38. 39. 40. 41. 42. 43. 44.

– ô (ñó ò ò ó )2 + õ coth( ï ) + ö . This is a special case of equation 5.2.6.1 with  ( ) =  coth( Œ  ) + ( .

ñ ó(5) = ôññóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò ó = ô cothú ( ï ) + õ . + This is a special case of equation 5.2.6.2 with  ( ) =  coth ( Œ  ) +  .

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò ó = ô cothú ( ƒñ ) + õ . + This is a special case of equation 5.2.6.3 with  ( ) =  coth ( Œ

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

)+ .

+ 10(ñó ò ò ó ò ó )2 = ô cothú ( ï ). + This is a special case of equation 5.2.6.6 with  ( ) =  coth ( Œ  ).

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó

= ô cothú ( ƒñ ) + õ . + This is a special case of equation 5.2.6.8 with  ( ) =  coth ( Œ

ñ ó( ÷

)

= ô ï –÷ cothú ( ƒñ ). + This is a special case of equation 5.2.6.11 with  ( ) =  coth ( Œ

ñ ó( ÷

)+ .

)

).

= ô cothú ( ï ) + õ . + This is a special case of equation 5.2.6.17 with  ( ) =  coth ( Œ  ) +  .

ññ ó(2÷ ñ ó(2÷

+1)

= ñ + ô (ñó ò – ñ coth ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = sinh  . )

© 2003 by Chapman & Hall/CRC

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717

5.2. NONLINEAR EQUATIONS

45. 46. 47.

48.

ñ ó(2÷

= ñ coth ï + ô (ñ„ó ò – ñ coth ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = sinh  . +1)

ò ñ ó(2÷ ) = ô cothú ( ï ) + õ . (2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ + This is a special case of equation 5.2.6.62 with  ( ) =  coth ( Œ  ) +  . ñ ó( ÷

)

=

ô

coth ( ƒñ )ñóÈ òñ

ó( ÷

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =  coth  ( Œ

).

– ñó ò ñ ó(÷ –1) = ô coth( ï )ñ 2. This is a special case of equation 5.2.6.64 with  ( ) =  coth( Œ  ).

ññ ó(÷

)

5.2.4. Equations Containing Logarithmic Functions 5.2.4-1. Equations of the form 1. 2. 3. 4. 5. 6. 7.

  =  ( , ). ( )

= ô lnú ( õÿñ ) + ö . + This is a special case of equation 5.2.6.8 with  ( ) =  ln (  ) + ( .

ñ ó( ÷

)

= ô lnú ( õ ï ). + This is a special case of equation 5.2.6.17 with  ( ) =  ln ( ' ).

ññ ó(2÷

+1)

= ñ (  ï + þ ln ñ + õ ). This is a special case of equation 5.2.6.25 with  (¨ ) = ln ¨ +  .

ñ ó( ÷

)

ñ ó( ÷

)

= ï –÷ ( ñ + þ ln ï + õ ). This is a special case of equation 5.2.6.26 with  (¨ ) = ln ¨ +  .

= ô ï –÷ lnú ( õÿñ ). + This is a special case of equation 5.2.6.11 with  ( ) =  ln (  ).

ñ ó( ÷

)

= ô ï –÷ –1 [ln ñ + (1 – ù ) ln ï ]. This is a special case of equation 5.2.6.13 with  (¨ ) =  ln ¨ .

ñ ó( ÷

)

= ô ï –÷ –  (ln ñ + ü ln ï ). This is a special case of equation 5.2.6.15 with  (¨ ) =  ln ¨ .

ñ ó( ÷

)

8.

= ôñ ï –÷ (þ ln ñ + ü ln ï ). This is a special case of equation 5.2.6.16 with  (¨ ) =  ln ¨ .

9.

ñ ó(2÷

= ô ï – 2 [2 ln ñ + (1 – 2ù ) ln ï ]. This is a special case of equation 5.2.6.14 with  (¨ ) = 2  ln ¨ .

10.

= ( ô ï 2 + ö ) 2 [2 ln ñ + (1 – ù ) ln( ô ï 2 + ö )]. This is a special case of equation 5.2.6.22 with  = 0 and  (¨ ) = 2 ln ¨ .

11.

ñ ó( ÷

ñ ó( ÷

)

÷

2 +1

)

–

)

÷

+1

ó

= õ ^ (ln ñ –  ï ). This is a special case of equation 5.2.6.24 with  (¨ ) =  ln ¨ .

ñ ó( ÷

)

© 2003 by Chapman & Hall/CRC

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HIGHER-ORDER DIFFERENTIAL EQUATIONS

5.2.4-2. Other equations. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

ñ ó( ÷

= ôñ ó ò ln ñ + õ ln ï . This is a special case of equation 5.2.6.34 with  ( ) =  ln and ( ) =  ln  . )

= ô ÷ ñ + õ ln ï (ñ ó ò – ôñ )  . This is a special case of equation 5.2.6.38 with  ( , ¨ ) =  ¨  ln  .

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

)

= ô ln ï ( ï ñó ò – ñ )  . This is a special case of equation 5.2.6.39 with  ( , ¨ ) =  ¨  ln  .

= ô ln ï ( ï ñó ò – 2ñ )  . This is a special case of equation 5.2.6.40 with

ñ ó( ÷

= 2 and  ( , ¨ ) =  ¨  ln  .

¡

= ôññó ò ò ó ò ò ó ó – ô (ñó ò ò ó )2 + õ ln ï + ö . This is a special case of equation 5.2.6.54 with  ( ) =  ln  + ( . )

ï ñ ó( ÷

+ ùñ ó(÷ –1) = ô ln ï + ô ln ñ . This is a special case of equation 5.2.6.59 with  (¨ ) =  ln ¨ . )

ï 2 ñ ó( ÷

+ 2ù ï ñ ó(÷ –1) + ù (ù – 1)ñ ó(÷ –2) = 2 ô ln ï + ô ln ñ . This is a special case of equation 5.2.6.61 with  (¨ ) =  ln ¨ .

ññ ó(÷

)

– ñó ò ñ ó(÷ –1) = ôñ 2 ln ï . This is a special case of equation 5.2.6.64 with  (¨ ) =  ln  . )

ò ñ ó(2÷ ) = ô lnú ( õ ï ) + ö . (2ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ + This is a special case of equation 5.2.6.62 with  ( ) =  ln ( ' ) + ( . ñ ó( ÷

)

=

ô

ln  ( õÿñ ) ñó òñ

ó( ÷

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =  ln  (  ).

= ôñ ú ñó ò ln ñ ó(÷ –1) . This is a special case of equation 5.2.6.73 with  ( ) =  "+ and (¨ ) = ln ¨ .

ñ ó( ÷

)

5.2.5. Equations Containing Trigonometric Functions 5.2.5-1. Equations with sine. 1.

– ô (ñó ò ò ó )2 + õ sin( ï ). 1 b . This is a special case of equation 5.2.6.1 with  ( ) =  sin( Œ  ). Integrating the equation twice, we obtain a third-order equation: 


 =  !

 −  (
 )2 + L 1  + L 2 − Œ −2 sin( Œ  ).

ñ ó(5) = ôññóò ò ó ò ò ó ó

2 b . Particular solution: 2.

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó Solution:

3.

2

+ 10ñ ó ò ò ó = L 4 4 + L 3

=−

Œ

ñ óò ò ó ò ó

3

+L

 L

4( 2

= 2

2

ô



2

+ Π2)

sin( +L

ï

1 + L

V L sin( Π ) + Πcos( Π )W + L . ). 0

− 2 Œ

−5

cos( Π ).

+ 10ñ ó ò ò ó ñ ó ò ò ó ò ó = ô sin ú ( ï ) + õ . This is a special case of equation 5.2.6.2 with  ( ) =  sin + ( Œ  ) +  .

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

© 2003 by Chapman & Hall/CRC

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5.2. NONLINEAR EQUATIONS

4. 5. 6. 7. 8. 9.

+ 10ñó ò ò óÈñó ò ò ó ò ó = ô sin ú ( ƒñ ) + õ . This is a special case of equation 5.2.6.3 with  ( ) =  sin + ( Œ

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

)+ .

+ 10(ñ ó ò ò ó ò ó )2 = ô sinú ( ï ). This is a special case of equation 5.2.6.6 with  ( ) =  sin + ( Œ  ).

ññ ó(6) + 6ñ óò ñ ó(5) + 15ñ óò ò ó ñ óò ò ó ò ò ó ó ñ ó( ÷

= ô sinú ( ƒñ ) + õ . This is a special case of equation 5.2.6.8 with  ( ) =  sin + ( Œ )

= ô ï –÷ sinú ( ƒñ ). This is a special case of equation 5.2.6.11 with  ( ) =  sin + ( Œ

ñ ó( ÷

)+ .

)

).

= ô sin ú ( ï ) + õ . This is a special case of equation 5.2.6.17 with  ( ) =  sin + ( Œ  ) +  .

ññ ó(2÷

+1)

= ôññó ò ò ó ò ò ó ó – ô (ñó ò ò ó )2 + õ sin( ï ). 1 b . This is a special case of equation 5.2.6.54 with  ( ) =  sin( Œ  ). Integrating the equation two times, we obtain an (B − 2)nd-order equation: (  −2) =  !

 −  (
 )2 + L 1  + L 2 − Œ −2 sin( Œ  ).

ñ ó( ÷

)

2 b . Particular solutions: = L sin( Œ  ) + =− 10. 11. 12.

Π2

−4



+  2L

(−1) ´J 2 p L Œ  − 2

Œ

if B is even number,

LpŒ 4 V L sin( Œ  ) + (−1)  4

−1 2

Π

−4

cos( Π )W + L

if B is odd number.

ò ñ ó(2÷ ) = ô sinú ( ï ) + õ . (2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ This is a special case of equation 5.2.6.62 with  ( ) =  sin + ( Œ  ) +  . ñ ó( ÷

= ô sin  ( ƒñ )ñ ó ò ñ ó(÷ –1) . This is a special case of equation 5.2.6.57 with  ( ) =  sin  ( Œ )

).

– ñó ò ñ ó(÷ –1) = ô sin( ï )ñ 2. This is a special case of equation 5.2.6.64 with  ( ) =  sin( Œ  ). Integrating yields an  (B − 1)st-order linear equation: (  −1) = L − cos( Œ  )S .

ññ ó(÷

)

Q

Œ

5.2.5-2. Equations with cosine. 13.

– ô (ñó ò ò ó )2 + õ cos( ï ). 1 b . This is a special case of equation 5.2.6.1 with  ( ) =  cos( Œ  ). Integrating the equation twice, we obtain a third-order equation: 


 =  !

 −  (
 )2 + L 1  + L 2 − Œ −2 cos( Œ  ).

ñ ó(5) = ôññóò ò ó ò ò ó ó

2 b . Particular solution: 14.

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó Solution:

15.

2

=−

Π4( 2 L

+ 10ñó ò ò óÈñó ò ò ó ò = L 4 4 + L 3 3 + L

ó

= 2

2

ô



2

+ Π2)

cos( +L

ï

1 + L

V L cos( Œ  ) − Œ sin( Œ  )W + L . ). 0

+ 2 Œ

−5

sin( Π ).

+ 10ñ ó ò ò ó ñ ó ò ò ó ò ó = ô cosú ( ï ) + õ . This is a special case of equation 5.2.6.2 with  ( ) =  cos + ( Œ  ) +  .

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

© 2003 by Chapman & Hall/CRC

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720 16. 17. 18. 19. 20. 21.

HIGHER-ORDER DIFFERENTIAL EQUATIONS + 10ñó ò ò óÈñó ò ò ó ò ó = ô cosú ( ƒñ ) + õ . This is a special case of equation 5.2.6.3 with  ( ) =  cos + ( Œ

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

)+ .

+ 10(ñ ó ò ò ó ò ó )2 = ô cosú ( ï ). This is a special case of equation 5.2.6.6 with  ( ) =  cos + ( Œ  ).

ññ ó(6) + 6ñ óò ñ ó(5) + 15ñ óò ò ó ñ óò ò ó ò ò ó ó ñ ó( ÷

)

= ô cosú ( ƒñ ) + õ . This is a special case of equation 5.2.6.8 with  ( ) =  cos + ( Œ

ñ ó( ÷

)

= ô ï –÷ cosú ( ƒñ ). This is a special case of equation 5.2.6.11 with  ( ) =  cos + ( Œ

)+ . ).

= ô cosú ( ï ) + õ . This is a special case of equation 5.2.6.17 with  ( ) =  cos + ( Œ  ) +  .

ññ ó(2÷

+1)

= ôññó ò ò ó ò ò ó ó – ô (ñó ò ò ó )2 + õ cos( ï ). 1 b . This is a special case of equation 5.2.6.54 with  ( ) =  cos( Œ  ). Integrating the equation two times, we obtain an (B − 2)nd-order equation: (  −2) =  !

 −  (
 )2 + L 1  + L 2 − Œ −2 cos( Œ  ).

ñ ó( ÷

)

2 b . Particular solutions: = L cos( Œ  ) + =− 22. 23. 24.

Π

2 −4



+ L 2

(−1) ´J 2 p L Œ  −

Œ

2 4

LpŒ 4 V L cos( Œ  ) + (−1) 

if B is even number, +1 2

Π

−4

sin( Π )W + L

if B is odd number.

ò ñ ó(2÷ ) = ô cosú ( ï ) + õ . (2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ This is a special case of equation 5.2.6.62 with  ( ) =  cos + ( Œ  ) +  . ñ ó( ÷

= ô cos  ( ƒñ )ñ ó ò ñ ó(÷ –1) . This is a special case of equation 5.2.6.57 with  ( ) =  cos  ( Œ )

).

– ñó ò ñ ó(÷ –1) = ô cos( ï )ñ 2. This is a special case of equation 5.2.6.64 with  ( ) =  cos( Œ  ). Integrating yields an  sin( Œ  ) + L3S . (B − 1)st-order linear equation: (  −1) =

ññ ó(÷

)

Q Œ

5.2.5-3. Equations with tangent. 25. 26. 27. 28.

– ô (ñó ò ò ó )2 + õ tan( ï ) + ö . This is a special case of equation 5.2.6.1 with  ( ) =  tan( Œ  ) + ( .

ñ ó(5) = ôññóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò ó = ô tanú ( ï ) + õ . This is a special case of equation 5.2.6.2 with  ( ) =  tan + ( Œ  ) +  .

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñ ó ò ò ó ñ ó ò ò ó ò ó = ô tanú ( ƒñ ) + õ . This is a special case of equation 5.2.6.3 with  ( ) =  tan + ( Œ

ññ ó(5) + 5ñ óò ñ óò ò ó ò ò ó ó

)+ .

+ 10(ñó ò ò ó ò ó )2 = ô tanú ( ï ). This is a special case of equation 5.2.6.6 with  ( ) =  tan + ( Œ  ).

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó

© 2003 by Chapman & Hall/CRC

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5.2. NONLINEAR EQUATIONS

29. 30. 31. 32. 33. 34. 35. 36.

ñ ó( ÷

)

= ô tanú ( ƒñ ) + õ . This is a special case of equation 5.2.6.8 with  ( ) =  tan + ( Œ

ñ ó( ÷

)

= ô ï –÷ tanú ( ƒñ ). This is a special case of equation 5.2.6.11 with  ( ) =  tan + ( Œ

)+ . ).

= ô tanú ( ï ) + õ . This is a special case of equation 5.2.6.17 with  ( ) =  tan + ( Œ  ) +  .

ññ ó(2÷

+1)

ñ ó(2÷

)

= (–1)÷ ñ + ô (ñó ò + ñ tan ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = cos  .

ñ ó(2÷

+1)

= (–1)÷ +1 ñ tan ï + ô (ñó ò + ñ tan ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = cos  .

ò ñ ó(2÷ ) = ô tanú ( ï ) + õ . (2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ This is a special case of equation 5.2.6.62 with  ( ) =  tan + ( Œ  ) +  . ñ ó( ÷

= ô tan ( ƒñ )ñóÈ ò ñ ó(÷ –1) . This is a special case of equation 5.2.6.57 with  ( ) =  tan  ( Œ )

).

– ñó ò ñ ó(÷ –1) = ô tan( ï )ñ 2. This is a special case of equation 5.2.6.64 with  ( ) =  tan( Œ  ).

ññ ó(÷

)

5.2.5-4. Equations with cotangent. 37. 38. 39. 40. 41. 42. 43. 44.

– ô (ñó ò ò ó )2 + õ cot( ï ) + ö . This is a special case of equation 5.2.6.1 with  ( ) =  cot( Œ  ) + ( .

ñ ó(5) = ôññóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò ó = ô cotú ( ï ) + õ . This is a special case of equation 5.2.6.2 with  ( ) =  cot + ( Œ  ) +  .

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò ó = ô cotú ( ƒñ ) + õ . This is a special case of equation 5.2.6.3 with  ( ) =  cot + ( Œ

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

)+ .

+ 10(ñó ò ò ó ò ó )2 = ô cotú ( ï ). This is a special case of equation 5.2.6.6 with  ( ) =  cot + ( Œ  ).

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó ñ ó( ÷

)

= ô cotú ( ƒñ ) + õ . This is a special case of equation 5.2.6.8 with  ( ) =  cot + ( Œ

ñ ó( ÷

)

= ô ï –÷ cotú ( ƒñ ). This is a special case of equation 5.2.6.11 with  ( ) =  cot + ( Œ

)+ . ).

= ô cotú ( ï ) + õ . This is a special case of equation 5.2.6.17 with  ( ) =  cot + ( Œ  ) +  .

ññ ó(2÷ ñ ó(2÷

+1)

= (–1)÷ ñ + ô (ñ ó ò – ñ cot ï )  . This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = sin  . )

© 2003 by Chapman & Hall/CRC

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722 45.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó(2÷

= (–1)÷

+1)

This is a special case of equation 5.2.6.47 with  ( , ‡ ) = ‡  and Y ( ) = sin  .

46.

(2 ù – 1)ññ

47.

ñ ó( ÷

48.

cot ï + ô (ñó ò – ñ cot ï )  .

ñ

ó(2÷

+ (2ù + 1)ñ

+1)

óò ñ ó(2÷

)

cotú (

ô

=

ï

)+ õ .

This is a special case of equation 5.2.6.62 with  ( ) =  cot + ( Π ) +  . )

=

ó( ÷

cot ( ƒñ )ñóÈ òñ

ô

–1)

.

This is a special case of equation 5.2.6.57 with  ( ) =  cot  ( Œ

ññ ó(÷

)

ó ò ñ ó( ÷

–ñ

–1)

=

ô

cot(

ï )ñ

2

).

.

This is a special case of equation 5.2.6.64 with  ( ) =  cot( Π ).

5.2.6. Equations Containing Arbitrary Functions 5.2.6-1. Fifth- and sixth-order equations. 1.

ñ ó(5) = ôññ óò ò ó ò ò ó ó

)2 + ! ( ï ).

óò ò ó

– ô (ñ

Integrating the equation two times, we obtain a third-order equation:






 =  ! 

 −  ( 
)2 + L 1  + L 2.

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

2 + X  ( − a )  (a ) a , M 0

3.

4.

=L

ó

4 3 2 4 + L 3 + L 2 + L 1 + L 0 +

ññ ó(5) + 5ñóò ñóò ò ó ò ò ó ó

+ 10ñó ò ò óÈñó ò ò ó ò

The substitution ¨ = 2 l A £ ¨  .

ññ ó(5) + ôñóò ñóò ò ó ò ò ó ó

2

0

is an arbitrary number.

= ! ( ï ).

+ 10ñó ò ò óÈñó ò ò ó ò

Solution: 2

where 

ó



1 X (  − a )4  ( a ) a , 12  0 M

where 

0

is an arbitrary number.

= ! (ñ ).

leads to an autonomous equation of the form 5.2.6.8:

+ (3 ô – 5)ñ„ó ò ò óÈñó ò ò ó ò

¨ (5)  =

= ! ( ï ).

ó

Integrating the equation three times, we obtain a second-order equation:

 ! 

 +  − 3 ( 
)2 = L  2 + L  + L + 1 X ( − a )2  (a ) a , 2 1 0 2 2  0 M 5.

( ô + ñ )ñ

ó(5) + õÿñóò ñóò ò ó ò ò ó ó

+ ö^ñó ò ò ó

where 

0

is an arbitrary number.

= ! ( ï ).

ñóò ò ó ò ó

Integrating yields a fourth-order equation: 1 (  + ) 



 + (  − 1) 



 + (1 −  + ( )( 

 )2 = X¯ ( )  + L . 2 M 6.

ññ ó(6) + 6ñóò ñ ó(5) + 15ñóò ò óÈñóò ò ó ò ò ó ó 2

Solution: 7.

ñ ó(6) = (ô ï

2

+

=L

õï

5

5

+L

+ ö )–7 )

4 2

4

+ 10(ñó ò ò ó ò ó )2 = ! ( ï ).

+L

!  ñ (ô ï

3

3 2

+L

+

õï

2

2

+L

1 + L

0+



1 X (  − a )5  ( a ) a . 60  0 M

+ ö )–5 ) 2  .

This is a special case of equation 5.2.6.22 with B = 6.

© 2003 by Chapman & Hall/CRC

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723

5.2. NONLINEAR EQUATIONS

(  ) =  ( , ).

5.2.6-2. Equations of the form 8.

ñ ó( ÷

= ! (ñ ).

)

Autonomous equation. This is a special case of equation 5.2.6.77.

1 b . The substitution ¨ ( ) = 
 leads to an (B − 1)st-order equation. 2 b . For even B = 2 , the first integral of the equation is:

¡

+½

−1



 + −  ) + 12 (−1) + V ( + ) W 2 + X  ( )

(−1)  ( 

) (2

=1

M

=L .

Furthermore, the order of the obtained equation can be reduced by one by the substitution ¨ ( ) = 
 . 9.

10.

11.

ñ ó( ÷

ï ú

= ! (ñ + ô

)

The substitution ¨ ¨ (  ) =  (¨ ).

ñ ó( ÷

=u ! 

)

ñ

+

þ

),

+  +

=

ô ÷ ï ÷

+

= 0, 1,

ô ÷ –1 ï ÷

)

ñ ó( ÷

)

ï –÷ ! (ñ

=

– 1.

leads to an autonomous equation of the form 5.2.6.8: –1

+

The substitution ¨ = +     +   form 5.2.6.8: ¨ (  ) =   B ! +  (¨ ).

ñ ó( ÷

uuu , ù

uuu

 −1 

+

ô

−1

. + ))) +  0

0

leads to an autonomous equation of the

).

The substitution a = ln | | leads to an autonomous equation of the form 5.2.6.77. 12.

ï 1–÷ ! (ñ  ï

=

).

Homogeneous equation. This is a special case of equation 5.2.6.83. The transformation a = ln  , ¨ =   leads to an autonomous equation of the form 5.2.6.77. 13.

ñ ó( ÷

)

ï –÷ –1 ! (ï 1–÷ ñ

=

).

The transformation  = a ¨ (g  ) = (−1)   (¨ ). 14.

ñ ó(2÷

)

=

ï

–

÷

2 +1 2

!IO ï

1–2 2

÷

−1

ñ g

=a

,

P

1−

 ¨ leads to an autonomous equation of the form 5.2.6.8:

.



2 −1

The transformation  =  , =  2 ¨ (a ) leads to an autonomous equation of the form 5.2.6.68, whose order can be reduced by two. 15.

ñ ó( ÷

)

=

ï –÷ –  ! (ñ ï 

).

This is a special case of equation 5.2.6.86. 1 b . The transformation a = ln  , H =   leads to an autonomous equation of the form 5.2.6.77. 2 b . The transformation H =   , ¨ =  16.

ñ ó( ÷

)

=

ñ ï –÷ ! (ï  ñ ú


 leads to an (B − 1)st-order equation.

).

This is a special case of equation 5.2.6.89. The transformation a =   + , ¨ to an (B − 1)st-order equation.

=


 leads

© 2003 by Chapman & Hall/CRC

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724 17.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ññ ó(2÷

+1)

= ! ( ï ).

Integrating yields a 2B th-order equation: 2

½

−1

(−1)

+

+

+ (

ñ ó( ÷

)

= ! ( ï , ñ ).

=H

1−

)

= (ô

ï

+

+ ö )1–÷

õÿñ

2

= 2 XZ ( )  + L ,

M

 ¨ ( H ) leads to an equation of the same form: ¨ (‘  ) =

ô ï  ï

+ õÿñ + ñ 1 b . For ý − ˜ƒ = 0, the substitution the form 5.2.6.8.

ñ ó( ÷

+ (−1) rV (  ) W

is used.

The transformation  = H −1 , (−1)  H −  −1  ( H −1 , H 1−  ¨ ). 19.

)

=0

 ≡ where the notation (0) 18.

  +

) (2 −

! O

+ö . + P  ¨ =  + 

+ ( leads to an autonomous equation of

2 b . For ý − ˜ƒ ≠ 0, the transformation

¨ = − , 0

H =  −  0, where 

0

and

0

are the constants which are determined by solving the linear algebraic system



0

+

+ ( = 0,

0

ƒ…

0

+ý

0

+ = = 0,

leads to a homogeneous equation of the form 5.2.6.12:

¨ (‘  ) = H 20.

ñ ó( ÷

)

= (ô

1

ï

+

õ 1ñ

1−

ö

+

¨

 O

1)

1–

,

HIP

 (ˆ ) = ( + 'ˆ )1−  *O

where

÷ !e” ô 2 ï ô 3ï

Let the following condition hold: ûû 



+ +

õ 2ñ

+ö +ö

õ 3ñ 

1

1

(

2 3

 + 'ˆ . ƒ + ý±ˆÑP

.

•

1

( 2 ûû = 0. û 3 3 (3û û û û û For  2  3 −  3  2 ≠ 0, the transformation 

2

2

¨ = − , 0

H =  −  0, where 

0

and

0

are the constants determined by the linear algebraic system

 2

+

0

+(

2 0

2

 3

= 0,

0

+

3 0

+(

3

= 0,

leads to a homogeneous equation of the form 5.2.6.12:

21.

(ô

ï

+ õ )÷ (ö

¨ (‘  ) = H

1−

 O

ï

ó( ÷

)

+ ý )ñ

¨

,

HIP

=e ! ”

(ö

 (ˆ ) = ( 1 +  1 ˆ )1−  *O

where

ï

 +  The transformation ˆ = ln û û ˜(  + M û the form 5.2.6.77.

ñ

+ ý )÷

û

û

û,

–1

¨ =

•

 2 +  2ˆ .  3 +  3Ñ ˆ P

. ( (˜ + ) 

M

−1

leads to an autonomous equation of

© 2003 by Chapman & Hall/CRC

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725

5.2. NONLINEAR EQUATIONS

22.

ñ ó( ÷

)

ï

= (ô

2

õï

+

+ö )

–

÷

!IOiñ (ô ï

1+ 2

2

õï

+

+ö )

÷

1– 2

1 b . The transformation

a =X





M

+ ' + (

2

P

.

¨ = ( 

,

leads to an autonomous equation with respect to ¨ by the substitution H (¨ ) = ¨ g
.

2

+ ' + ( )



1− 2

(1)

= ¨ (a ), which admits reduction of order

2 b . Let B = 2 be an even integer ( = 1, 2, 3,  ). In this case, transformation (1) yields ¡ ¡ an equation of the form 5.2.6.68, whose order can be reduced by two.

.

Setting


. by ¨  =

+ ' + ( , = ¨ . 
+ 1 − 2¡ O 2

= 

2

1+2 + − 2

.

O

.

.

+

2




−1 2

, we obtain

P

1−2 ¡ 2


+

and multiplying both sides of the original equation

.




 + (2

P

)

=  (¨ )¨ 
.

Integrating both sides of this equality with respect to  (the left-hand side is integrated by parts), we have

+½

−2



 +

(−1)  j ( 

 + (−1) +

−1− )

) (2

−1

X j ( +

+

−1) (

+1)

=0

where

 j (  ) = M  O  M

.

1−2 ¡ 2


+

.




=

P

.



( +1)

+ O, −

¡

+

M

 = X  (¨ ) ¨ + L , M

1 2P

.

(2)


(  ) + , ( , − 2 ) (  ¡

−1)

(remember that B = 2 ). It can be shown that the integrand on the left-hand side of (2) is a ¡ total differential. Finally, we arrive at the first integral:

+½ 

−2

(−1) 

=0

Q

.



+ (−1) +

( +1)

−1

1 2

¦

+, −

.

¡

.

+ 12 

V ( + ) W 2 −

1 2


(  ) + , ( , − 2 ) (  −1) S (2  + −1−  ) ¡ 
( + −1) ( + ) +  (1 − 2 ) ( + −2) ( + ) + 12  ¡ ¡

.

= Xv (¨ ) ¨ + L . 23.

ñ ó( ÷

)

=

ñ

1+ 1–

M

÷

÷ !IOiñ (ô ï

1 b . Setting  (‡ ) = ‡ 



2

õï

+

+1 −1 1 (

+ö )

÷

1– 2

P

2

V ( +

−1) 2

W §

.

‡ ), we have equation 5.2.6.22 with the function  1 (instead of  ). ‘ 2 b . The transformation  = H −1 , = H 1−  ¨ ( H ) leads to an equation of similar form: ¨ (  ) = 1−  1+  (−1)  ¨ 1−  *O ¨ ( (H 2 + ˜H +  ) 2 . P 24. 25. 26.

ñ ó( ÷

ó



ó

=  ^ ! (ñ – ^ ).  The substitution ¨ ( ) =  − 9 leads to an autonomous equation of the form 5.2.6.77. )

ó

= ñ! (  ^ ñ ú ).  The transformation H =  9 + , ¨ ( H ) = 
  leads to an (B − 1)st-order equation.

ñ ó( ÷

)

ñ ó( ÷

)

= ï –÷ ! ( ïƒú  ^ ] ). The transformation H =  +  9  , ¨ ( H ) =  
 leads to an (B − 1)st-order equation.

© 2003 by Chapman & Hall/CRC

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726 27.

28.

29.

30.

31.

32.

33.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó( ÷

ó

÷  ó

= ! (ñ + ô ) – ô The substitution ¨ ( ) = ¨ (  ) =  (¨ ). )

ñ ó(2÷

)

ñ ó(2÷

)

ñ ó(2÷

+1)

ñ ó(2÷

+1)

+  ´



leads to an autonomous equation of the form 5.2.6.8:

= ! (ñ + ô cosh ï ) – ô cosh ï . The substitution ¨ ( ) = +  cosh  leads to an autonomous equation of the form 5.2.6.8: ¨ (2  ) =  (¨ ). = ! (ñ + ô sinh ï ) – ô sinh ï . The substitution ¨ ( ) = +  sinh  leads to an autonomous equation of the form 5.2.6.8: ¨ (2  ) =  (¨ ).

= ! (ñ + ô cosh ï ) – ô sinh ï . The substitution ¨ ( ) = +  cosh  leads to an autonomous equation of the form 5.2.6.8: ¨ (2  +1) =  (¨ ). = ! (ñ + ô sinh ï ) – ô cosh ï . The substitution ¨ ( ) = +  sinh  leads to an autonomous equation of the form 5.2.6.8: ¨ (2  +1) =  (¨ ).

= ! (ñ + ô cos ï ) – ô cos  ï + The substitution ¨ ( ) = +  cos  ¨ (  ) =  (¨ ).

ñ ó( ÷

)

= ! (ñ + ô sin ï ) – ô sin  ï + The substitution ¨ ( ) = +  sin  ¨ (  ) =  (¨ ).

ñ ó( ÷

)

ñ ó( ÷

)

= ! (ñ )ñ

óò

1 2

® ù 

. leads to an autonomous equation of the form 5.2.6.8:

® ù 

1 2

. leads to an autonomous equation of the form 5.2.6.8:

  =  ( , , 
 ). ( )

5.2.6-3. Equations of the form 34.

.

+ " ( ï ).

Integrating yields an (B − 1)st-order equation: 35. 36.

37.

ñ ó( ÷

)

ñ ó( ÷

)

ñ ó( ÷

)

39.

= X¯ ( )

M

+ Xw ( )  + L .

M

= ! ( ï , ñó ò ). The substitution ¨ ( ) = 
leads to an (B − 1)st-order equation: ¨ ( 

−1)

=  ( , ¨ ).

= ! (ñ , ñó ò ). Autonomous equation. This is a special case of equation 5.2.6.77. The substitution ¨ ( ) = 
leads to an (B − 1)st-order equation. =

ñ ï –÷ ! (ï ñ óò ¦ñ

).

The transformation H = 

38.



( −1)


 , ¨ = 

2



  leads to an (B − 2)nd-order equation.

= ô ÷ ñ + ! ( ï , ñó ò – ôñ ). The substitution ¨ = 
−  leads to an (B − 1)st-order equation:

ñ ó( ÷ ñ ó( ÷

)

)

= ! (ï , ï

¨ ( 

ñ óò

−1)

+  ¨ ( 

−2)

+ ))) +  

−1

¨ =  ( , ¨ ).

– ñ ).

The substitution ¨ = 


−

leads to an (B − 1)st-order equation:

M

¨ 
 −2 2 =  ( , ¨ ). M  −2 O   P

© 2003 by Chapman & Hall/CRC

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727

5.2. NONLINEAR EQUATIONS

40.

41.

42.

ñ ó( ÷

)

!  ï , ï ñ óò

=

– þxñ  .

  is a positive integer and B ≥ + 1. The substitution ¨ =  "
 − Here, ¡ ¡ ¡ ¨ (  − + −1) =  ( , ), where = ¨ ( + )  . (B − 1)st-order equation: ïƒ÷ ñ ó(÷

= ! (ï , ï

ñóò

+

ôñ

!  ï , g ú ñóò

–

g úò ñ 

)

44.

45.

46.

47.

ñ ó( ÷

)

=

ñ ó(2÷

)

ñ ó(2÷

)

ñ ó(2÷

)

ñ ó(2÷

)

g ú

,

.


− +


+ ! (ï , ñ

=

ñ

+ ! ( ï , ñó ò sinh ï – ñ cosh ï ).

½ú



=0

ô  ü ï –1 , ù

>

þ

.

The substitution ¨ = 
sinh  − cosh  leads to an (2B − 1)st-order equation. = (–1)÷

ñ

+ ! ( ï , ñó ò sin ï – ñ cos ï ).

= (–1)÷

ñ

+ ! ( ï , ñó ò cos ï + ñ sin ï ).

The substitution ¨ = 
sin  − cos  leads to an (2B − 1)st-order equation. The substitution ¨ = 
cos  + sin  leads to an (2B − 1)st-order equation.

ñ ó÷

( )

=

r (ó ÷ ) r ñ

+ !IO ï ,  ñ óò

–

ñ ó( ÷

= ! (ï , ï

)

ñóò

r óò r ñ P

,

r

=

r (ï

).

leads to an (B − 1)st-order equation.

  =  ( , , 
 , 

 ). ( )

– ñ , ñó ò ò ó ).

This is a special case of equation 5.2.6.78. The substitution ¨ ( ) = 
 − (B − 1)st-order equation.

ñ ó( ÷

= ! (ï , ï

)

2

ñóò ò ó

– 2ï

ñóò

+ 2ñ ).

This is a special case of equation 5.2.6.81. The substitution ¨ ( ) =  to an (B − 2)nd-order equation.

ñ ó(2÷

)

=

ô ÷ ñ

The substitution ¨ = 

 − 

ñ ó(2÷

)

=

2

leads to an



 − 2 
 + 2 leads

+ ! ( ï , ñó ò ò ó – ôñ ).

¨ (2  51.

=


The substitution ¨ =  cosh  − sinh  leads to an (2B − 1)st-order equation.

5.2.6-4. Equations of the form

50.

=0

g úò

,

cosh ï – ñ sinh ï ).

Y

49.



ô  ï

leads to an (B − 1)st-order equation.

ñ

óò

½ú

=

=

Y
 The substitution ¨ = 
−

48.

.

Here, (  )  =  (  + 1)  ( + B − 1) is the Pochhammer symbol. The substitution ¨ = 
 +  leads to an (B − 1)st-order equation.

. The substitution ¨ = + 43.

ñ

) – ( ô )÷

leads to an

ñ!  ñ ñóò ò ó

leads to an (2B − 2)nd-order equation: −2)

  +  ¨ (2

−4)

+ ))) +  

−1

¨ =  ( , ¨ ).

– ñó ò 2  .

This is a special case of equation 5.2.6.52.

© 2003 by Chapman & Hall/CRC

Page 727

728 52.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ñ ó(2÷

= ñó ò ò ó'!  ññó ò ò ó – ñó ò 2  + ñ"  1 b . Particular solution: =L )

where the constants L

1

– ñó ò 2  .

exp( L

3

 )+L

2

exp(− L

3

 ),

L 2 , and L 2 are related by the constraint L 32  − L 32  (4 L 1 L 2 L 32 ) − (4 L 1 L 2 L 32 ) = 0.

1,

2 b . Particular solution: where the constants L

ññóò ò ó

=L

1

cos( L

3

 )+L

2

sin( L

3

 ),

L 2 , and L 2 are related by the constraint (−1)  L 32  + L 32  (− L 12 L 32 − L 22 L 32 ) − (− L 12 L 32 − L

53.

ñ ó( ÷

)

=

1,

ñ óò ò ó ñ óò !IO ó , ñ óò ñ ò

ñ óò ò ó ñ óò

–ñ

2 2

L 32 ) = 0.

.

P

Particular solution: = L 1 exp( L 2  )+ L 3 , where L 1 is an arbitrary constant and the constants L 2 and L 2 are related by the constraint L 2 −1 =  ( L 2 , − L 2 L 3 ). 5.2.6-5. Equations of the form  ( , ) (  ) + ( , , 
 ) (  54.

ñ ó( ÷

−2)

.

= ôññ ó ò ò ó ò ò ó ó – ô (ñ ó ò ò ó )2 + ! ( ï ). Integrating the equation two times, we obtain an (B − 2)nd-order equation:



56.

= %   , , 
 ,  , ( 

)

( −2)

55.

−1)

ñ ó(2÷




=  !   −  (  )2 + L

1



 +L

2

+ X  ( − a )  (a ) a , M 0

where 

= ô 2 ñ + !u ï , ñ ó(÷ ) + ôñ  . The substitution ¨ = (  ) +  leads to an B th-order equation: ¨ ( 

ñ ó( ÷

0

is an arbitrary number.

)

=  ¨ +  ( , ¨ ).

)

= !u ñ ó(÷ –2)  . Having set ‡ ( ) = (  the form: )

−2)

 =X

M

, we obtain a second-order equation ‡

 =  (‡ ), whose solution has

‡

+L

2,

Y (‡ ) = A L Q

where

1

+2X

J

1 2

 (‡ ) ‡ S M

.

Y (‡ ) Expressing ‡ in terms of  and integrating the resulting relation B − 2 times, we find . Solution in parametric form:

57.

ñ ó( ÷

 ‡  =X k M , 2 Y (‡ )

=X

‡

1

k 3 Y (‡ 1 ) X k M



1 4

‡

n n

−4 −1

‡ M  Y (‡ 

−3

X −3 ) k

n n

−3

‡ 

−2 ‡  −2 . M Y (‡  −2 )

= ! (ñ )ñ ó ò ñ ó(÷ –1) . Integrating yields an (B − 1)st-order autonomous equation of the form 5.2.6.8:



ñ ó( ÷

)

= V ! (ñ )ñó ò + " ( ï )WNñ

=  ( ),

ó( ÷

–1)

where

ï ñ ó( ÷

 ( ) = L exp X  ( ) S . Q M

.

Integrating yields an (B − 1)st-order equation: 59.

2

'X k Y (‡ 2 ) M

)

( −1)

58.





( −1)

= L exp X

Q

 ( ) M

+ X ( )  S .

M

+ ùñ ó(÷ –1) = ! ( ï ñ ). The substitution ¨ ( ) =  leads to an autonomous equation of the form 5.2.6.8: ¨ (  ) =  (¨ ). )

© 2003 by Chapman & Hall/CRC

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729

5.2. NONLINEAR EQUATIONS

60. 61. 62.

ï ñ ó( ÷

+ ( ô + ù – 1)ñ ó(÷ –1) = ! ( ï , ï ñó ò + ôñ ). The substitution ¨ =  
+  leads to an (B − 1)st-order equation: ¨ (  )

−1)

=  ( , ¨ ).

+ 2ù ï ñ ó(÷ –1) + ù (ù – 1)ñ ó(÷ –2) = ! ( ï 2 ñ ). The substitution ¨ ( ) =  2 leads to an autonomous equation of the form 5.2.6.8: ¨ (  ) =  (¨ ).

ï 2 ñ ó( ÷

)

(2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ ó ò ñ ó(2÷ ) = ! ( ï ). Having integrated the equation, we obtain

  )+2 (2B − 1)! (2

½



−1

(−1) 



   + (−1) 

+1 ( ) (2 − )

+1

=1

V (  ) W 2 = X  ( )  + 2 L 2 . M

The second integration leads to an (2B − 1)st-order equation:

½

−1



(2B − 1 − 2 , )(−1)  ( 

 

 = 2L 2  + L

) (2 −1− )



+ X  ( − a )  (a ) a . M 0

1

=0

The third integration leads to an (2B − 2)nd-order equation:

½

−2



( , + 1)(2B − , − 1)(−1)  ( 

 + 1 (−1) 

 

) (2 −2− )

2

=0

=L 63.

2

2

ññ ó(÷

64.

)

ó( ÷

– ñó òñ

–1)

½

−1



(−1) 



   + (−1) 

+1 ( ) (2 − )

+1

ññ ó ÷

( )

65.

= ! ( ï )ñ 2 .

= ñó ò ñ ó÷

( –1)

+ ! ( ï ) ñ ñ ó( ÷

–1)

.

Integrating yields an (B − 1)st-order linear equation:

68.

+L

V ( 

−1) 2

1 + L

W

0+

ññ ó ÷



1 X (  − a )2  ( a ) a . 2  0 M

V (  ) W 2 = Xv ( )

=1

Integrating yields an (B − 1)st-order linear equation:

67.

B

ò ñ ó(2÷ ) = ! (ñ )ñóò + " (ï ). (2 ù – 1)ññ ó(2÷ +1) + (2ù + 1)ñ„óÈ Integrating yields an (B − 1)st-order equation:   )+2 (2B − 1)! (2

66.

−1 2

+ ( ! – 1)ñ ó ò ñ ó ÷

+ XÈ ( )  + L .

M

M

( −1)



= X¯ ( )  + L2S .



= L exp X¯ ( )  S .

( −1)

Q

M

Q

! = ! (ï

" = " (ï

M

+ !"Iññ ó ò + " ó ò ñ = 0, ), ). This equation is solved by the functions that are solutions of the (B − 1)st-order linear equation (  −1) + ( ) = 0. ( )

( –1)

2

[ñ + ! (ï )]ñ ó(÷ ) = [ñ ó ò + ! ó ò ( ï )]ñ ó(÷ –1) + ô! ( ï )ñ ó ò – ô! ó ò ( ï )ñ . Integrating yields an (B − 1)st-order constant coefficient nonhomogeneous linear equation: (  −1) −L = ( L −  )  ( ). There is also the trivial solution = 0.

½

ú

÷

=1

ô ú ñ ó(2ú

)

= ! (ñ ).

The first integral has the form:

½ 

+

 + =1

™

+½

Á

−1

(−1) =1

Á ( Á

1  + − Á ) + (−1) + V ( + ) W

) (2

2

2

¢

+X

 ( ) M

=L ,

where L is an arbitrary constant. Furthermore, the order of the obtained equation can be reduced by one by the substitution ¨ ( ) = 
 .

© 2003 by Chapman & Hall/CRC

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730

HIGHER-ORDER DIFFERENTIAL EQUATIONS

÷ ½

69.

ú

ô ú ïƒú ñ ó(ú

=1

= ! (ñ ).

)

The substitution a = ln | | leads to an autonomous equation of the form 5.2.6.77.

½

ñ

70.

÷

ú

=0

ô ú ñ ó(2ú

= ! ( ï ).

+1)

Integrating yields a 2B th-order equation:

½  +

™

 + =0

2

+½

−1

Á

Á Á (−1) (

 + − Á ) + (−1) + V ( + ) W

) (2

2

=0

¢

=2X

 ( )  + L , M

 stands for . where (0) 71.

½

ú

÷

ô ú ñ ó(ú ) ñ ó(2÷

=0

+1–

ú

)

= ! ( ï ).

The first integral has the form: 2 where # + =

7+ 

=0

½

−1

+

( +

# +

  +

) (2 −

)

(−1) +

=0 +

   =  + − +

−1 +

+ #  V (  ) W

 +

−2 −

2

=2X

 ( )  + L , M 7

))) . If the condition #  = 2 +

−1

(−1) 

−1+

=0

+ # +

is satisfied, the obtained equation can be integrated two times more (see, in particular, equation 5.2.6.62).

 −1)  .   =    , , 
 ,  , (? ( )

5.2.6-6. Equations of the form 72.

ñ ó( ÷

= !  ñ ó(÷ –1)  . Having set ‡ ( ) = (  )

relation  = X

−1)

‡

, we obtain a first-order equation ‡
 =  (‡ ). Further, find ‡ from the

+ L 1 . Then the (B − 1)-fold integration yields . M  (‡ )

Solution in parametric form:

73.

ñ ó( ÷

 ‡  =X k M , 1  (‡ )

=X



‡

1

k 2  (‡ 1 ) X k M



1 3

‡

2

'X k  (‡ 2 ) M

= ! (ñ )ñó ò "  ñ ó(÷ –1)  . Integrating yields an (B − 1)st-order equation: )

X

¨ M ¨ = X¯ ( ) ( ) M

+L ,

n n

−3 −1

where

‡ M   (‡ 

−2

X −2 ) k

¨ = ( 

n n

−1)

−2

‡ 

‡ M  −1 .  (‡  −1 ) −1

.

Furthermore, the order of this equation can be reduced by one by the substitution H ( ) = /
 . 74.

ñ ó( ÷

= V ! (ñ )ñó ò + " ( ï )W²% (ñ ó(÷ –1) ). Integrating yields an (B − 1)st-order equation: )

ù 75.

¨ M ¨ =ù % ( )

 ( ) M

+ù

( )  + L , M

¨ = ( 

−1)

.

= !  ï , ñ ó(÷ –2) , ñ ó(÷ –1)  . ¨ 

 =  ( , ¨ , ¨2
). The substitution ¨ ( ) = (  −2) leads to a second-order equation: u

ñ ó( ÷

)

© 2003 by Chapman & Hall/CRC

Page 730

731

5.2. NONLINEAR EQUATIONS ( )  5.2.6-7. Equations of the general form    , , 
 ,  ,  = 0.

76.

H  ï , ñ óò , ñ óò ò ó , u uu , ñ ó(÷ ) 

= 0.

explicitly. Hence, the substitution ¨ ( ) =
 leads to an

The equation does not depend on (B − 1)st-order equation:

   , ¨ , ¨ 
,  , ¨ ( 

77.

78.

H  ñ , ñ óò , ñ óò ò ó , u uu , ñ ó(÷ ) 

Autonomous equation. It does not depend on  explicitly. The substitution ¨ ( ) = /
 leads to an (B − 1)st-order equation. The derivatives of the original equation and the transformed one are related by

H  ï , ï ñ óò



 = ¨*¨
, 




 = ¨

2

¨

+ ¨ ( ¨
)2 ,  

ñóò ò ó , ñ óò ò ó ò ó , u uu , ñ ó(÷ )  = 0. The substitution ¨ ( ) =  
 −  leads to  an (B

−1)


 . 

−2)

− 1)st-order equation: 

 = 0,

= ¨ 
 .

where

– 2ñ ,

H  ï , ï ñ óò

– þxñ ,

ñ ó( ú

+1)

The substitution ¨ =  


ñ ó( ú − ¡  ,

+2)

,

H  ï , ï 2 ñ óò ò ó

– 2ï

ñóò



(–1)  =0

, !(

¡

= ¨ 

  .

where

ù

= 0,

≥

þ

+ 1.

leads to an (B − 1)st-order equation:  −1)

ñóò ò ó ò ó , u uu , ñ ó(÷ 
2 

   − 2  + 2

ü ! = ú  ï ú –  ñ ó( ú –  ) = H ( ï , ñ ó( ú

Here, L +  =

 = 0,

+ 2ñ ,

The substitution ¨ ( ) = 

½ú

−3)

uuu , ñ ó(÷ ) 

  , ¨ , , 
,  ,  (  82.

( )

ñ óò ò ó ò ó , ñ óò ò ó ò ò ó ó , u uu , ñ ó(÷ )  = 0. The substitution ¨ =  
−  2  leads to an (B − 1)st-order equation:  HÊ ï , ï ñ óò

   , ¨ , , 
,  ,  (  − + 81.

  = ¨  ( 

 ,

–ñ ,

   , ¨ , , 
,  ,  ( 

80.

 = 0.

= 0.

   , ¨ , , 
,  ,  ( 

79.

−1)

)

 = 0,

= ¨ ( +

where

)

 .

 = 0.

leads to an (B − 2)nd-order equation: 

−3)

 = 0,

+1)

,

where

uuu , ñ ó(÷

)

=

−2

¨ 
.

).

! are binomial coefficients. − , )!

¡ 7+ The substitution ¨ ( ) = (−1)  , ! L +   + 

=0

−

 ( + −  ) leads to an (B − )th-order equation; ¡

the derivatives on the right-hand side are calculated in consecutive manner using the formula ( + +1) =  − +3¨2
. 83.

HwO ï

ñ , ñ óò , ï ñ óò ò ó , u uu , ïƒ÷ –1 ñ ó(÷

)

P

= 0.

Homogeneous equation. The transformation a = ln  , ¨ equation of the form 5.2.6.77.

=  

leads to an autonomous

© 2003 by Chapman & Hall/CRC

Page 731

732 84.

HIGHER-ORDER DIFFERENTIAL EQUATIONS

ô ï  ï

+ õÿñ + ö , ñ ó ò , uuu , ( ô ï + õÿñ + ö )÷ –1 ñ ó(÷ ) = 0. P + ñ + ¨ 1 b . For ý − ˜ƒ = 0 , the substitution  =  +  + ( leads to an autonomous equation of the form 5.2.6.77.

HwO

2 b . For ý − ˜ƒ ≠ 0, the transformation

¨ = − , 0

H =  −  0, where 

and

0

0

are the constants determined by the linear algebraic system



0

+

0

+ ( = 0,

ƒ…

0

+ý

0

+ = = 0,

−1

H 

leads to a homogeneous equation of the form 5.2.6.83:

 O 85.

HwO

ô 1ï

+ +

ô 2ï

õ 1ñ õ 2ñ

+ö +ö

1

,

2

¨ H  + u ¨u  ¨ ‘
¨ H , ,  , (  +  H ) ƒ +ý u

ñóò , u uu , (ô 3 ï

Let the following condition hold: ûû 



+

õ 3ñ

1



+ ö 3 )÷ 1

(

ñ ó÷

–1 ( )

−1

¨ (‘ 

)

= 0.

P

= 0.

P

1

( 2 ûû = 0. û 3 3 (3û û û û û For  1  2 −  2  1 ≠ 0, the transformation H =  −  0, ¨ = − 0,

where 

and

0

0



2

2

are the constants determined by the linear algebraic system

 1

0

+

1 0

+(

1

= 0,

 2

0

+

2 0

+(

2

H 

−1

= 0,

leads to a homogeneous equation of the form 5.2.6.83:

 O 86.

87.

¨ H  1 +  1u ‘
, ¨ ,  , (  3 +  3 ¨u H )   2 +  2 ¨u H

−1

H  ï  ñ , ï +1 ñ óò , u uu , ï +÷ ñ ó(÷ ) 

)

P

= 0. Generalized homogeneous equation. The transformation a = ln  , ¨ autonomous equation of the form 5.2.6.77.

H ”

ï ñ óò ñ

ï 2 ñ óò ò ó ï ÷ ñ ó( ÷ ,  u uu , ñ ñ

,

)

•

Hv”Èñóò

ñ óò ò ó ñ óò

–ñ

,

ñ óò ò ó ñ óò

,

= 0.

=  

leads to an

= 0.

Generalized homogeneous equation. The transformation H =  an (B − 2)nd-order equation. 88.

¨ (‘ 

ñ óò ò ó ò ó ñ ó( ÷ ,  u uu , ñ óò ñ óò

)

•


 , ¨ = 

2



  leads to

= 0.

Autonomous equation. Particular solution: = L 1 exp( L 2  ) + L 3 , where L 1 is an arbitrary constant and the constants L 2 and L 3 are related by  (− L 2 L 3 , L 2 , L 22 ,  , L 2 −1 ) = 0. 89.

H ” ï ñ ú

,

ï ñ óò ñ

,

ï 2 ñ óò ò ó ï ÷ ñ ó( ÷ ,  u uu , ñ ñ

)

•

= 0.

Generalized homogeneous equation. The transformation a =   + , H =  (B − 1)st-order equation.


 leads to an

© 2003 by Chapman & Hall/CRC

Page 732

5.2. NONLINEAR EQUATIONS

90.

Hv”

ñ ó( ÷ ñ óò

)

ñ ó( ÷ ñ óò

ñ

,

)

–ñ

ó( ÷

–1)

733

= 0.

•

A solution of this equation is any function that satisfies the (B − 1)st-order constant coefficient linear equation (  −1) = L 1 + L 2 , where the constants L 1 and L 2 are related by the constraint  ( L 1 , − L 2 ) = 0. 91.

Hv”

ñ ó(÷ ) ï 1– ñ , ñ ñ ó(  ) ñ

ó(÷ ) ï 1– (÷ – ) – ñó • ó(  )

ù

= 0,

>ü .

A solution of this equation is any function that satisfies the (B − , )th-order linear equation (  −  ) = L 1 + L 2   −1 , where the constants L 1 and L 2 are related by  ( L 1 , − L 2 ) = 0. 92. 93.

H  ï , ñ ó( ÷

– ñ , ñ ó(ú ) – ñ  = 0. The substitution ¨ = 
− reduces the order of the equation by one.

H  ñ ó( ÷

)

– ñ ,ñ

ó(2÷

–ñ ,ñ

ó(2÷

ó( ÷

)

ñ , ñ ó(2÷

)

–ñ

 = 0.  The substitution ‡ =  − leads to an B th-order autonomous equation   ‡ , ‡ (  )+ ‡ , ‡ (  )  = 0. )

)

)

( )

94.

H  ï , ñ ó( ÷

)

+

ôñ , ñ ó(2÷

)

–ô

The substitution ‡ =   +  ( )

95.

96.

2

+

ôñ ó(÷

)

 = 0.

 )  = 0. leads to an B th-order equation    , ‡ , ‡ (  ) − ‡ , ‡ (v

ó ó ó ó HÊp ^ ñ ,  ^ ñ óò ,  ^ ñ óò ò ó , u uu ,  ^ ñ ó(÷ ) 

= 0.  Equation invariant under “translation–dilatation” transformation. The substitution ‡ =  9 leads to an autonomous equation of the form 5.2.6.77.

ó Hv”- ^ ñ ú

,

ñ óò ñ

,

ñ óò ò ó ñ ó( ÷ ,  u uu , ñ ñ

)

•

= 0.

Equation invariant under “translation–dilatation” transformation. The transformation  H =  9 + , ¨ = 
  leads to an (B − 1)st-order equation. See also Paragraph 0.5.2-7. 97.

H  ïƒú  ^ ] , ï ñ óò , ï 2 ñ óò ò ó , u uu , ƒï ÷ ñ ó(÷ ) 

= 0. Equation invariant under “dilatation–translation” transformation. The transformation H =  +  9  , ¨ =  
leads to an (B − 1)st-order equation. See also Paragraph 0.5.2-8.

© 2003 by Chapman & Hall/CRC

Page 733

Supplements S.1. Elementary Functions and Their Properties ]

Throughout Section S.1 it is assumed that B is a positive integer unless otherwise specified.

S.1.1. Trigonometric Functions S.1.1-1. Simplest relations. sin2  + cos2  = 1, sin(− ) = − sin  , sin  , tan  = cos  tan(− ) = − tan  , 1 , 1 + tan2  = cos2 

tan  cot  = 1, cos(− ) = cos  , cos  cot  = , sin  cot(− ) = − cot  , 1 1 + cot2  = . sin2 

S.1.1-2. Relations between trigonometric functions of single argument. tan  1 sin  = A £ 1 − cos2  = A £ =A £ , 2 1 + tan  1 + cot2  1 cot  =A £ , cos  = A £ 1 − sin2  = A £ 2 1 + tan  1 + cot2  £ 1 − cos2  1 sin  =A = , tan  = A £ 2 cos  cot  1 − sin  cot  = A

£ 1 − sin2  1 cos  =A £ = . 2 sin  tan  1 − cos 

S.1.1-3. Reduction formulas. ) = (−1)  sin  , 2B + 1 = A (−1)  cos  , sin OuA 2  P tan(pAIB ) = tan  ,  2B + 1 = − cot  , tan OpA 2  P

) = (−1)  cos  , 2B + 1 = T (−1)  sin  , cos OpA 2  P cot(pAIB ) = cot  ,  2B + 1 = − tan  , cot OpA 2  P

sin OuA}

cos OpA}

sin(uACB



4P

=

£ 2

(sin pA cos  ),

2 tan pA 1 = , tan OpA} 4P 1 T tan 

cos(pAIB



£ 2

(cos uT sin  ), 2 cot uT 1 = . cot OpA} 4P 1 A cot  4P

=

© 2003 by Chapman & Hall/CRC

Page 735

736

SUPPLEMENTS

S.1.1-4. Addition and subtraction of trigonometric functions. sin  + sin = 2 sin O sin  cos  cos  sin2 

 +

cos O

 −

, 2 P 2 P  −  + − sin = 2 sin O cos O , 2 P 2 P  +  − + cos = 2 cos O cos O , 2 P 2 P  +  − − cos = −2 sin O sin O , 2 P 2 P − sin2 = cos2 − cos2  = sin( + ) sin( − ),

sin2  − cos2

= − cos( + ) cos( − ), sin(pA ) sin( AI ) , cot pA cot = , tan pA tan = cos  cos sin  sin  cos  +  sin  = & sin( + Y ) = & cos( − j ).

Here, & = £ 

2

+  2 , sin Y =  & , cos Y =  & , sin j =  & , and cos j =  & .

S.1.1-5. Products of trigonometric functions. sin  sin = 12 [cos( − ) − cos( + )],

cos  cos = 12 [cos( − ) + cos( + )],

sin  cos = 12 [sin( − ) + sin( + )]. S.1.1-6. Powers of trigonometric functions. cos2  =

1 2

cos 2 + 12 ,

cos3  =

1 4

cos 3 +

3 4

cos  ,

sin3  = − 14 sin 3 +

cos4  =

1 8

cos 4 +

1 2

cos 2 + 38 ,

sin4  =

1 8

cos5  =

1 16

sin5  =

1 16

cos 5 +

5 16

cos2   =

sin2  = − 12 cos 2 + 12 ,

cos 3 + 1



22 −1

½

5 8



−1 =0

cos  ,

cos 4 − sin 5 −

3 4 1 2

sin  ,

cos 2 + 38 ,

5 16

sin 3 +

5 8

sin  ,

1 L 2  cos[2(B − , ) ] + 2  L 2  , 2

 1 ½ L 2  +1 cos[(2B − 2 , + 1) ], cos2  +1  = 2  2  =0  −1 1 ½ 1 2 (−1)  −  L 2  cos[2(B − , ) ] + 2  L 2  , sin  = 2  −1 2 2  =0  ½ 1 (−1)  −  L 2  +1 sin[(2B − 2 , + 1) ]. sin2  +1  = 2  2  =0 Here, L +  =

!

are binomial coefficients (0! = 1). ¡ , ! ( − , )! ¡

© 2003 by Chapman & Hall/CRC

Page 736

737

S.1. ELEMENTARY FUNCTIONS AND THEIR PROPERTIES

S.1.1-7. Addition formulas. sin(uA

cos(uA

) = sin  cos A cos  sin , tan uA tan )= , 1 T tan  tan

tan(pA

) = cos  cos T sin  sin , 1 T tan  tan )= . tan pA tan

cot(uA

S.1.1-8. Trigonometric functions of multiple arguments. cos 2 = 2 cos2  − 1 = 1 − 2 sin2  ,

sin 2 = 2 sin  cos  ,

cos 3 = −3 cos  + 4 cos3  ,

sin 3 = 3 sin  − 4 sin3  ,

cos 4 = 1 − 8 cos2  + 8 cos4  ,

sin 4 = 4 cos  (sin  − 2 sin3  ),

cos 5 = 5 cos  − 20 cos3  + 16 cos5  ,

sin 5 = 5 sin  − 20 sin3  + 16 sin5  ,

cos(2B/ ) = 1 +

½  

(−1) 

B 2 (B

=1

cos[(2B +1) ] = cos 

™

1+

½ 

(−1)  =1

½

sin(2 B/ ) = 2B cos w¬ sin  +



sin[(2B +1) ] = (2B +1) ™ sin  + tan 2 =

2 tan  , 1 − tan2 

− 1)  [B 2 − ( , − 1)2 ]  4 sin2   , (2 , )!

2

[(2B +1)2 −1][(2B +1)2 −32 ]  [(2B +1)2 −(2 , −1)2] sin2  ˆ¢ , (2 , )!

(−4) 

(B

(−1) 

[(2B +1)2−1][(2B +1)2−32]  [(2B +1)2−(2 , −1)2] sin2  (2 , +1)!

2

− 1)(B

=1

½  

=1

tan 3 =

− 22 )  (B (2 , − 1)!

2

3 tan  − tan3  , 1 − 3 tan2 

2

tan 4 =

− , 2)

sin2 

−1

 ® , +1

^¢ ,

4 tan  − 4 tan3  . 1 − 6 tan2  + tan4 

S.1.1-9. Trigonometric functions of half argument.



1 − cos  1 + cos   , cos2 = , 2 2 2 sin  sin  1 − cos  1 + cos    = , cot = = , tan = 2 1 + cos  sin  2 1 − cos  sin     2 tan 2 1 − tan2 2 2 tan 2 sin  =  , cos  =  , tan  =  . 2 1 + tan 2 1 + tan2 2 1 − tan2 2 sin2

2

=

S.1.1-10. Euler and de Moivre formulas. Relationship with hyperbolic functions.

   + © =   (cos  + ™ sin  ), (cos  + ™ sin  )  = cos(B/ ) + ™ sin(B/ ), ™ 2 = −1, sin(™Ý ) = ™ sinh  , cos(™Ý ) = cosh  , tan(™Ý ) = ™ tanh  , cot(ݙ  ) = −™ coth  . S.1.1-11. Differentiation formulas.

M

sin 

M



= cos  ,

M

cos 

M



= − sin  ,

M

tan 

M



=

1 , cos2 

M

cot 

M



=−

1 . sin2 

© 2003 by Chapman & Hall/CRC

Page 737

738

SUPPLEMENTS

S.1.1-12. Expansion into power series.



cos  = 1 − sin  =  −



2



+

2! 3



4



4!

−

5



6

6!

+ )))

(| | < F ),

7

+ ))) 5! 7!  3 2 5 17 7 + + + ))) tan  =  + 3 15 315 1   3 2 5 − − ))) cot  = − −  3 45 945 +

3!

−

(| | < F ), (| | < 2),



(| | < ).



S.1.2. Hyperbolic Functions S.1.2-1. Definitions. sinh  =





− 2

−

 ,

cosh  =





+ 2

−

 ,





− − tanh  =   ,  + −







+ − coth  =   .  − −



S.1.2-2. Simplest relations. cosh2  − sinh2  = 1, sinh(− ) = − sinh  , sinh  , tanh  = cosh  tanh(− ) = − tanh  , 1 , 1 − tanh2  = cosh2 

tanh  coth  = 1, cosh(− ) = cosh  , cosh  , coth  = sinh  coth(− ) = − coth  , 1 . coth2  − 1 = sinh2 

S.1.2-3. Relations between hyperbolic functions of single argument ( ≥ 0). tanh 

sinh  = ü cosh2  − 1 = cosh  = ü sinh2  + 1 =

ü coth  = ü

ü 1 − tanh  1 ü

1 − tanh2 

1

= =

2 ü coth  − 1 coth 

ü

coth2  − 1

, ,

1 cosh2  − 1 = ü = , 2 cosh  coth  sinh  + 1 sinh 

tanh  =

2

sinh2  + 1 = sinh 

cosh  2

ü cosh  − 1

=

1 . tanh 

S.1.2-4. Addition formulas. sinh(uA

) = sinh  cosh A sinh cosh  , tanh pA tanh , tanh(uA ) = 1 A tanh  tanh

cosh(pA coth(uA

) = cosh  cosh A sinh  sinh , coth  coth A 1 )= . coth A coth 

© 2003 by Chapman & Hall/CRC

Page 738

739

S.1. ELEMENTARY FUNCTIONS AND THEIR PROPERTIES

S.1.2-5. Addition and subtraction of hyperbolic functions. sinh uA sinh = 2 sinh O

pA

uT

cosh O 2 P 2 P  +  − cosh O cosh  + cosh = 2 cosh O 2 P 2  +  − sinh O cosh  − cosh = 2 sinh O 2 P 2 P 2 2 2 2 sinh  − sinh = cosh  − cosh = sinh( 2

, ,

P ,

+ ) sinh( − ),

2

sinh  + cosh

= cosh( + ) cosh( − ), sinh(uA ) sinh(pA ) , coth pA coth = A . tanh pA tanh = cosh  cosh sinh  sinh

S.1.2-6. Products of hyperbolic functions. sinh  sinh = 12 [cosh( + ) − cosh( − )],

cosh  cosh = 12 [cosh( + ) + cosh( − )],

sinh  cosh = 12 [sinh( + ) + sinh( − )]. S.1.2-7. Powers of hyperbolic functions. cosh2  = cosh3  = cosh4  = cosh5  =

Here, L + 

sinh2  =

1 2 cosh 2 1 4 cosh 3 1 8 cosh 4 1 16 cosh 5

 + 12 ,

1 2 cosh 2 1 4 sinh 3 1 8 cosh 4 1 16 sinh 5

 − 12 ,

 + 34 cosh  ,  − 34 sinh  , sinh3  =  + 12 cosh 2 + 38 ,  − 12 cosh 2 + 38 , sinh4  =  + 165 cosh 3 + 58 cosh  , sinh5  =  − 165 sinh 3 + 58 sinh  ,  −1 1 ½ 1 L 2  cosh[2(B − , ) ] + 2  L 2  , cosh2   = 2  −1 2 2  =0  ½ 1 L 2  +1 cosh[(2B − 2 , + 1) ], cosh2  +1  = 2  2  =0  −1 ½ 1 (−1)  2 (−1)  L 2  cosh[2(B − , ) ] + 2  L 2  , sinh  = 2  −1 2 2  =0  1 ½ (−1)  L 2  +1 sinh[(2B − 2 , + 1) ]. sinh2  +1  = 2  2  =0

are binomial coefficients.

S.1.2-8. Hyperbolic functions of multiple arguments. cosh 2 = 2 cosh2  − 1,

sinh 2 = 2 sinh  cosh  ,

cosh 3 = −3 cosh  + 4 cosh  ,

sinh 3 = 3 sinh  + 4 sinh3  ,

cosh 4 = 1 − 8 cosh2  + 8 cosh4  ,

sinh 4 = 4 cosh  (sinh  + 2 sinh3  ),

cosh 5 = 5 cosh  − 20 cosh3  + 16 cosh5  ,

sinh 5 = 5 sinh  + 20 sinh3  + 16 sinh5  .

3

© 2003 by Chapman & Hall/CRC

Page 739

740

SUPPLEMENTS cosh(B/ ) = 2 

−1

[ ´J B ½

cosh   +

2



½

=0

J

[( −1) 2]

sinh(B/ ) = sinh  Here, L + 



(−1)  +1  −2  L 2 , + 1  −  −2

2]

2 − 

−1

=0

L   −



(cosh  ) 

 )

−2 −1

−2 −2

−1 (cosh





−2 −2

,

.

are binomial coefficients and [# ] stands for the integer part of a number # .

S.1.2-9. Relationship with trigonometric functions. sinh(™Ý ) = ™ sin  ,

cosh(™Ý ) = cos  ,

tanh(™Ý ) = ™ tan  ,

coth(™Ý ) = −™ cot  ,

™ 2 = −1.

S.1.2-10. Differentiation formulas.

M

sinh 

M



= cosh  ,

M

cosh 

M



= sinh  ,

tanh 

M

M

=



1 , cosh2 

M

coth 

M



=−

1 . sinh2 

S.1.2-11. Expansion into power series.



cosh  = 1 + sinh  =  +



2

2! 3



+



4

4! 5



+



6

6!

+ )))

7

(| | < F ),

+ ))) 5! 7!  3 2 5 17 7 + − + ))) tanh  =  − 3 15 315 1   3 2 5 + − ))) coth  = + −  3 45 945 3!

+

+

(| | < F ), (| | < 2),



(| | < ).



S.1.3. Inverse Trigonometric Functions S.1.3-1. Definitions and some properties. sin(arcsin  ) =  , tan(arctan  ) =  ,

cos(arccos  ) =  , cot(arccot  ) =  .

Principal values of inverse trigonometric functions are defined by the inequalities: − z2 ≤ arcsin  ≤ − z2 < arctan 
−1.

S.1.3-5. Differentiation formulas. 1

1

, M arcsin  = £  1− 2 M

, M arccos  = − £  1− 2 M

M

M

1

, M arctan  =  1+ 2

1

. M arccot  = −  1+ 2

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SUPPLEMENTS

S.1.3-6. Expansion into power series. arcsin  =  + arctan  =  −

1  3 1×3  5 1×3×5  7 + + + ))) 2 3 2×4 5 2×4×6 7



3



5



(| | < 1),

7

+ ))) (| | ≤ 1), 5 7 1 1 (| | > 1). arctan  =  − + 3 − 5 + ))) 2  3 5 The expansions for arccos  and arccot  can be obtained with the aid of the formulas arccos  = z2 − arcsin  and arccot  = z2 − arctan  . 3 1

+

−

S.1.4. Inverse Hyperbolic Functions S.1.4-1. Relationships with logarithmic functions. arcsinh  = ln   + £ 

2

+1 ,

arccosh  = ln  + £ 

2

−1 ,

1 1+ ln , 2 1− 1 1+ , arccoth  = ln 2  −1 arctanh(− ) = −arctanh  , arccoth(− ) = −arccoth  . arctanh  =

arcsinh(− ) = −arcsinh  , arccosh(− ) = arccosh  ,

S.1.4-2. Relations between inverse hyperbolic functions. arcsinh  = arccosh £ 

2

+ 1 = arctanh £ 

arccosh  = arcsinh £ 

2

− 1 = arctanh



arctanh  = arcsinh £ 1−

2



+1 £  2−1 2



1 = arccosh £ 1−

, ,

2

= arccoth



1

.

S.1.4-3. Addition and subtraction of inverse hyperbolic functions.

£ 1+

, V ( 2 − 1)( 2 − 1) W , arccosh pA arccosh = arccosh  A ü ( 2 + 1)( 2 − 1) W , arcsinh pA arccosh = arcsinh V  A ü pA  A 1 , arctanh pA arccoth = arctanh . arctanh pA arctanh = arctanh 1 AC A  I arcsinh pA arcsinh = arcsinh  

ü

1+

2

A

2

S.1.4-4. Differentiation formulas.

M arcsinh  = £   M M

1 2

1

M arctanh  =  1−

+1 2

M arccosh  = £   M

, (

2

< 1),

M

1 2

1

M arccoth  =  1−

−1 2

, (

2

> 1).

© 2003 by Chapman & Hall/CRC

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S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

S.1.4-5. Expansion into power series. arcsinh  =  − arctanh  =  +

^_

1  3 1×3  5 1×3×5  7 + − + ))) 2 3 2×4 5 2×4×6 7



3

3

+



5

5

+



7

7

+ )))

(| | < 1), (| | < 1).

References for Section S.1: H. B. Dwight (1961), M. Abramowitz and I. A. Stegun (1964), G. A. Korn and T. M. Korn (1968), W. H. Beyer (1991).

S.2. Special Functions and Their Properties ]

Throughout Section S.2 it is assumed that B is a positive integer unless otherwise specified.

S.2.1. Some Symbols and Coefficients S.2.1-1. Factorials. Definitions and some properties: 0! = 1! = 1, B ! = 1 × 2 × 3  (B − 1)B , (2B )!! = 2 × 4 × 6  (2B − 2)(2B ) = 2  B !,

B = 2, 3,  ,

2 (2B + 1)!! = 1 × 3 × 5  (2B − 1)(2B + 1) = £

B !! =

™

(2 , )!! if B = 2 , , (2 , + 1)!! if B = 2 , + 1,

+1



w

OB +

3 , 2P

0!! = 1.

S.2.1-2. Binomial coefficients. Definition:

B ! , where , = 1,  , B , L  = , !(B − , )! (−  )   (  − 1)   (  − , + 1) = , L   = (−1)  , ! , !

where

, = 1, 2, 

General case: (  + 1) , L  = w (  + 1) (  −  + 1) Properties:

w

w

where

w

( ) is the gamma function.

L   = 0 for , = −1, −2,  or , > B ,   − +1 +1 +1   =  + 1 L   −1 =  + 1 L   , L   + L   = L   +1 , (−1)   (2B − 1)!!  , L 2  = (−1)  −1 J 2 = 22  (2B )!! (−1)  −1  −1 (−1)  −1 (2B − 3)!!  , L =  −2 2 1J 2 = B 22  −1 B (2B − 2)!! 2  +1  −4  −1 L  , L  −2  L 42  +1 , 2  +1 J 2 = (−1) 2 2  +1 J 2 = 2 2 22  +1 ´J 2 = 2 L (  −1) J 2 . 1J 2 = , L      L 2 

L 0 = 1, L L L L L

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SUPPLEMENTS

S.2.1-3. Pochhammer symbol. Definition and some properties ( , = 1, 2,  ): ( + B ) (1 −  ) = (−1)  w , ( ) (1 −  − B ) w w (B + , − 1)! , (  )0 = 1, (  )  +  = (  )  (  + B )  , (B )  = (B − 1)! (−1)  ( − B ) = , where  ≠ 1,  , B ; (  )−  = w ( ) (1 −  )  w (2B )! (2B + 1)! , (3 2)  = 2−2  , (1)  = B !, (1 2)  = 2−2  B ! B ! (  ) +  + ´ (  )2  ( )  ( + , ) , ( + B ) = , ( + B )  = . (  + , ) ´ = ¡ ( )+  ( ) ( ) (  )  =  (  + 1)  ( + B − 1) =

w

S.2.2. Error Functions and Exponential Integral S.2.2-1. Error function and complementary error function. Definitions: erf  = £



2

ù



exp(−a 2 ) a ,

erfc  = 1 − erf  = £

0

M

Expansion of erf  into series in powers of  as þD erf  = £

2 ½

 

÷ (−1)  =0

2

ùÈ ÷ exp(−a 2 ) M a .



0:

½ 2  2  +1 2   2  +1 = £ . exp  − 2  ÷ , ! (2 , + 1) (2 , + 1)!! =0  

Asymptotic expansion of erfc  as þDGF : erfc  = £

1

exp  − 2  ¬ 6



½

−1

(−1)

+

=0

1 +  2  + + [þ | |−2 6  2 + +1

−1

® , 8

= 1, 2, 

S.2.2-2. Exponential integral. Definition: Ei( ) = ù

 −

 g a aRM

−

÷

Ei( ) = lim ª ” ù +0

Other integral representations:



−

÷

ª

 g  g  a +ù a aÊM a M • ª R

 sin a + a cos a a  2+a 2 M 0   sin a − a cos a a Ei(− ) =  − ùÈ÷  2+a 2 M 0 g Ei(− ) = − ù ÷  − ln a a M 1

Ei(− ) = − 

−

ùÈ÷

for

 < 0,

for

 > 0.

for

 > 0,

for

 < 0,

for

 > 0.

© 2003 by Chapman & Hall/CRC

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S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

Expansion into series in powers of  as þD






Ei(


 z ) =





0:

 

½

+ ln(− ) + ÷



½

z

745

+ ln  + ÷



if  < 0,

, ×, !   , ×, !

=1

=1

if  > 0,

where z = 0.5772  is the Euler constant. Asymptotic expansion as EDGF : Ei(− ) = 

−

 ½ 

( , − 1)!

(−1)  =1

B ! ²  <  . 

+²  ,

 

S.2.2-3. Logarithmic integral.





Definition: li( ) =




 ù

M

a

= Ei(ln  )

ln a

0

+0 ” ª lim ù

1− ª

M

0

For small  ,

a

ln a li( ) ≈

Asymptotic expansion as ED

if 0 <  < 1,



+ù

a

M

1+ ª

if  > 1.

ln a•



ln(1  )

.

1:

½

li( ) = z + ln |ln  | + ÷



=1

ln   . , ×, !

S.2.3. Gamma and Beta Functions S.2.3-1. Gamma function. The gamma function, ( H ), is an analytic function of the complex argument H everywhere, except w for the points H = 0, −1, −2,  For Re H > 0,

w

(H ) = ù ÷ a

‘

−1 −

0



g a. M

For −(B + 1) < Re H < −B , where B = 0, 1, 2,  ,

g ½ (H ) = ù ÷ ¬  − w 0 + −

Euler formula:

 =0

‘ (−1) + ®a ! ¡

‘ B !B ( H ) = lim  w H ( H + 1)  ( H + B ) ÷

−1

M

a.

( H ≠ 0, −1, −2,  ).

Simplest properties:

w

( H + 1) = H

w

(H ),

w

(B + 1) = B !,

w

(1) = (2) = 1.

w

© 2003 by Chapman & Hall/CRC

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SUPPLEMENTS Symmetry formulas: ( H ) (− H ) = −

w

w

Multiple argument formulas:

w



w



w

(3 H ) =

33



sin( H )



,



‘

−1

‘

J

(H ) O H +

w

w

−1 2

2

w



1 , 2P 1 3P

( H ) OH +

w

(B±H ) = (2 )(1−  ) J 2 B 

 J »

‘

−1 2



w

( H ) (1 − H ) =

w

1 1 O −H =  . +H 'P w 2 P cos( H ) 2

O

22 (2 H ) = £

w

,

H sin( H )

Fractional values of the argument:

−1

w

 =0 w

OH +

2 , 3P

OH +

, . BpP

£ 1 1 OB + =   (2B − 1)!!, =£ , w 2P 2 w 2P  1 1 2 £ O− = −2 £ , O − B = (−1)   . w 2P  w 2 P (2B − 1)!! Asymptotic expansion (Stirling ‘ ‘ formula): O

w

(H ) = £ 2 



−

J V1+

−1 2

H

1 −1 12

H

+

1 −2 288

H

+ [ (H

−3

)W

(|arg | H < ).



S.2.3-2. Logarithmic derivative of the gamma function. Definition:

j (H ) = M Functional relations:

ln ( H )

M

w

H

=


‘ ( H ) . (H )

w w

1

j ( H ) − j (1 + H ) = − , H j ( H ) − j (1 − H ) = − cot( H ),





1

j ( H ) − j (− H ) = − cot( H ) − ,   H jç 12 + H  − jç 12 − H  = tan( H ),  + −1  1 ½ , j ( H ) = ln + j\OH + . ¡ ¡ P =0  ¡ ¡ Integral representations (Re H > 0): ‘ j ( H ) = ùÈ÷ V  −g − (1 + a )− Wa −1 a , M 0 ‘ j ( H ) = ln H + ùÈ÷ V a −1 − (1 −  −g )−1 W  −g a , M 0 ‘ 1 −1 1−a a, j ( H ) = −z + ù 1−a M 0 where z = −j (1) = 0.5772  is the Euler constant. Values for integer argument:

j (1) = −z ,

j ( B ) = −z +

½ 

−1

,

−1

(B = 2, 3,  ).

=1

© 2003 by Chapman & Hall/CRC

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S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

S.2.3-3. Beta function. Definition:

1

$ ( , ) = ù

0

 a

−1

(1 − a ) 

−1

a, M

where Re  > 0 and Re > 0. Relationship with the gamma function:

$ ( , ) = w

( ) ( ) w . ( + )

w

S.2.4. Incomplete Gamma and Beta Functions S.2.4-1. Incomplete gamma function. Definitions (integral representations):

= (ƒ ,  ) = ù



 −g a 9

0

(ƒ ,  ) = È ù  ÷ 

w Recurrence formulas:

Asymptotic expansions as þD

w

−

−1

ga9

a, M

−1

M

Re ƒ > 0,

a = ( ƒ ) − = ( ƒ ,  ).

w

 = ( ƒ + 1,  ) = ƒ±= (ƒ ,  ) −  9  − ,  ( ƒ + 1,  ) = ƒ ( ƒ ,  ) +  9  − .

w

0:

½ (−1)   9 +  , = (ƒ ,  ) = ÷  =0 B ! ( ƒ + B ) ½ (−1)   9 + . (ƒ ,  ) = (ƒ ) − ÷ w w  =0 B ! ( ƒ + B ) Asymptotic expansions as þDGF :

= (ƒ ,  ) = (ƒ ) −  9

−1 −



w

w

(ƒ ,  ) =  9

−1 −





¬ 6

½ +

−1 =0



¬ 6 +

½

−1

(1 − ƒ ) + (− ) +

=0

(1 − ƒ ) + (− ) +

+ [  | |−  ® ,

6

+ [  | |−  ®

 − 32 < arg  < 32  .



6

Integral functions related to the gamma function: erf  = £

1



=2O

1 2 , , 2 P

1

erfc  = £



w

O

1 2 , , 2 P

Ei(− ) = − (0,  ).

w

S.2.4-2. Incomplete beta function. Definition: where Re  > 0 and Re  > 0.

$  ( ,  ) = ù

 0

aK

−1

(1 − a ) &

−1

M

a,

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SUPPLEMENTS

S.2.5. Bessel Functions S.2.5-1. Definitions and basic formulas. The Bessel function of the first kind, ø Á ( ), and the Bessel function of the second kind, n…Á ( ) (also called the Neumann function), are solutions of the Bessel equation



 +  
+ (

2



2

− @ 2) = 0

and are defined by the formulas:

Á ½ (−1)  ( 2) +2  ø Á ( ) cos  @ − ø − Á ( ) . , n Á ( ) = Áø ( ) = ÷ (1) sin @  =0 , ! w ( @ + , + 1)  The formula for n/Á ( ) is valid for @ ≠ 0, A 1, A 2,  (the cases @ ≠ 0, A 1, A 2,  are discussed in

the following). The general solution of the Bessel equation has the form called the cylindrical function. Some relations:

:

M

Á ( ) = L

1

ø

Á ( ) + L 2 n Á ( ) and is

:

2 @ Á ( ) =  [ Á −1 ( ) + Á +1 ( )], : : 1 : @ : Á ( ) = [ Á −1 ( ) − Á +1 ( )] = A Á ( ) − Á³1 1 ( )S , 2 Q 

: 

:

:

M : : : : M [ Á Á ( )] =  Á Á −1 ( ), M [ − Á Á ( )] = − − Á Á +1 ( ),   M  M  1 1 Á Á Á Á ” [ ø Á ( )] =  −  ø Á −  ( ), ” [ − ø Á ( )] = (−1)   − −  ø Á +  ( ), M M  2•  *• M M ø − ( ) = (−1) ø ( ), n − ( ) = (−1) n  ( ), B = 0, 1, 2,  1 2,

S.2.5-2. Bessel functions for @ = A"BçA

ø

J  )=

ø

1 2(

J  )=

2



ø

3 2(

2

J  )=

™





B

ø

2

J  )=

− −1 2 (







2



 J  ) = (−1)

+1 2 (



B

ø



½

 2

 2

P

 J

 B

=0

½

P n



½ ´J

=0

J  )=

2]

=0

where [ # ] is the integer part of a number # .

1

” − cos  − sin  ,  •

(−1)  (B + 2 , + 1)! (2 , + 1)! (B − 2 , − 1)! (2 )2 

J  )= n −



¢

+1

,

(−1)  (B + 2 , )! (2 , )! (B − 2 , )! (2 )2 

−1 2 (

J  ),



2

(−1)  (B + 2 , + 1)! (2 , + 1)! (B − 2 , − 1)! (2 )2 

P  J

(−1)  (B + 2 , )! (2 , )! (B − 2 , )! (2 )2 

2]

[

 2

cos  ,



−3 2 (

=0

[( −1) 2]

− −1 2 (



½ ´J

[

2 P

 ø

,

•

[( −1) 2]

cos  , +1

B

cos O +

− sin O +

n 1 J 2 ( ) = − n 

™

2

J  )=

−1 2 (

sin O −

+ cos O −



sin  − cos 



2

+1 2 (



1

”





ø

sin  ,



where B = 0, 1, 2, 



2



¢

,

sin  ,

 J  ) = (−1) ø 

−1 2 (

+1

J  ),

+1 2 (

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S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

S.2.5-3. Bessel functions for @ = A"B , where B = 0, 1, 2,  Let @ = B be an arbitrary integer. The relations ø − ( ) = (−1) ø ( ), n − ( ) = (−1) n  ( ) are valid. The function ø! ( ) is given by the first formula in (1) with @ = B , and n  ( ) can be obtained from the second formula in (1) by proceeding to the limit @\D B . For nonnegative B , the function n  ( ) can be represented in the form:

n  ( ) =



2



ø ( ) ln

 1 ½

−

2



−1



=0

(B − , − 1)! 2  O , ! 3P

where j (1) = −z , j (B ) = −z +

7

−1



=1

,

−1



1 ½

−





÷ (−1)  O =0





+2

2P

 j ( , + 1) + j (B + , + 1) , , ! (B + , )!

= 0.5772  is the Euler constant, j ( ) = [ln ( )]
 is

z

,

−2

w

the logarithmic derivative of the gamma function. S.2.5-4. Wronskians and similar formulas.

ú

( ø Á , ø −Á ) = −

2



ú

sin( @ ),





2 sin( @ )  ) + ø − Á ( ) ø Á −1 ( ) =  ,  ú

 Here, the notation (  , ) =   −   is used.

ø

Á ( ) ø − Á

+1 (

( ø Á , n Á ) =

ø

Á ( ) n Á

2





,

 )− ø Á

+1 (

+1 (

 ) n Á ( ) = −



2



.

S.2.5-5. Integral representations. The functions ø Á ( ) and n Á ( ) can be represented in the form of definite integrals (for  > 0):

 ø

Á ( ) = ù

n Á ( ) = ù



For | @ | < 12 ,  > 0,

z

0

z

0



Á˜g +  − ˜Á g cos @ )  − 

− ù ÷ (



0

Á

w Á

Á

ù ÷ sin( cosh a ) M a , 0  For integer @ = B = 0, 1, 2,  ,

ø ø ø

2





 )=

ù

0

2

 ( ) =

2 +1 (

z

1

( ) =

 

2

ù ù

0

0

z´J 2

2

n 0 ( ) = −

cos(B/a −  sin a ) a

z´J 2

g a. M

cos(a ) a M . (a 2 − 1) Á +1 J 2

w

2

sinh

sin(a ) a M , (a 2 − 1) Á +1 J 2

(Poisson’s formula).

w

 )=

M

0

Á z´J 2 2( 2) Á cos( cos a ) sin2 a a ù 1 1J 2 ( + @ ) M 0 2

 > 0,

0(



21+  − 1 J 2 ( 1 − @ ) ùÈ÷ 1 2



Á ( ) =

ø

− sin @ ù ÷ exp(− sinh a − @!a ) a ,

9

M

Á

Á ( ) =

9

M

21+  − n Á ( ) = − 1 J 2 1 ù ÷ (2 − @ ) 1

For @ > − 21 ,

For @ = 0, 

9

9

)

sin( sin − @ )

ø

ø

9

9

cos( sin − @

M



ù ÷ cos( cosh a ) M a . 0

(Bessel’s formula),

cos( sin a ) cos(2B/a ) a ,

M

sin( sin a ) sin[(2B + 1)a ] a .

M

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SUPPLEMENTS

S.2.5-6. Asymptotic expansions. Asymptotic expansions as | | DGF :

ø

2

Á ( ) =

™





½ 4 − 2 @ − 4  P ¬ 6 +

cos O

2

n Á ( ) =



™



½ 4 − 2 @ − 4  P ¬ 6 +

sin O

)=

+

1

2

(4 @

−1

£ £

2

− 1)(4 @

  ø 2  (   ø 2 +1 (

Asymptotic behavior for large @

ø

Á ( ) ¬

Á (@ ) ¬

J

+

−1

+ [ (| |−2

+ 1)(2 )−2 +

−1

+ [ (| |−2

+ 1)(2 )−2

¡

6

−1

)®ƒ¢ ,

−1

)®ƒ¢ ,

6

¡

− 32 )  [4 @

−1

(−1) + ( @ , 2

=0 2

− (2

¡

) = (−1) 

+1

¡

¡

w

n Á ( ) ¬ −

n Á ( @ ) ¬ − 1 J 3

2

−2

@

O



6

( 21 + @ + ) ¡ . ! ( 21 + @ − )

w

¡

), −2

(cos  − sin  ) + [ (

( @\DGF ):

1 21 J 3 , (2 3) @ 1 J 3

w

− 1)2 ] =

) = (−1)  (cos  + sin  ) + [ (



32 3

+

(−1) ( @ , 2

=0

=0

1   Á O £ 2 @ 2 @lP ,

where  is fixed, and

ø

−1

(−1) + ( @ , 2 )(2 )−2 + + [ (| |−2 )®

¡ ! ¡ For nonnegative integer B and large  , 22

6

¡

=0

½ 4 − 2 @ − 4  P ¬ 6 +

+ cos O where ( @ ,

(−1) + ( @ , 2 )(2 )−2 + + [ (| |−2 )®

½ 4 − 2 @ − 4  P ¬ 6 +

− sin O

−1

 

2 @lP

).

−

Á

,

1 21 J 3 . (2 3) @ 1 J 3

6

w

S.2.5-7. Zeros and orthogonality properties of the Bessel functions. Each of the functions ø Á ( ) and n Á ( ) has infinitely many real zeros (for real @ ). All zeros are simple, except possibly for the point  = 0. The zeros = + of ø 0 ( ), i.e., the roots of the equation ø 0 (= + ) = 0, are approximately given by:

= + = 2.4 + 3.13 ( − 1) ¡

(

¡

= 1, 2,  ),

with a maximum error of 0.2%. Let “ = “ + be positive roots of the Bessel function ø Á (“ ), where @ > −1 and = 1, 2, 3,  ¡ Then the set of functions ø Á (“ + &  ) is orthogonal on the interval 0 ≤ & ≤  with weight & :

ù



0

ø

Á…O

“ + &  P

ø

Á…O

“  & & & = †P M

™

0

2  V ø Á
(“ + )W = 12 

1 2 2

ø

if ¡ Á +1 (“ + ) if

2 2

¡

≠, , =, .

S.2.5-8. Hankel functions (Bessel functions of the third kind). The Hankel functions of the first kind and the second kind are related to the Bessel functions by:

5 Á (1) ( H ) = ø Á ( H ) + ™Ön Á ( H ),

5 Á (2) ( H ) = ø Á ( H ) − ™Ön Á ( H ),

™ 2 = −1.

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751

S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

Asymptotics for HCD

0: 2™

5

(1) 0 (

H )¬

5

(2) 0 (

H )¬ −

 2™

Asymptotics for | H | DGF :

5 Á (H ) ¬ (1)

5 Á (2) ( H ) ¬

ln H ,



2



Q

2

1 2

exp −™  H −

H



5 Á (2) ( H ) ¬

exp ™  H −

H

™

(@ ) w ( H 2) Á  (@ ) ™ w ( H 2) Á

5 Á (1) ( H ) ¬ −

ln H ,

Q



@ −

 1 2



1 4

@ −

S

 1 4



(Re @ > 0), (Re @ > 0).

(− < arg H < 2 ),



S



(−2 < arg H < ).





S.2.6. Modified Bessel Functions S.2.6-1. Definitions. Basic formulas.

The modified Bessel functions of the first kind, ®Á ( ), and the second kind, ; Á ( ) (also called the Macdonald function), of order @ are solutions of the modified Bessel equation

 and are defined by the formulas:

®Á (

½

)= ÷



=0



 +  
− (

2

2

Á ( 2)2  + , , ! ( @ + , + 1)

+ @ 2) = 0

; Á (

w

)=

(see below for ; Á ( ) with @ = 0, 1, 2,  ). The modified Bessel functions possess the properties:

; − Á (

M

) = ; Á ( ); 2 @¦®Á ( ) =  [ ®Á −1 ( ) − ®Á 1 M ®Á ( ) = [ ®Á −1 ( ) + ®Á  2

® 3 J 2 ( ® ®

® 1 J 2 ( 2

1

)=



2



,

( ) = (−1)  ®  ( ), B = 0, 1, 2,  2 @ ; Á ( ) = − [ ; Á −1 ( ) − ; Á +1 ( )], +1 (  )], 1 M ; Á ( ) = − [ ; Á −1 ( ) + ; Á +1 ( )]. +1 (  )],  2

® −

M

S.2.6-2. Modified Bessel functions for @ = A"BçA

− Á − ®Á 2 ® sin  @

sinh  ,

1 2,

where B = 0, 1, 2, 

® −1 J 2 (

)=



2



2

cosh  , 1

” − sinh  + cosh  , ® −3 J 2 ( ) = ” − cosh  + sinh  ,    • •     ½  1 (−1)  (B + , )! (B + , )!   − ½ ¬ ® , − (−1)  +1 J 2 ( ) = £  , ! (B − , )! (2 ) , ! (B − , )! (2 )  2   =0 =0     ½  1 (−1)  (B + , )! (B + , )!   − ½ ¬ ® , + (−1) −  −1 J 2 (  ) = £  , ! (B − , )! (2 ) , ! (B − , )! (2 )  2   =0 =0  

 1 − ; 1 1 J 2 ( ) = 2   , ; 1 3 J 2 ( ) = 2  O 1 + 3P  − ,

 ½ (B + , )! ;  +1 J 2 ( ) = ; − −1 J 2( ) = 2   −  .  =0 , ! (B − , )! (2 ) )=



© 2003 by Chapman & Hall/CRC

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S.2.6-3. Modified Bessel functions for @ = B , where B = 0, 1, 2,  If @ = B is a nonnegative integer, then

;| (

) = (−1) 

+1

®



( ) ln

 1½ 2+

+

2

−1



(−1) + O

=0

2

2P

½   1 + (−1)  ÷ O 2 + =0 2 P

+ −  (B − +2

− 1)! !

¡

¡ + j (B + ¡

+ 1) + j ( + 1) ; ¡ ! (B + )!

¡

¡

B = 0, 1, 2,  ,

where j ( H ) is the logarithmic derivative of the gamma function; for B = 0, the first sum is dropped. S.2.6-4. Wronskians and similar formulas.

ú

®Á ( )® − Á +1 ( where

ú

( ®Á , ® − Á ) = −

) − ® − Á ( ) ®Á

−1 (

2

sin( @ ),





 )=−



2 sin( @ )



(  , ) =  
−  
.

 

,

ú

1 ( ®Á , ; Á ) = − ,

®Á

( ) ; Á



+1 (

 ) + ®Á

+1 (

 ) ; Á ( ) = 

1

,

S.2.6-5. Integral representations.

The functions ® Á ( ) and

®Á

( ) =



; Á (

) can be represented in terms of definite integrals:

 Á

J

1 22

ù

Á (@ + 1 ) 2

w

1 −1

exp(−a )(1 − a 2 )

; Á (

) = ùÈ÷ exp(− cosh a ) cosh( @!a ) a

; Á (

)=

0

; Á ( ) =

1 2



1 1 2

sin 

For integer @ = B ,

® ;

M

a



( > 0, @ > − 12 ), ( > 0),

ùÈ÷ cos( sinh a ) cosh( @!a ) M a @  0

( > 0, −1 < @ < 1),

ùÈ÷ sin( sinh a ) sinh( @!a ) M a @  0

( > 0, −1 < @ < 1).

1

( ) = 0(

J

−1 2

M

1

cos 

Á

z



ù

exp( cos a ) cos(B/a ) a

M

0

 ) = ùÈ÷ cos( sinh a ) a = ùÈ÷ M 0 0

(B = 0, 1, 2,  ),

cos(a ) £ a 2+1 M a

( > 0).

S.2.6-6. Asymptotic expansions as þDGF .

®Á





( ) = £ 2 

; Á (

)=



 2

½

™

(−1) +

1+ 6



+

−

The terms of the order of [  



™ −

2

− 1)(4 @

2

½

+

−1

− 32 )  [4 @ ! (8 ) +

2

− (2

¡

=1

(4 @

1+ 6

6

(4 @

=1

2

− 1)(4 @

2

− 32 )  [4 @ ! (8 ) +

¡

2

− (2

¡

¡

− 1)2 ] − 1)2 ]

¢

¢

,

.

 are omitted in the braces.

© 2003 by Chapman & Hall/CRC

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S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

S.2.7. Degenerate Hypergeometric Functions S.2.7-1. Definitions. The Kummer’s series. The degenerate hypergeometric functions ` (  ,  ;  ) and i (  ,  ;  ) are solutions of the degenerate hypergeometric equation:  

 + (  −  ) 
−  = 0. In the case  ≠ 0, −1, −2, −3,  , the function ` (  ,  ;  ) can be represented as Kummer’s series: ½ ( )    , ` ( ,  ;  ) = 1 + ÷ ( ) , !



=1

where (  )  =  (  + 1)  ( + , − 1), (  )0 = 1. Table 14 (see Subsection 2.1.2) presents some special cases where ` of simpler functions.

can be expressed in terms

S.2.7-2. Some transformations and linear relations. Kummer transformations:

 ` (  ,  ;  ) =  ` (  −  ,  ; − ),

i ( ,  ;  ) = 

1−

i

(1 +  −  , 2 −  ;  ).

Linear relations for ` : (  −  ) ` (  − 1,  ;  ) + (2  −  +  ) ` (  ,  ;  ) −  ` (  + 1,  ;  ) = 0,  (  − 1) ` (  ,  − 1;  ) −  (  − 1 +  ) ` (  ,  ;  ) + (  −  )/` (  ,  + 1;  ) = 0, (  −  + 1) ` ( ,  ;  ) −  ` (  + 1,  ;  ) + (  − 1) ` (  ,  − 1;  ) = 0, ` (  ,  ;  ) − ` ( − 1,  ;  ) − /` (  ,  + 1;  ) = 0,  (  +  ) ` (  ,  ;  ) − (  −  )/` (  ,  + 1;  ) − ` ( + 1,  ;  ) = 0, (  − 1 +  ) ` ( ,  ;  ) + (  −  ) ` (  − 1,  ;  ) − (  − 1) ` (  ,  − 1;  ) = 0. Linear relations for i :

i (  − 1,  ;  ) − (2 −  +  ) i (  ,  ;  ) +  ( −  + 1) i (  + 1,  ;  ) = 0, (  −  − 1) i ( ,  − 1;  ) − (  − 1 +  ) i ( ,  ;  ) + /i (  ,  + 1;  ) = 0, i (  ,  ;  ) −  i ( + 1,  ;  ) − i (  ,  − 1;  ) = 0, (  −  ) i (  ,  ;  ) − /i (  ,  + 1;  ) + i (  − 1,  ;  ) = 0, (  +  ) i (  ,  ;  ) +  (  −  − 1) i (  + 1,  ;  ) − /i (  ,  + 1;  ) = 0, (  − 1 +  ) i ( ,  ;  ) − i (  − 1,  ;  ) + ( − ( + 1) i (  ,  − 1;  ) = 0. S.2.7-3. Differentiation formulas and Wronskian. Differentiation formulas:

 ( ) ` (  + B ,  + B ;  ), M  ` ( ,  ;  ) =  ( ) M  M  i (  ,  ;  ) = (−1)  (  )  i (  + B ,  + B ;  ). 

M

 M ` (  ,  ;  ) = ` (  + 1,  + 1;  ),   M

M i (  ,  ;  ) = − i ( + 1,  + 1;  ), 

Wronskian:

ú

M

( ) ( ` , i ) = `‹i
 − `
 i = − w  ( )

w

−





.

© 2003 by Chapman & Hall/CRC

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SUPPLEMENTS

S.2.7-4. Degenerate hypergeometric functions for B = 0, 1, 2,  (−1)  −1 ™ ` ( , B +1;  ) ln  B ! ( − B )

i (  , B + 1;  ) =

w

½

+ ÷

Ÿ =0

Ÿ

( )Ÿ V j (  + & ) − j (1 + & ) − j (1 + B + & )W  (B + 1) Ÿ &!

¢

+

 (B − 1)! ½ ( ) Ÿ

w

−1 =0

Ÿ ( − B )Ÿ  − , (1 − B ) Ÿ & !

where B = 0, 1, 2,  (the last sum is dropped for B = 0), j ( H ) = [ln ( H )]
‘ is the logarithmic w derivative of the gamma function,

j (1) = −z ,

j ( B ) = −z +

½ 

−1

−1

,

,

=1

where z = 0.5772  is the Euler constant. If  < 0, then the formula 1−

i ( ,  ;  ) = 

 i (

−  + 1, 2 −  ;  )

is valid for any  . For  ≠ 0, −1, −2, −3,  , the general solution of the degenerate hypergeometric equation can be represented in the form: = L 1 ` (  ,  ;  ) + L 2 i (  ,  ;  ), and for  = 0, −1, −2, −3,  , in the form: =

1−

 V L 1 ` (

−  + 1, 2 −  ;  ) + L

2

i (  −  + 1, 2 −  ;  )W .

S.2.7-5. Integral representations.

` ( ,  ;  ) = i ( ,  ;  ) = where

w

1  ( )  g a  −1 (1 − a )  −  ù ( ) (  −  ) 0 w  1  − g a  −1 (1 + a )  −  −1 a È ù ÷ ( ) 0 M

w

w w

−1

M

a

(for  >  > 0), (for  > 0,  > 0),

(  ) is the gamma function.

S.2.7-6. Asymptotic expansion as | | DGF .

` ( ,  ;  ) = w

w

` ( ,  ;  ) =

w

( )     ( )

/

−1

/ ½ 

=0

½ ( ) (− )−  ¬ ( −  )

½ i ( ,  ;  ) =   ¬ −

 ¬

w

/

−

where = [  

−



(−1)  =0



/

(  −  )  (1 −  )  −   + ® , B !

=0

 > 0,

(  )  (  −  + 1)  (− )−  + ® , B !

(  )  (  −  + 1)  −   + ® , B !

−F

 < 0,

0, the hypergeometric function can be expressed in terms of a definite integral:

 (ƒ , ý , = ;  ) =

w

(= ) ù ( ý ) (= − ý )

w

w

1 0

a:

−1

(1 − a ); −:

−1

(1 − aÖ )− 9

M

a,

where (ý ) is the gamma function. w See M. Abramowitz and I. Stegun (1964) and H. Bateman and A. Erd e´ lyi (1953, Vol. 1) for more detailed information about hypergeometric functions.

© 2003 by Chapman & Hall/CRC

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S.2.9. Legendre Functions and Legendre Polynomials S.2.9-1. Definitions. Basic formulas.

.

The associated Legendre functions Á  ( H ) and Æ Á ( H ) of the first and the second kind are linearly independent solutions of the Legendre equation: (1 − H 2 )

‘'

‘ − 2 H ‘
+ [ @ ( @ + 1) − “ 2 (1 − H 2 )−1 ] = 0,

where the parameters @ and “ and the variable H can assume arbitrary real or complex values. For |1 − H | < 2, the formulas

.  Á Æ Á ( H

H −1 ) = #þO H +1P



(H ) = 2

w

1 H + 1 žJ 2 1−H O  O −@ , 1 + @ , 1 − “ ,  , (1 − “ ) H − 1 P 2 P

# =  © z w



1−H H +1 2 1−H + $ÊO O − @ , 1 + @ , 1 − “ , , 2 P H −1P 2 P (−“ ) (1 + @ + “ ) (“ ) © w , $ =  z w , ™ 2 = −1, 2 (1 + @ − “ ) 2

O − @ , 1 + @ , 1 + “ ,

w

are valid, where  (  ,  , ( ; H ) is the hypergeometric series (see Subsection S.2.8). For | H | > 1,

.  Á

2− (H ) = £

Á

−1

§w

(− 21 − @ ) − Á H (− @ − “ )

w



+ −1

(H

2

− 1)− žJ 2   O

2+@ −“ 2@ + 3 1 1+@ −“ , , , 2 2 2 2 H •

Á (1 + @ ) 1 @ + “ 1 − @ − “ 1 − 2@ w 2 , , , 2 , H Á +  ( H 2 − 1)− žJ 2 O − 2 2 2 H • §w (1 + @ − “ ) £ 1 + 2 1 ( @ + “ + 1) 2 + @ + “ @ + “ @ + 3 , , , 2 . Æ Á ( H ) =  © zf oÁ +1 w H − Á −  −1 ( H 2 − 1) žJ 2 O 3 2 2 2 H • 2 (@ + 2 ) 2

+ £

w

. The functions

.

Á ( H ) ≡ Á 0 ( H ) and ÆIÁ ( H ) ≡ Æ 0Á ( H ) are called the Legendre functions. The modified associated Legendre functions, on the cut H =  , −1 <  < 1, of the real axis, are

defined by the formulas:

.  Á

V  © z . Á  ( + ™ 0) +  − 21 © z . Á  ( − ™ 0)W , 1 1 Æ Á ( ) =  © z·V  − 2 ©  z Æ Á ( + ™ 0) +  2 © z Æ Á ( − ™ 0)W . ( ) =

1 1 2 2 1 − 2

S.2.9-2. Trigonometric expansions. For −1 <  < 1, the modified associated Legendre functions can be represented in the form of trigonometric series as:

.  Á

9 2 (cos ) = £ 9

Æ Á (cos ) = £ where 0
0, Á ù M w ( @ +( @ 1)+ B +0 1) z Á Á ( H 2 − 1)− ´J 2 ù ( H + cos a )  − −1 (sin a )2 +1 a , Æ Á ( H ) = (−1)  w Á +1 2 ( @ + 1) M 0 w Note that H ≠  , −1 <  < 1, in the latter formula.

Re @ > −1.

S.2.9-5. Legendre polynomials. . . The Legendre polynomials  =  ( ) and the Legendre functions Æ  ( ) are solutions of the linear differential equation:


(1 −  2 )  − 2  + B (B + 1) = 0. . The Legendre polynomials  ( ) and the Legendre functions Æ  ( ) are defined by the formulas: . ( ) = 1 M  ( 2 − 1) , Æ ( ) = 1 . ( ) ln 1 +  − ½  1 . + ( ). + ( ).  −1    − B ! 2   2 1− +

=1 ¡ M .  =  ( ) can be calculated recursively using the relations: . ( ) = 1, . ( ) =  , . ( ) = 1 (3 2 − 1),  , . ( ) = 2B + 1  . ( ) − B . 0 1 2  +1 2 B +1  B +1  The first three functions Æ  = Æ  ( ) are given by: 1+ 1 1+ 3 2 − 1 1 +  3  ln Æ 0 ( ) = ln , Æ 1 ( ) = ln − 1, Æ 2 ( ) = −  . 2 .1 −  2 1− 4 1− 2 The polynomials  ( ) have the explicit representation: . ( ) = 2− [½ ´J 2](−1)+ L + L   −2 + ,   2  −2 +  + =0 where [ # ] is the integer part of a number # .

The polynomials

.

−1 (

 ).

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S.2.9-6. Zeros of the Legendre polynomials and the generating function.

.

All zeros of  ( ) are real and lie on the interval −1 <  < +1; the functions orthogonal system on the interval −1 ≤  ≤ +1, with

ù

+1

.

 (

−1

a

. )

+ ( )  = M

0

2 2B + 1

if B ≠ if B =

.

 ( ) form an

,

¡ ¡

.

The generating function is:

.

½

1

£ 1 − 2 °˜ + °

2

= ÷



=0

 ( ) ° 

(| ° | < 1).

S.2.9-7. Associated Legendre functions. The associated Legendre functions

.

+  ( ) of order ¡ are defined by the formulas:

+ . B = 1, 2, 3,  , = 0, 1, 2,  M +  ( ), ¡  . M . It is assumed by definition that  0 ( ) =  ( ). . + The functions  ( ) form an orthogonal system on the interval −1 ≤  ≤ +1, with +1 . + ( ). + ( )  =   0 2 (B + )! if B ≠ , , ù   ¡ if B = , . M −1 2B + 1 (B − )! ¡ . ≠ 0) are orthogonal on the interval −1 ≤  ≤ +1 with weight The functions  + ( ) (with ¡ (1 −  2 )−1 , that is, . if B ≠ , ,  0 +1 . + ( )  + ( )  B + )! (  = ù if B = , . ¡ (1 −  2 ) M −1 (B − )! ¡ ¡

.

+ 2 + J  ( ) = (1 −  )

2

S.2.10. Parabolic Cylinder Functions S.2.10-1. Definitions. Basic formulas. The Weber parabolic cylinder function c‚Á ( H ) is a solution of the linear differential equation:

‘'

‘ +  − 1 H 2 + @ + 1  4 2

= 0,

where the parameter @ and the variable H can assume arbitrary real or complex values. Another linearly independent solution of this equation is the function c − Á −1 (™ÖH ); if @ is noninteger, then c\Á (− H ) can also be taken as a linearly independent solution. The parabolic cylinder functions can be expressed in terms of degenerate hypergeometric functions as:

c\Á ( H ) = 21 J 2 exp  − 14 H 2  ¬

w

1



w 1  2 Á `  − 2Á , 12 , 12 H 2   2 − 2

For nonnegative integer @ = B , we have

+ 2−1 J

2

w

−1 w  2Á H`   −2

1 2

Á − 2 , 32 , 12 H 2  ® .

c  ( H ) = 2− ´J 2 exp  − 41 H 2  5   2−1 J 2 H  , B = 0, 1, 2,  ;  5  ( H ) = (−1)  exp  H 2  M  exp  − H 2  , H M where 5  ( H ) is the Hermite polynomial of order B . © 2003 by Chapman & Hall/CRC

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S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

S.2.10-2. Integral representations.

c\Á ( H ) = c\Á ( H ) =

2

ü

exp 



H  ùÈ÷ a Á exp  − 21 a 2  cos  H´a −

1 2 4

1 Á exp  − 41 H 2  È ù ÷ a− (− @ ) 0

w

1 2

0

−1

exp  − H´a − 12 a 2 

M

@  a M



for Re @ > −1,

a

for Re @ < 0.

S.2.10-3. Asymptotic expansion as | H | DGF .

½ c\Á ( H ) = H Á exp  − 14 H 2  ¬ 

/

Á

=0

(−2)   − 2    B !

1 2

−

Á  2 

1

+ [þ | H |−2 H 2

/

−2

®

for | arg H |
−1) satisfy the linear differential equation  

 + ( ƒ + 1 −  ) 
+ B = 0

© 2003 by Chapman & Hall/CRC

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SUPPLEMENTS

and are defined by the formulas:

0 9 ( 

½  1 −    9  M     +9  −  =  B ! + M

)=



(− ) + . L   +− + 9 ! =0 ¡

The first two polynomials have the form:

0 09

0 19

= 1,

= ƒ +1− .

0 9 ( ) for B ≥ 2, one can use the recurrence formulas: To calculate l 0 9

+1 (

1

 )=

B +1

V (2B + ƒ + 1 −  )0 9 ( ) − (B + ƒ )0 9  

−1 (

 )W .

0 9 ( ) form an orthogonal system on the interval 0 <  < F The functions l ù ÷  9 

−



0

a

0 9 (  ) 0 +9 (  )  M

0

=

w

if B ≠

( ƒ + B + 1) B !

if B =

with weight h9 

−



:

,

¡

¡

.

The generating function is: (1 − ° )− 9

−1

exp O −

½

°˜

0 9 (  ) ° 

= ÷

1 − °±P



S.2.11-2. Chebyshev polynomials.

=0

{

1 b . The Chebyshev polynomials of the first kind

| ° | < 1.

,

{

 =  ( ) satisfy the linear differential equation (1 −  2 ) 

 −  
+ B 2 = 0

and are defined by the formulas:

{

 (−2)  B ! £ 1  ( ) = cos(B arccos  ) = (2B )! 1 −  2 M   V (1 −  2 )  − 2 W M [ ´J 2] B ½ (B − − 1)! + (2 )  −2 + = (−1) ¡ 2 + ! (B − 2 )! =0 ¡ ¡

where [ # ] stands for the integer part of a number # . The first five polynomials are:

{

0

= 1,

{

1

= ,

The recurrence formulas:

{

The functions

{

{

2

2

= 2

{

− 1,

{

3

= 4

3

− 3 ,

{

4

(B = 0, 1, 2,  ),

= 8

4

− 8

2

+ 1.

{

 ) = 2  ( ) −  −1 ( ), B ≥ 2; { { { { 2  ( ) + ( ) =  + + ( ) +  − + ( ), B ≥ . ¡ 

+1 (

 ( ) form an orthogonal system on the interval −1 <  < +1, with a 0 if B ≠ , { +1 {  ( ) + ( )  = 1 ¡ if B = ≠ 0, ù 2 £ 2 M ¡ 1 −  −1 if B = = 0.  ¡

The generating function is:

1 − °˜ 1 − 2 °˜ + °

{

½

2

= ÷



=0

  ( ) °

(| ° | < 1).

© 2003 by Chapman & Hall/CRC

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S.2. SPECIAL FUNCTIONS AND THEIR PROPERTIES

2 b . The Chebyshev polynomials of the second kind ’  = ’  ( ) satisfy the linear differential equation (1 −  2 ) 

 − 3 
+ B (B + 2) = 0 and are defined by the formulas:

’  ( ) =

sin[(B + 1) arccos  ]

£ 1−

2

½ ´J

[

=

+

2]

(−1) +

¡

=0

(B − )! (2 )  ¡ ! (B − 2 )!

−2

+

(B = 0, 1, 2,  ).

¡

The recurrence formulas:

’ 

+1 (

 ) = 2/’  ( ) − ’ 

−1 (

 ),

B ≥ 2.

The generating function is:

½

1

1 − 2 °˜ + °

2

= ÷



=0

’  ( ) ° 

(| ° | < 1).

S.2.11-3. Hermite polynomials. The Hermite polynomial 5  = 5  ( ) satisfies the linear differential equation



 − 2 
+ 2B

=0

and is defined by the formulas: [ ´J 2]  ½ (−1) + B ! (2 )  5  ( ) = (−1)  exp   2  M  exp  − 2  =  ! ( B − 2 )! + =0 ¡ ¡ M

−2

+

(B = 0, 1, 2,  ),

where [ # ] stands for the integer part of a number # . The first five polynomials are:

5

0

= 1,

5

1

= ,

5

2

= 4

2

− 2,

5

3

= 8

3

− 12 ,

5

4

= 16

4

− 48

2

+ 12.

The recurrence formulas:

5 

+1 (

 ) = 2"5  ( ) − 2B±5 

−1 (

 ),

B ≥ 2.

The functions 5  ( ) form an orthogonal system on the interval − F

 .

13. 14.

15.

16.



sinh  coshK

−1

 +

1



ù coshK

B = 1, 2, 

−2

 M

 .

S.3.4-2. Integrals containing sinh  . 1

17.

ù sinh(  + ' ) M  =  cosh(  + ' ).

18.

ù

 sinh 

19.

ù

 2 sinh 

20.

ù

 2  sinh 

 =  cosh  − sinh  . M

 = ( M M

2

+ 2) cosh  − 2 sinh  .

 = (2B )! ¬

½ 

=0

½  2  2  −1 cosh  − sinh  ® . (2 , )!  =1 (2 , − 1)!

© 2003 by Chapman & Hall/CRC

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S.3. TABLES OF INDEFINITE INTEGRALS

½

ù

 2

22.

ù

 K sinh 

23.

2 ù sinh  M  = − 21  + 14 sinh 2 .

21.

+1

sinh 

M

M

 = (2B + 1)!



¬ =0

 =  K cosh  −  K

−1

 2  +1  2 cosh  − sinh  ® . (2 , + 1)! (2 , )! sinh  +  ( − 1) ù

 K

−2

sinh 

M

 .

ù sinh3  M  = − cosh  + 13 cosh3  .  −1  1 ½ sinh[2(B − , ) ] , B = 1, 2,  (−1)  L 2  25. ù sinh2    = (−1)  L 2  2  + 2  −1 2 2 2(B − , ) M  =0   1 ½ cosh[(2B − 2 , + 1) ] ½ cosh2  +1  = , 26. ù sinh2  +1   = 2  (−1)  L 2  +1 (−1)  +  L   2 2B − 2 , + 1 2, + 1 M  =0  =0 B = 1, 2,  24.

1

−1

ù sinhK  M  =

28.

ù sinh  sinh ' M  =  2 −  2   cosh  sinh ' −  cosh ' sinh   .  1  = ln û tanh . M ù sinh   2 ûû û û û  cosh  1 = ¬− M ù 2B − 1 sinh2  −1  sinh2   ½  −1 1 2  (B − 1)(B − 2)  (B − , ) ® , B = 1, 2,  + (−1)  −1 (2B − 3)(2B − 5)  (2B − 2 , − 1) sinh2  −2  −1   =1 ½  −1 1  cosh  1 (2B −1)(2B −3)  (2B −2 , +1) ¬ − 2  + (−1)  −1 ® M 2  +1 = ù 2  2B 2 (B −1)(B −2)  (B − , ) sinh  −2   sinh  sinh   =1 (2B − 1)!!  ln tanh , B = 1, 2,  + (−1)  (2B )!! 2  1  tanh( 2) −  + £  2 +  2 . ù  +  M sinh  = £ 2 2 ln  +  tanh( 2) −  − £  2 +  2 #* + $ sinh  $ #3 − $C  tanh( 2) −  + £  2 +  2  =  + £ ln . ù  +  sinh  M    2 +  2  tanh( 2) −  − £  2 +  2

29. 30.

31.

32. 33.



sinhK



27.

−1

 cosh  − 1

S.3.4-3. Integrals containing tanh  34.

ù tanh  M  = ln cosh  .

35.

ù tanh2  M  =  − tanh  .



ù sinhK

−2

 M

 .

or coth  .

ù tanh3  M  = − 12 tanh2  + ln cosh  . ½  tanh2  −2  +1  , B = 1, 2,  37. ù tanh2    =  − M  =1 2B − 2 , + 1 ½ ½  tanh2  −2  +2  (−1)  L   , 38. ù tanh2  +1   = ln cosh  − = ln cosh  − 2 M  =1 2 , cosh   =1 2B − 2 , + 2 36.

B = 1, 2, 

© 2003 by Chapman & Hall/CRC

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SUPPLEMENTS 1

39.

ù tanhK  M  = −  − 1 tanhK

40.

ù coth  M  = ln |sinh  |.

41.

ù coth2  M  =  − coth  .

−1

 + ù tanhK

−2



 . M

ù coth3  M  = − 12 coth2  + ln |sinh  |. ½  coth2  −2  +1  , B = 1, 2,  43. ù coth2    =  − M  =1 2B − 2 , + 1 ½  ½  coth2  −2  +2  L  , 44. ù coth2  +1   = ln |sinh  |− = ln |sinh  |− 2 M  =1 2 , sinh   =1 2B − 2 , + 2 42.

1

ù cothK  M  = −  − 1 cothK

45.

−1

 + ù cothK

−2

 M

B = 1, 2, 

 .

S.3.5. Integrals Containing Logarithmic Functions 1.

ù ln  M  =  ln  −  .

2.

ù

 ln  M

ù

 K ln 

3.

 = 12  2 ln  − 14  2 . M

 =

 

1 2

1  K +1 ln2 

+1

ln  −

1  K ( + 1)2

+1

if



≠ −1,

if



= −1.

ù (ln  )2 M  =  (ln  )2 − 2 ln  + 2 .

4.

 (ln  )2  = 12  2 (ln  )2 − 12  2 ln  + 14  2 . M 2K +1 2K +1 K +1  2 (ln  ) − ln  + if  ( + 1)2 ( + 1)3 6. ù  K (ln  )2  =  + 1 M 3 1 if 3 ln   ½  (−1)  (B + 1)Be (B − , + 1)(ln  )  −  , 7. ù (ln  )   = B + 1  =0 M 5.

ù



≠ −1,



= −1.

B = 1, 2, 

ù (ln  ) & M  =  (ln  ) & −  ù (ln  ) & −1 M  ,  ≠ −1. +   +1 ½ (−1)  ( + 1)  ( − , + 1)(ln  ) + −  , B , = 1, 2,  9. ù   (ln  ) +  = ¡ ¡ ¡ +1 (B + 1)  +1 ¡ M  =0 ¡ 1  +1 −1 10. ù  K (ln  ) &  =  + 1  K (ln  ) & −  + 1 ù  K (ln  ) & M  ,  ,  ≠ −1. M 8.

1

ù ln(  + ' ) M  =  (  +  ) ln( +  ) −  . 1 1  2 12. ù  ln(  + ' )  = ” 2 − 2 ln(  + ' ) − ” M 2  • 2 3 1 1  13. ù  2 ln(  + ' )  = ” 3 − 3 ln(  + ' ) − M 3  • 3

11.

”



2

−

2



3

3

−

 

 

2

.

• 2

+

 2 . 2 •

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773

S.3. TABLES OF INDEFINITE INTEGRALS

15.

ù

1 ln   + ln .  (  + ' )   + ' 1 ln  1   + 2 ln =− + . 3 2 )


2  ( + ' ) 2 ( + '  ) 2£   + £' 2  + ' +  ® ¬ (ln  − 2) £  + ' + £  ln £   ln    + ' − £ 
£  +M ' =

 2 £  + ' ¬ (ln  − 2) £  + ' + 2 £ −  arctan £ ®  − ln( 2 +  2 )  =  ln( 2 +  2 ) − 2 + 2  arctan(  ). M  ln( 2 +  2 )  = 12 V ( 2 +  2 ) ln( 2 +  2 ) −  2 W . M  2 ln( 2 +  2 )  = 13 V  3 ln( 2 +  2 ) − 23  3 + 2  2  − 2  3 arctan( M

ù

ln  M (  + ' ln  M (  + '

14.

ù

16.

17.

ù

18.

ù

19.

ù



=−

)2

if  > 0, if  < 0.

 )W .

S.3.6. Integrals Containing Trigonometric Functions S.3.6-1. Integrals containing cos 

(B = 1, 2,  ).

1

1.

ù cos(  + ' ) M  =  sin(  + ' ).

2.

ù

 cos 

3.

ù

 2 cos  ù

  cos  ù

 2

6.

ù

 K cos 

7.

ù cos2  M  = 12  + 14 sin 2 .

8.

ù cos3  M  = sin  − 13 sin3  .

5.

+1

ù

10.

ù

11.

ù

12.

ù

13.

ù

M

cos 

cos2  

M

M

2

 = 2 cos  + ( M

2

4.

9.

 = cos  +  sin  . M

 = (2B )! ¬ M

½ 

− 2) sin  .

(−1)  =0

 = (2B + 1)!

½ 

=0

 =  K sin  +  K

−1

 =

½  −1  2  −2   sin  + (−1)  (2B − 2 , )! (2B  =0 2  −2  +1   sin  + ¬ (−1)  (2B − 2 , + 1)! (2B cos  −  ( − 1) ù

 1  1 ½ L  + 22  2  22  −1

−1

 K

−2

cos 

M





2 −2 −1

− 2 , − 1)!



2 −2



− 2 , )!

cos  ® .

cos  ® .

 .

sin[(2B − 2 , ) ] L 2  . 2B − 2 ,

 =0  ½ 1 sin[(2B − 2 , + 1) ] . cos2  +1   = 2  L 2  +1 2 2B − 2 , + 1 M  =0   +  = ln û tan O . M cos  2 4 P ûû û û û  M 2 = tan  . cos   sin  1  +  = + ln û tan O . M cos3  2 cos2  2 û 2 4 P ûû û û

© 2003 by Chapman & Hall/CRC

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774

SUPPLEMENTS

ù

15.

ù

M

14.





M 2



+

ù

16.

ù

17.

18.

ù

19.

ù ù

20.

2

ù

   cos2  ù

   cos  

23.

=0

−1

+



B −2

(B − 1)!   tan  + ln |cos  |  . (2B − 1)!! sin V (  +  ) W sin V (  −  ) W + , cos  cos '  = 2(  −  ) 2(  +  ) M


(  −  ) tan( 2) 2 arctan  £  2− 2 £  2− 2  =


 M £  +  cos   2 −  2 + (  −  ) tan( 1 £  2 −  2 ln ûû £  2 −  2 − (  −  ) tan( û   sin û  − 2 = 2 M ù  2 2 (  +  cos  ) (  −  )(  +  cos  )  −  2  1  tan  = £ arctan £ . M 2 2  2 +  2 cos2    +  2+ 2

1  tan    £  2 −  2 arctan £  2 −  2  =

 M £  2 −  2 −  tan   2 −  2 cos2  1 û ln 2  £  2 −  2 ûû £  2 −  2 +  tan 

   cos '

22.

−1



−1

ù

21.

½

=



cos

 , B > 1. M B − 1 ù cos −2  (2B − 2 , ) sin  − cos  (2B − 2)(2B − 4)  (2B − 2 , + 2) (2B − 1)(2B − 3)  (2B − 2 , + 3) (2B − 2 , + 1)(2B − 2 , ) cos2  −2 

sin  (B − 1) cos

=

cos 

M M

S.3.6-2. Integrals containing sin 



 ≠ A/ . if 

2

>  2,

2)

2 2

2) ûû if  >  . û û . M +  cos 

û û û

if 

2

>  2,

if 

2

>  2.

û û     =   ” 2 2 sin ' + 2 2 cos ' .  +  + •  2    = 2 ” cos2  + 2 sin  cos  + .  +4 r•  B (B − 1)   cos  −1  (  cos  + B sin  ) + 2  = 2 2  +B  +B 2 ù

M

+1

   cos

−2

 M

 .

(B = 1, 2,  ).

1

24.

ù sin(  + ' ) M  = −  cos(  + ' ).

25.

ù

 sin 

26.

ù

 2 sin 

27.

ù

 3 sin 

28.

ù

 2  sin  ù

 2

30.

ù

 K sin 

31.

ù sin2  M  = 12  − 14 sin 2 .

29.

+1

 = sin  −  cos  . M

2

 = 2 sin  − ( M

 = (3 M M

sin 

M

2

3

− 6) sin  − (

 = (2B )! ¬ M

− 2) cos  .

½ 

(−1)  =0

 = (2B + 1)!

½ 

=0

 = − K cos  +  K

− 6 ) cos  .

½  −1  2  −2  −1  2  −2  cos  + sin  ® . (−1)  (2B − 2 , )! (2B − 2 , − 1)!  =0  2  −2   2  −2  +1 cos  + (−1)  sin  ® . ¬ (−1)  +1 (2B − 2 , + 1)! (2B − 2 , )! +1

−1

sin  −  ( − 1) ù

 K

−2

sin 

M

 .

© 2003 by Chapman & Hall/CRC

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S.3. TABLES OF INDEFINITE INTEGRALS

 sin2 

 = 14 

2

− 14  sin 2 −

32.

ù

33.

ù sin3  M  = − cos  + 13 cos3  .

M

1 8

775

cos 2 .

 −1 1  (−1)  ½ sin[(2B − 2 , ) ]  (−1)  L 2  , 34. ù sin   = 2  L 2   + 2  −1 2 2 2B − 2 , M  =0 ! are binomial coefficients (0! = 1). ¡ where L +  = , ! ( − , )! ¡  1 ½ cos[(2B − 2 , + 1) ] 2  +1 .   = 2 (−1)  +  +1 L 2  +1 35. ù sin 2 2B − 2 , + 1 M  =0   = ln û tan û . 36. ù M sin  2û û û û  37. ù M 2 = − cot  . sin   cos  1  M =− + ln û tan û . 38. ù sin3  2 sin2  2 û 2û û B −û 2  cos   + 39. ù M  =− M  −2 , B > 1. ù  −1 sin  (B − 1) sin  B − 1 sin   −1 ½ sin  + (2B − 2 , ) cos    (2B − 2)(2B − 4)  (2B − 2 , + 2) =− 40. ù M sin2   (2B − 1)(2B − 3)  (2B − 2 , + 3) (2B − 2 , + 1)(2B − 2 , ) sin2  −2  +1   =0 2  −1 (B − 1)! +  ln |sin  | −  cot   . (2B − 1)!! sin[(  −  ) ] sin[(  +  ) ] − ,  ≠ A/ . 41. ù sin  sin '  = 2(  −  ) 2(  +  ) M


2  +  tan  2 arctan £ if  2 >  2 ,  £ 2− 2 2− 2    =


 42. ù M  − £  2 −  2 +  tan  2  +  sin  1 2 2 ln û £  2 −  2 û  + £  2 −  2 +  tan  2 ûû if  >  . û û û    cos û  + = . 43. ù M M ù (  +  sin  )2 (  2 −  2 )(  +  sin  )  2 −  2  +  sin  £  2 +  2 tan   1 = arctan . 44. ù M  2 +  2 sin2   
£  2 +  2

£  2 −  2 tan  1 arctan if  2 >  2 ,  £ 2 2   M =

   −  45. ù £  2 −  2 tan  +   2 −  2 sin2  1 û if  2 >  2 . ln û 2  £  2 −  2 ûû £  2 −  2 tan  −  ûû û 1 sin   , cosû  . 46. ù £ M2 2 = − arcsin £ , 1 + , sin  1+, 2 1 sin   = − ln , cos  + £ 1 − , 2 sin2  . 47. ù £ M û û , 1 − , 2 sin2  û û cos 1+, 2  , cos  £ £ 2 2 2 2 1 + , sin   = − 1 + , sin  − arcsin £ . 48. ù sin  2 2, M 1+, 2 cos  £ 1−, 2 1 − , 2 sin2  − ln , cos  + £ 1 − , 2 sin2  . 49. ù sin  £ 1 − , 2 sin2   = − û û 2 2, M û û 2

© 2003 by Chapman & Hall/CRC

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776

SUPPLEMENTS

    =   O 2 2 sin ' − 2 2 cos ' .  +  + P  2  O  sin2  − 2 sin  cos  + . sin2   = 2  +4  P M   sin  −1  B (B − 1)  sin    = (  sin  − B cos  ) + 2 2 2  +B  +B 2 ù M

50.

ù

   sin '

51.

ù

  

52.

ù

  

M

S.3.6-3. Integrals containing sin 

54. 55. 56. 57.

−2

 M

 .

and cos  .

cos[(  +  ) ] cos V (  −  ) W ù sin  cos ' M  = − 2(  +  ) − 2(  −  )  ( 1 ù  2 cos2  M + ( 2 sin2  = ˜( arctan O  tan  P .  ( tan  +  1 ù  2 cos2  M − ( 2 sin2  = 2 ˜( ln û ( tan  −  û . û û û û + −1  + 2  −2 + +1 ½  tan  , L   + + −1 ù cos2   M sin2 +  = 2, − 2 + 1  =0 ¡  ++  + ln |tan  | + ½ L  = L M ù cos2  +1  sin2 + +1   ++  ++  =0

53.

   sin 

 ≠ A/ .

,

B , ¡

= 1, 2, 

tan2  −2 +  , 2, − 2

¡

B , ¡

= 1, 2, 

S.3.6-4. Reduction formulas. The parameters  and  below can assume any values, except for those at which the denominators on the right-hand side vanish. sinK

−1

ù sinK  cos &  M  = −

59.

ù

60.

ù

61.

ù

62.

ù

63.

ù

64.

ù

S.3.6-5. Integrals containing tan  65.

 cos & +1 



−1 −2  +  +  ù sinK  cos&  M  . sinK +1  cos & −1   − 1 sinK  cos& −2   . sinK  cos &   = +  +  + ù M M −1 −1 sinK  cos &   −1 O sin2  − sinK  cos &   =  +  + −2P M ( − 1)(  − 1) + ù sinK −2  cos & −2  M  . ( +  )( +  − 2) sinK +1  cos & +1   +  + 2 sinK +2  cos &   . + sinK  cos &   =  +1  +1 ù M M +1 +1 sinK  cos &   +  + 2 sinK  cos & +2   . + sinK  cos &   = −  +1  +1 ù M M −1 +1 sinK  cos &   − 1 sinK −2  cos& +2   . + sinK  cos &   = −  +1  +1 ù M M +1 −1 sinK  cos &   −1 +2 −2 + sinK  cos &   =  +1  + 1 ù sinK  cos &  M  . M

58.

+

and cot  .

ù tan  M  = − ln |cos  |.

© 2003 by Chapman & Hall/CRC

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777

S.3. TABLES OF INDEFINITE INTEGRALS

66.

ù tan2  M  = tan  −  .

67.

ù tan3  M  = 12 tan2  + ln |cos  |. ½  (−1)  (tan  )2  −2  +1 2  , B = 1, 2,  tan = (−1) −    ù 2B − 2 , + 1 M  =1 ½  (−1)  (tan  )2  −2  +2 2  +1  +1 , B = 1, 2,    = (−1) ln |cos  | − ù tan 2B − 2 , + 2 M  =1  1 ù  + M  tan  =  2 +  2  +  ln |  cos  +  sin  |  .

1 tan    = £ arccos ” 1 − cos  ,  >  ,  > 0. M ù £  •  −  +  tan2 

68. 69. 70. 71. 72.

ù cot  M  = ln |sin  |.

73.

ù cot2  M  = − cot  −  .

ù cot3  M  = − 21 cot2  − ln |sin  |. ½  (−1)  (cot  )2  −2  +1 , B = 1, 2,  75. ù cot2    = (−1)   + 2B − 2 , + 1 M  =1 ½  (−1)  (cot  )2  −2  +2 , B = 1, 2,   76. ù cot2  +1   = (−1)  ln |sin  | + 2B − 2 , + 2 M  =1  1 77. ù = 2 2  −  ln |  sin  +  cos  |  . M  +  cot   + 74.

S.3.7. Integrals Containing Inverse Trigonometric Functions 1. 2. 3. 4. 5. 6. 7. 8. 9.

  £ ù arcsin  M  =  arcsin  +  2 −  2 .  2  2  £ ù O arcsin ‹P M  = ‹O arcsin ‹P − 2 + 2  2 −  2 arcsin  1    £ ù  arcsin  M  = 4 (2 2 −  2 ) arcsin  + 4  2 −  2 . 1   3  £ ù  2 arcsin þM  = 3 arcsin  + 9 ( 2 + 2  2 )  2 −  2 .   £ ù arccos þM  =  arccos  −  2 −  2 .  2  2 £ ù O arccos ‹P M  = ‹O arccos *P − 2 − 2  2 −  2 arccos 1    £ ù  arccos  M  = 4 (2 2 −  2 ) arccos  − 4  2 −  2 . 1   3  £ ù  2 arccos  M  = 3 arccos  − 9 ( 2 + 2  2 )  2 −  2 .    ù arctan þM  =  arctan  − 2 ln(  2 +  2 ). 

.



.

© 2003 by Chapman & Hall/CRC

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778

SUPPLEMENTS

ù

11.

ù

12.

ù

13.

ù

14.

ù

^_

1    = ( 2 +  2 ) arctan − .  M 2  2 3 2      3  = arctan − + ln(  2 +  2 ).  2 arctan þM 3  6 6     =  arccot + ln(  2 +  2 ). arccot  M  2 1 2    2  = ( +  ) arccot + .  arccot þM 2  2 3 2      3  = arccot + − ln(  2 +  2 ).  2 arccot þM 3  6 6

 arctan

10.



References for Subsection S.3.3: H. B. Dwight (1961), I. S. Gradshteyn and I. M. Ryzhik (1980), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1986, 1988).

© 2003 by Chapman & Hall/CRC

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REFERENCES Abramowitz, M. and Stegun, I. A. (Editors), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics, Washington, 1964. Akulenko, L. D. and Nesterov, S. V., Accelerated convergence method in the Sturm–Liouville problem, Russ. J. Math. Phys., Vol. 3, No. 4, pp. 517–521, 1996. Akulenko, L. D. and Nesterov, S. V., Determination of the frequencies and forms of oscillations of non-uniform distributed systems with boundary conditions of the third kind, Appl. Math. Mech. (PMM), Vol. 61, No. 4, pp. 531–538, 1997. Alexeeva, T. A., Zaitsev, V. F., and Shvets, T. B., On discrete symmetries of the Abel equation of the 2nd kind [in Russian], In: Applied Mechanics and Mathematics, MIPT, Moscow, pp. 4–11, 1992. Anderson, R. L. and Ibragimov, N. H., Lie–B a¨ cklund Transformations in Applications, SIAM Studies in Applied Mathematics, Philadelphia, 1979. Arnold, V. I., Kozlov, V. V., and Neishtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, Dynamical System III, Springer-Verlag, Berlin, 1993. Baker, G. A. (Jr.) and Graves–Morris, P., Pad´e Approximants, Addison–Wesley, London, 1981. Bakhvalov, N. S., Numerical Methods [in Russian], Nauka, Moscow, 1973. Bateman, H. and Erd´elyi, A., Higher Transcendental Functions, Vols. 1–2, McGraw-Hill, New York, 1953. Bateman, H. and Erd´elyi, A., Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York, 1955. Berkovich, L. M., Factorization and Transformations of Ordinary Differential Equations [in Russian], Saratov University Publ., Saratov, 1989. Birkhoff, G. and Rota, G. C., Ordinary Differential Equations, John Wiley & Sons, New York, 1978. Blaquiere, A., Nonlinear System Analysis, Academic Press, New York, 1966. Bluman, G. W. and Cole, J. D., Similarity Methods for Differential Equations, Springer-Verlag, New York, 1974. Bluman, G. W. and Kumei, S., Symmetries and Differential Equations, Springer-Verlag, New York, 1989. Bogolyubov, N. N. and Mitropol’skii, Yu. A., Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow, 1974. Boyce, W. E. and DiPrima, R. C., Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons, New York, 1986. Chicone, C., Ordinary Differential Equations with Applications, Springer-Verlag, Berlin, 1999. Chowdhury, A. R., Painlev´e Analysis and its Applications, Chapman & Hall/CRC Press, Boca Raton, 2000. Cole, G. D., Perturbation Methods in Applied Mathematics, Blaisdell Publishing Company, Waltham, MA, 1968. Collatz, L., Eigenwertaufgaben mit Technischen Anwendungen, Akademische Verlagsgesellschaft Geest & Portig, Leipzig, 1963.

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Conte, R., The Painlev´e approach to nonlinear ordinary differential equations, In: The Painlev e´ Property (ed. by R. Conte), pp. 77–180, CRM Series in Mathematical Physics, Springer-Verlag, New York, 1999. Dobrokhotov, S. Yu., Integration by quadratures of 2B -dimensional linear Hamiltonian systems with B known skew-orthogonal solutions, Russ. Math. Surveys, Vol. 53, No. 2, pp. 380–381, 1998. El’sgol’ts, L. E., Differential Equations, Gordon & Breach Inc., New York, 1961. Fedoryuk, M. V., Asymptotic Analysis. Linear Ordinary Differential Equations, Springer-Verlag, Berlin, 1993. Finlayson, B. A., The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972. Fokas, A. S. and Ablowitz M. J., On unified approach to transformation and elementary solutions of Painlev´e equations, J. Math. Phys., Vol. 23, No. 11, pp. 2033–2042, 1982. Gambier, B., Sur les e´ quations differentielles du second ordre et du premier degr´e dont l‘integrale g´en´erale est a` points critiques fixes, Acta Math., Vol. 33, pp. 1–55, 1910. Giacaglia, G. E. O., Perturbation Methods in Non-Linear Systems, Springer-Verlag, New York, 1972. Godunov, S. K. and Ryaben’kii, V. S., Differences Schemes [in Russian], Nauka, Moscow, 1973. Golubev, V. V., Lectures on Analytic Theory of Differential Equations [in Russian], GITTL, Moscow, 1950. Gradshteyn, I. S. and Ryzhik, I. M., Tables of Integrals, Series, and Products, Academic Press, New York, 1980. Gromak, V. I. and Lukashevich, N. A., Analytical Properties of Solutions of Painlev e´ Equations [in Russian], Universitetskoe, Minsk, 1990. Hartman, P., Ordinary Differential Equations, John Wiley & Sons, New York, 1964. Hill, J. M., Solution of Differential Equations by Means of One-parameter Groups, Pitman, Marshfield, MA, 1982. Ibragimov, N. H. (Editor), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1, CRC Press, Boca Raton, 1994. Ibragimov, N. H., Transformation Groups Applied to Mathematical Physics, (English translation published by D. Reidel), Dordrecht, 1985. Ince, E. L., Ordinary Differential Equations, Dover Publications, New York, 1964. Its, A. R. and Novokshenov, V. Yu., The Isomonodromic Deformation Method in the Theory of Painlev´e Equations, Springer-Verlag, Berlin, 1986. Kalitkin, N. N., Numerical Methods [in Russian], Nauka, Moscow, 1978. Kantorovich, L. V. and Krylov, V. I., Approximate Methods of Higher Analysis [in Russian], Fizmatgiz, Moscow, 1962. Kamke, E., Differentialgleichungen: Lo¨ sungsmethoden und L¨osungen, I, Gew¨ohnliche Differentialgleichungen, B. G. Teubner, Leipzig, 1977. Keller, H. B., Numerical Solutions of Two Point Boundary Value Problems, SIAM, Philadelphia, 1976. Kevorkian, J. and Cole, J. D., Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1981. Kevorkian, J. and Cole, J. D., Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York, 1996. Korn, G. A. and Korn, T. M., Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York, 1968.

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INDEX A Abel equation first kind, 9, 138, first kind, canonical (normal) form, 9 second kind, 10, 107 second kind, canonical (normal) form, 11 Airy equation, 213 Airy functions, 25, 214 first kind, 214 second kind, 214 algebraic form of Mathieu equation, 233 approximation function, 47 associated Legendre functions, 756, 758 first kind, 756 second kind, 756 asymptotic solutions Duffing equation, 302 fourth-order linear equations, 55 nth-order linear equations, 55, 56 second-order linear equations, 24–27 second-order nonlinear equations, 299, 305 autonomous equation, 295, 486, 608, 625, 637, 678, 686, 687 averaging method, 41, 42

B B¨acklund transformation, 66, 424, 433 Bernoulli equation, 4, 81 Bessel equation, 227 nonhomogeneous, 286 Bessel function, 227, 748 first kind, 227, 748 orthogonality properties, 750 second kind, 227, 748 third kind, 750 zeros, 750 Bessel’s formula, 749 beta function, 747 incomplete, 747 binomial coefficients, 736, 743 boundary conditions, 29 first kind, 30 mixed, 33 second kind, 32 third kind, 33 boundary value problem, 27 first kind, 30 mixed, 33 second kind, 32 third kind, 33 Bubnov–Galerkin method, 48

C canonical form Abel equation of first kind, 11 Abel equation of second kind, 9 Riccati equation, 8 second-order linear equation, 21 Cauchy problem, 1, 34 C-discriminant curve, 2 characteristic equation, 51, 692, 703 characteristic index, 262 characteristic polynomial, 503, 645 Chebyshev polynomials, 230, 760 first kind, 231 second kind, 232 Clairaut equation, 15, 204 collocation method, 48 complete elliptic integral first kind, 302 second kind, 304 complementary error function, 744 conservation laws, 71, 312 constant coefficient linear equation, 692, 703 contact transformations, 15, 65 Coulomb law, 162 critical point, 432 curve C-discriminant, 2 singular integral, 2 t-discriminant, 2 cylindrical function, 227, 748

D Darboux equation, 5, 196 degenerate hypergeometric equation, 220, 753 de Moivre formula, 737 differential equation, see equation discrete transformations of generalized Emden– Fowler equation, 355 Duffing equation, 38, 42, 302 dynamical system, 144, 145

E eigenfunctions, 29 eigenvalue problems, 29, 50 eigenvalues, 29 elliptic integral of first kind, 582 elliptic Weierstrass function, 297 Emden–Fowler equation, 63–65, 77, 306 conservation laws, 312 first integrals, 312 equation Abel, 9–13, 107–142 Abel, first kind, 9–10, 138–142 Abel, second kind, 10–13, 107–137

Airy, 213 autonomous, 34, 57, 295, 486, 608, 625, 637, 678, 686, 687 Bernoulli, 4, 81 Bessel, 227 characteristic, 51, 692, 703 Clairaut, 15, 204 Darboux, 5, 196 Duffing, 38, 42, 302 Emden–Fowler, 63–65, 77, 306 Euler, 226, 511, 647, 694 Euler, nonhomogeneous, 286, 556, 704 exact, 5, 36, 197 first-order linear, 4, 81 general Riccati, 6, 8 generalized Emden–Fowler, 74, 76, 336 generalized homogeneous, 35, 58, 137, 195, 462, 487, 623, 637, 660, 679, 687, 710, 732 generalized Yermakov, 462 Halm, 241 Heun, 238 homogeneous, 3, 35, 57, 82, 143, 462, 686, 723, 731 homogeneous linear, 21, 51, 53, 213 hypergeometric, 234, 755 invariant under “dilatation–translation” transformation, 197, 465, 488, 638, 733 invariant under “translation–dilatation” transformation, 197, 465, 487, 638, 687, 733, 733 Jacobi, 134 Lagrange–d’Alembert, 15, 204 Lam´e, 283 Laplace, 693 Legendre, 231, 242 Mathieu, 96, 262 modified Bessel, 228 nonhomogeneous Bessel, 286 nonhomogeneous Euler, 286, 556, 704 nth-order linear, 51–56, 689–705 nth-order nonlinear, 56–59, 706–733 of damped vibrations, 215 of forced oscillations, 260, 262 of forced oscillations with friction, 286 of forced oscillations without friction, 285 of free oscillations, 213, of loaded rigid spherical shell, 654 of oblate spheroidal wave functions, 243 of oscillations of mathematical pendulum, 303 of prolate spheroidal wave functions, 243 of theory of diffusion boundary layer, 218 of transverse vibrations of bar, 658 of transverse vibrations of pointed bar, 647 Painlev´e, 432 quasi-homogeneous, 136 quasilinear, 37 Rayleigh, 304 Riccati, 6, 82 Riemann, 244 separable, 81 special Riccati, 83 Van der Pol, 305

Weber, 214 Whittaker, 226, 755 Yermakov, 461 equations admitting reduction of order, 34, 56 containing arbitrary functions, 102, 203, 285, 461, 550, 655, 678, 701, 722 containing exponential and hyperbolic functions, 197, 648, 695 containing exponential functions, 89, 162, 246, 418, 438, 512, 595, 608, 666, 711 containing hyperbolic functions, 92, 166, 252, 421, 445, 516, 611, 668, 713 containing inverse trigonometric functions, 100, 271, 539 containing logarithmic functions, 94, 168, 199, 257, 450, 525, 616, 651, 672, 698, 717 containing power functions, 159, 195, 213, 425, 496, 592, 601, 641, 659, 689, 705 containing trigonometric functions, 95, 169, 200, 260, 452, 528, 618, 652, 674, 698, 718 first-order, 1–20, 81–212 fourth-order, 641–688 fourth-order linear, 55, 641–658 fourth-order nonlinear, 659–688 not solved for derivative, 2, 14 of combustion theory, 136 of theory of chemical reactors, 136 of theory of nonlinear oscillations, 136, 299 second-order, 21–51, 213–494 second-order linear, 21–33, 213–294 second-order nonlinear, 33–51, 295–494 third-order, 495–639 third-order linear, 495–558 third-order nonlinear, 559–639 error function, 221, 744 Euler constant, 227, 746, 749, 754 Euler equation, 226, 511, 647, 694 nonhomogeneous, 286, 556, 704 Euler formula, 737, 745 exact differential equation, 5, 36, 197 existence theorem, 1, 34, 56 exponential integral, 744

F factorial, 743 factorization principle, 68 fifth Painlev´e transcendent, 436 first integral, 71, 312 first-order differential equations, 1–20, 81–212 linear, 4, 81 first-order linear differential operator, 61 first Painlev´e transcendent, 432 fixed singular point, 432 formal operator, 66 formula Bessel, 749 de Moivre, 737 Euler, 737, 745 Liouville, 21, 54 Poisson, 749

Stirling, 746 Weierstrass, 762 fourth-order constant coefficient linear equation, 645 fourth-order differential equations, 641–688 linear, 55, 641–658 nonlinear, 659–688 fourth Painlev´e transcendent, 435 function Airy, 25, 214 approximation, 47 Bessel, 227, 748 beta, 747 complementary error, 744 cylindrical function, 227, 748 error, 221, 744 gamma, 220, 745 Gegenbauer, 230 Green, 29 Hankel, 750 Hermite, 761 hyperbolic, 738 hypergeometric, 234, 755 incomplete beta, 747 incomplete gamma, 221, 747 inverse hyperbolic, 742 inverse trigonometric, 740 Jacobi elliptic, 283, 304 Legendre, 231, 757 lemniscate, 330 Mathieu, 262, 263 Mittag-Leffler type, 690 modified associated Legendre, 756 modified Bessel, 221, 751 modified Emden–Fowler, 356 modified Mathieu, 253 parabolic cylinder, 758 special, 743 fundamental system of solutions, 24, 25, 55

G Galerkin method, 47 gamma function, 220, 745 incomplete, 221, 747 Gaussian hypergeometric equation, 234, 755 Gegenbauer function, 230 Gegenbauer polynomials, 232, 762 general Riccati equation, 6, 8 general solution, 21, 33 generalized Emden–Fowler equation, 74, 76, 336 discrete transformations, 355 generalized homogeneous equations, 35, 58, 137, 195, 462, 487, 623, 637, 660, 679, 687, 710, 732 generalized Yermakov’s equation, 462 Green’s function, 29 group analysis of second-order equations, 62 group methods, 61

H Halm equation, 241 Hankel functions, 750 Hermite functions, 761 Hermite polynomials, 216, 221, 761 Heun’s equation, 238 higher-order differential equations, 689 higher-order linear equations, 55 hodograph transformation, 74 homogeneous equation, 3, 35, 57, 82, 143, 462, 686, 723, 731 homogeneous linear equation, 21, 51, 53, 213 hyperbolic functions, 738 hypergeometric equation, 234, 755 hypergeometric functions, 755 hypergeometric series, 234, 755

I incomplete beta function, 747 incomplete elliptic integral of second kind, 117, 297, 310, 323, 587 incomplete gamma function, 221, 747 infinitesimal generator, 61 infinitesimal operator, 61 initial condition, 1, 34 inner expansion, 41 integrating factor, 5 invariance condition, 61 invariant of group, 61 of operator, 61 inverse hyperbolic functions, 742 inverse trigonometric functions, 740 iteration methods, 49

J Jacobi elliptic function, 283, 304 Jacobi equation, 134 Jacobi polynomials, 233, 762

K Krylov–Bogolyubov–Mitropolskii method, 41 Kummer–Liouville transformation, 23 Kummer series, 220, 753 Kummer transformation, 753

L Lagrange–d’Alembert equation, 15, 204 Laguerre polynomials, 221, 759 generalized, 759 Lam´e equation in form of Jacobi, 283 in form of Weierstrass, 283 Laplace equation, 693 least squares method, 48 Legendre equation, 231, 242 Legendre function, 757 second kind, 231 Legendre polynomials, 231, 756, 757 Legendre transformation, 16, 178

lemniscate functions, 330 Lie group of transformations, 61 Lienard equations, 301 linear equations, 4, 81, 213, 495, 641, 689 asymptotic solutions, 55 first-order, 4, 81 fourth-order, 55, 641–658 nth-order, 51–56, 689–705 second-order, 21–33, 213–294 second-order, canonical form, 21 second-order, self-adjoint form, 28 third-order, 495–558 two-term asymptotic expansions, 25 with constant coefficients, 51, 53 Lindstedt–Poincar´e method, 42 Liouville formula, 21, 54 logarithmic derivative of gamma function, 221, 227, 228, 746, 749, 752 logarithmic integral, 745

M Macdonald function, 751 Mathieu equation, 96, 262 algebraic form, 233 Mathieu functions, 262, 263 method averaging, 41, 42 Bubnov–Galerkin, 48 for construction of contact transformations, 16 Galerkin, 47 moment, 48 least squares, 48 least squared error, 49 Lindstedt–Poincar´e, 42 Krylov–Bogolyubov–Mitropolskii, 41 multiple scales, 41 of accelerated convergence, 50 of composite expansions, 41 of discrete-group analysis, 73 of expansion in powers of independent variable, 37 of Euler polygonal lines, 20 of group analysis, 61 of “integration by differentiation”, 14 of matched asymptotic expansions, 41, 45 of partitioning the domain, 48 of regular expansion in small parameter, 19, 38 of regular series expansions with respect to independent variable, 37 of regular series expansions with respect to small parameter, 37 of RF-pairs, 75 of scaled parameters, 41, 42 of strained coordinates, 41 of successive approximations, 18, 49 of Taylor series expansion in independent variable, 18 of two-scale expansions, 41, 43 projection, 47 Runge–Kutta, 49 Mittag-Leffler type special function, 690

modified associated Legendre functions, 756 modified Bessel equation, 228 modified Bessel function, 221, 751 first kind, 228, 751 second kind, 228, 751 modified Emden–Fowler equation, 356 modified Mathieu equation, 253 moment method, 48 movable critical point, 432 multiple scales method, 41

N Neumann function, 748 nonhomogeneous Bessel equation, 286 nonhomogeneous Euler equation, 286, 556, 704 nonhomogeneous linear equations, 52 existence theorem, 22 general solution, 22, 54 nonlocal exponential operator, 68 nonlocal variable, 66 nth-order equations linear, 51–56, 689–705 nonlinear, 56–59, 706–733 numerical integration of differential equations, 20 numerical methods, 49

O operator canonical form, 67 first-order linear differential, 61 formal, 66 infinitesimal, 61 nonlocal exponential, 68 of total derivative, 62 prolonged, 62 orthogonal polynomials, 759 outer expansion, 41

P Pad´e approximant, 39 Painlev´e equations, see Painlev´e transcendents Painlev´e transcendents, 432–437 fifth, 436 first, 432 fourth, 435 second, 433 sixth, 437 third, 434 parabolic cylinder function, 758 perturbation method, 40 Pochhammer symbol, 744 point transformation, 2, 73 Poisson’s formula, 749 polynomial characteristic, 503, 645 Chebyshev, 230, 760 Chebyshev, first kind, 231 Chebyshev, second kind, 232 Gegenbauer, 232, 762 Hermite, 216, 221, 761

Jacobi, 233, 762 Laguerre, 221, 759 Legendre, 231, 756, 757 orthogonal, 759 problem Cauchy, 1, 34 Sturm–Liouville, 29 with boundary conditions of first kind, 30 with boundary conditions of second kind, 32 with boundary conditions of third kind, 33 with mixed boundary conditions, 33 projection methods, 47 prolonged operator, 62

Q quasi-homogeneous equation, 136 quasilinear equation, 37

R Rayleigh equation, 304 Rayleigh–Ritz principle, 31, 50 RF-pair of transformations, 75 first, 76 second, 76 Riccati equation, 6, 82 canonical form, 8 general, 6, 8 special, 83 Riemann’s equation, 244 Runge–Kutta method, 20, 49

S second-order constant coefficient linear equation, 215 second-order differential equations, 21–51, 213– 494 linear, 21–33, 213–294 nonlinear, 33–51, 295–494 second Painlev´e transcendent, 433 secular terms, 39, 41 separable equation, 81 shooting method, 50 single-step methods, 20 singular integral curve, 2 singular point, 432 algebraic branch, 432 essential singularity, 432 fixed, 432 logarithmic branch, 432 movable, 432 movable critical, 432 pole, 432 singular solution, 2 sixth Painlev´e transcendent, 437 solution asymptotic, 24–27, 55, 56 explicit form, 33 general, 1, 21, 33

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implicit form, 1, 295 parametric form, 1, 295, 306 particular, 1 singular, 2 trivial, 21 space of nth prolongation, 61 special functions, 743 special Riccati equation, 83 Stirling formula, 746 Sturm–Liouville problem, 29

T t-discriminant curve, 2 theorem existence, 1, 34, 56 uniqueness, 1, 34, 56 third-order constant coefficient linear equation, 503 third-order differential equations, 495–639 linear, 495–558 nonlinear, 559–639 third Painlev´e transcendent, 434 total differential equation, see exact equation transformation B¨acklund, 66, 424, 433 contact, 15, 65 “dilatation–translation”, 197, 465, 488, 638, 733 discrete, 355 hodograph, 74 Kummer, 753 Kummer–Liouville, 23 Legendre, 16, 178 Lie group, 61 point, 2, 73 “translation–dilatation”, 197, 465, 487, 638, 687, 733, 733 trivial solution, 21

U uniqueness theorem, 1, 34, 56

V Van der Pol equation, 305 Van der Pol oscillator, 299

W wave functions prolate spheroidal, 243 Weber equation, 214 Weierstrass function, 762 Whittaker equation, 226, 755 Whittaker functions, 755 Wronskian, see Wronskian determinant Wronskian determinant, 21, 29, 54

Y Yermakov’s equation, 461