Handbook of First-Order Partial Differential Equations

Differential and Integral Equations and Their Applications A series edited by A.D. Polyanin

Institute for Problems in Mechanics. Moscow. Russia Volume 1 Handbook of First Order Partial Differential Equations A.D. Polyallin, Y.F Zaitsev and A. MOllss;a!lX

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A Handbook of First Order Partial Differential Equations

A.D. Polyanin Institute for Problem,,}' in lYfechanics Russian Academy of Sciences Mosco·w, Ru,\'/~da

V.F. Zaitsev Russian State Pedagogical University Sf Petersburg, Russia and

A. Moussiaux University de Namur Namul' Belgium

London and New York

First published 2002 by Taylor & Francis 11 New Fetter Lane, London EC4P 4EE Simultaneously published in the USA and Canada by Taylor & Francis Inc. 29 West 35th Street, New York, NY 10001

Taylor & Frallcis is an imprillt oj the Taylor & Frallcis Group

© 2002 Taylor & Francis

PubLisher's Note This book has been prepared from camera-ready copy provided by the authors. Printed and bound in Great Britain by TJ International Ltd. Padstow. Cornwall All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic. mechanical, or other means, now known or hereafter invented. including photocopying and recording. or in any information storage or retrieval system. without permission in writing from the publishers. Every effort has been made to ensure that the advice and information in this book is true and accurate at the time of going to press. However, neither the publisher nor the authors can accept any legal responsibility or liability for any errors or omissions that may be made. In the case of drug administration, any medical procedure or the use of technical equipment mentioned within this book. you arc strongly advised to consult the manufacturer's guidelines. British Library Cataloguing ill Publicatioll Data A catalogue record for this book is available from the British Library Library oj Congress Cataloging ill PllbLication Data A catalog record has been requested. ISBN 0-415-27267-X

CONTENTS Preface

xv

xvi Authors Annotation ............................................................... xvii Some Notation and Remarks ............................................... xviii

Part I. Linear Equations With Two Independent Variables

1

1. Equations Containing One Derivative .......................................

3

2. Linear Equations of the Form f(x, y) ~: + g(x,y) ~~ = 0 . . .. .. .. .. . . .. .. . . . . .. 2.1. Preliminary Remarks ..................................................... 2.1.1. Solution Method .................................................. 2.1.2. Cauchy Problem (Initial Value Problem) ............................... 2.1.3. Examples .................................................... : . . . 2.2. Equations Containing Power-Law Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Coefficients of Equations Are Linear in x and y ......................... 2.2.2. Coefficients of Equations Are Quadratic in x and y ....................... 2.2.3. Coefficients of Equations Contain Integer Powers of x and y ............... 2.2.4. Coefficients of Equations Contain Fractional Powers ..................... 2.2.5. Coefficients of Equations Contain Arbitrary Powers of x and y ............. 2.3. Equations Containing Exponential Functions .................................. 2.3.1. Coefficients of Equations Contain Exponential Functions .................. 2.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions ..... 2.4. Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.4.1. Coefficients of Equations Contain Hyperbolic Sine ....................... 2.4.2. Coefficients of Equations Contain Hyperbolic Cosine ..................... 2.4.3. Coefficients of Equations Contain Hyperbolic Tangent .................... 2.4.4. Coefficients of Equations Contain Hyperbolic Cotangent ............ _ . . . .. 2.4.5. Coefficients of Equations Contain Different Hyperbolic Functions .. . . . . . . . .. 2.5. Equations Containing Logarithmic Functions .................................. 2.5.1. Coefficients of Equations Contain Logarithmic Functions . . . . . . . . . . . . . . . . .. 2.5.2. Coefficients of Equations Contain Logarithmic and Power-Law Functions . . . . . 2.6. Equations Containing Trigonometric Functions ................................ 2.6.1. Coefficients of Equations Contain Sine _............................... 2.6.2. Coefficients of Equations Contain Cosine .............................. 2.63. Coefficients of Equations Contain Tangent ............................. 2.6.4. Coefficients of Equations Contain Cotangent. . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.6.5. Coefficients of Equations Contain Different Trigonometric Functions ......... 2.7. Equations Containing Inverse Trigonometric Functions .......................... 2.7.1. Coefficients of Equations Contain Arcsine. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.7.2. Coefficients of Equations Contain Arccosine ... . . . . . . . . . . . . . . . . . . . . . . . .. 2.7.3. Coefficients of Equations Contain Arctangent ........................... 2.7.4. Coefficients of Equations Contain Arccotangent .........................

5 5 5 6 6 7 7 9 12 13 14 24 24 25 30 30 30 31 32 33 34 34 34 37 37 39 40 42 43 45 45 46 48 49

vi

CONTENTS

2.8. Equations Containing Arbitrary Functions of x ................................ 2.8.1. Equations Contain Arbitrary and Power·Law Functions ................... 2.8.2. Equations Contain Arbitrary and Exponential Functions ................ . .. 2.8.3. Equations Contain Arbitrary and Hyperbolic Functions ................... 2.8.4. Equations Contain Arbitrary and Logarithmic Functions . . . . . . . . . . . . . . . . . .. 2.8.5. Equations Contain Arbitrary and Trigonometric Functions ................. 2.8.6. Equations Contain Arbitrary Functions and Their Derivatives. . . . . . . . . . . . . .. 2.9. Equations Containing Arbitrary Functions of Different Arguments ............... . 2.9.1. Equations Contain Arbitrary Functions of x and Arbitrary Functions of y ..... 2.9.2. Equations Contain One Arbitrary Function of Complicated Argument. . . . . . .. 2.9.3. Equations Contain Several Arbitrary Functions ..........................

5l 5l 53 54 55 55 56 58 58 59

3. LinearEquationsoftheFormf(x,y)~-:: +g(x,y)~; =h(x,y) ................

65

3.1. Preliminary Remarks ..................................................... 3.1.1. Solution Methods ................................................. 3.1.2. Cauchy Problem .................................................. 3.1.3. Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Equations Containing Power-Law Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.2.1. Coefficients of Equations Are Linear in x and y ......................... 3.2.2. Coefficients of Equations Are Quadratic in x and y ....................... 3.2.3. Coefficients of Equations Contain Other Power-Law Functions ............. 3.2.4. Coefficients of Equations Contain Arbitrary Powers of x and]l ............. 3.3. Equations Containing Exponential Functions .................................. 3.3.1. Coefficients of Equations Contain Exponential Functions ............. . . . .. 3.3.2. Coefficients of Equations Contain Exponential and Power· Law Functions " . .. 3.4. Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.4.1. Coefficients of Equations Contain Hyperbolic Sine .............. . . . . . . . .. 3.4.2. Coefficients of Equations Contain Hyperbolic Cosine ..................... 3.4.3. Coefficients of Equations Contain Hyperbolic Tangent .................... 3.4.4. Coefficients of Equations Contain Hyperbolic Cotangent .................. 3.4.5. Coefficients of Equations Contain Different Hyperbolic Functions .. . . . . . . . .. 3.5. Equations Containing Logarithmic Functions .................................. 3.5. I. Coefficients of Equations Contain Logarithmic Functions. . . . . . . . . . . . . . . . .. 3.5.2. Coefficients of Equations Contain Logarithmic and Power-Law Functions. . . . . 3.6. Equations Containing Trigonometric Functions ................................ 3.6.1. Coefficients of Equations Contain Sine ................................ 3.6.2. Coefficients of Equations Contain Cosine .............................. 3.6.3. Coefficients of Equations Contain Tangent ............................. 3.6.4. Coefficients of Equations Contain Cotangent. . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.6.5. Coefficients of Equations Contain Different Trigonometric Functions ........

65 65 66 67 68 68 69 70 70 72 72 73 74 74 75 76 76 77 77 77 78 79 79 79 80 80 81

3.7. Equations Containing Inverse Trigonometric Functions .......................... 3.7.1. Coefficients of Equations Contain Arcsine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2. Coefficients of Equations Contain Arccosine. . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.7.3. Coefficients of Equations Contain Arctangent ........................... 3.7.4. Coefficients of Equations Contain Arccotangent .........................

82 82 83 83 84

3.8. Equations Containing Arbitrary Functions .................................... 3.8.1. Coefficients of Equations Contain Arbitrary Functions of x ................ 3.8.2. Equations Contain Arbitrary Functions of x and Arbitrary Functions of y ..... 3.8.3. Equations Contain Arbitrary Functions of Complicated Arguments .......... 3.8.4. Equations Contain Arbitrary Functions of Two Variables ......... ,........

85 85 87 88 89

6l

CONTENTS

vii

4. LinearEquationsoftheFormf(x,y)~,: +g(x,y)~~ =h(x,y)w ...............

91

4.1. Preliminary Remarks ..................................................... 4.1.1. Solution Methods ................................................. 4.1.2. Examples . . . . .. .................................................

91 91 92

4.2. Equations Containing Power-Law Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Coefficients of Equations Are Linear in x and y ......................... 4.2.2. Coefficients of Equations Are Quadratic in x and y .... . . . . . . . . . . . . . . . . . . 4.2.3. Coefficients of Equations Contain Other Power-Law Functions ............. 4.2.4. Coefficients of Equations Contain Arbitrary Powers of x and y .............

93 93 94 95 96

4.3. Equations Containing Exponential Functions .................................. 4.3.1. Coefficients of Equations Contain Exponential Functions .................. 4.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions . . . . .

97 97 98

4.4. Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Coefficients of Equations Contain Hyperbolic Sine ................ . . . . . .. 4.4.2. Coefficients of Equations Contain Hyperbolic Cosine ..................... 4.4.3. Coefficients of Equations Contain Hyperbolic Tangent .................... 4.4.4. Coefficients of Equations Contain Hyperbolic Cotangent .................. 4.4.5. Coefficients of Equations Contain Different Hyperbolic Functions . . . . . . . . . ..

99 99 100 100 101 102

4.5. Equations Containing Logarithmic Functions ......................... ~........ 102 4.5.1. Coefficients of Equations Contain Logarithmic Functions . . . . . . . . . . . . . . . . .. 102 4.5.2. Coefficients of Equations Contain Logarithmic and Power-Law Functions. . . .. 103 4.6. Equations Containing Trigonometric Functions ................................ 4.6.1. Coefficients of Equations Contain Sine .............................. . 4.6.2. Coefficients of Equations Contain Cosine .............................. 4.6.3. Coefficients of Equations Contain Tangent ............................ 4.6.4. Coefficients of Equations Contain Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.6.5. Coefficients of Equations Contain Different Trigonometric Functions ........

104 104 104 105 106 106

4.7. Equations Containing Inverse Trigonometric Functions .......................... 4.7.1. Coefficients of Equations Contain Arcsine. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.7.2. Coefficients of Equations Contain Arccosine . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.7.3. Coefficients of Equations Contain Arctangent ........................... 4.7.4. Coefficients of Equations Contain Arccotangent .........................

107 107 108 108 109

4.8. Equations Containing Arbitrary Functions .................................... 4.8.1. Coefficients of Equations Contain Arbitrary Functions of x ................ 4.8.2. Equations Contain Arbitrary Functions of x and Arbitrary Functions of y ..... 4.8.3. Equations Contain Arbitrary Functions of Complicated Arguments .......... 4.8.4. Equations Contain Arbitrary Functions of Two Variables ..................

109 109 112 113 114

5. LinearEquationsoftheFormf(x,y)~,: +g(x,y)~~ =ht(x,y)w+ho(x,y) ..... 115 5.1. Preliminary Remarks ..................................................... 115 5.1.1. Solution Methods ................................................. 115 5.1.2. Examples ............................................ : ........... 116 5.2. Equations Containing Power~Law Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.2.1. Coefficients of Equations Are Linear in x and y ......................... 5.2.2. Coefficients of Equations Are Quadratic in x and 11 .••••.•....••••••••.•.• 5.2.3. Coefficients of Equations Contain Square Roots ......................... 5.2.4. Coefficients of Equations Contain Arbitrary Powers of x and y .............

117 117 118 119 120

viii

CONTENTS

5.3. Equations Containing Exponential Functions .................................. 122 5.3.1. Coefficients of Equations Contain Exponential Functions .................. 122 5.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions. . . .. 123

504. Equations Containing Hyperbolic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.4.1. Coefficients of Equations Contain Hyperbolic Sine ....................... 5.4.2. Coefficients of Equations Contain Hyperbolic Cosine ..................... 5.4.3. Coefficients of Equations Contain Hyperbolic Tangent .................... 5.404. Coefficients of Equations Contain Hyperbolic Cotangent .................. 5.4.5. Coefficients of Equations Contain Different Hyperbolic Functions. . . . . . . . . .. 5.5. Equations Containing Logarithmic Functions .................................. 5.5.1. Coefficients of Equations Contain Logarithmic Functions . . . . . . . . . . . . . . . . .. 5.5.2. Coefficients of Equations Contain Logarithmic and Power-Law Functions. . . .. 5.6. Equations Containing Trigonometric Functions ................................ 5.6.1. Coefficients of Equations Contain Sine ................................ 5.6.2. Coefficients of Equations Contain Cosine .............................. 5.6.3. Coefficients of Equations Contain Tangent ............................. 5.604. Coefficients of Equations Contain Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.6.5. Coefficients of Equations Contain Different Trigonometric Functions ........ 5.7. Equations Containing Inverse Trigonometric Functions .......................... 5.7.1. Coefficients of Equations Contain Arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.7.2. Coefficients of Equations Contain Arccosine. . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.7.3. Coefficients of Equations Contain Arctangent ........................... 5.704. Coefficients of Equations Contain Arccotangent ......................... 5.8. Equations Containing Arbitrary Functions .................................... 5.8.1. Coefficients of Equations Contain Arbitrary Functions of x ................ 5.8.2. Equations Contain Arbitrary Functions of x and Arbitrary Functions of y ..... 5.8.3. Equations Contain Arbitrary Functions of Two Variables ..................

125 125 125 126 126 127

Part II. Linear Equations With Three or More Independent Variables

139

6. Linear Equations of the Form I(x, y, z} ~': + g(x, y, z) ~~ + hex, y, z) ~~ == 0 .... 6.1. Preliminary Remarks ..................................................... 6.1.1. Solution Methods ................................................. 6.1.2. Cauchy Problem (Initial Value Problem) ............................... 6.1.3. Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

141 141 141 142 142

6.2. Equations Containing Power-Law Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.2.1. Coefficients of Equations Are Linear in x, y, and z ....................... 6.2.2. Coefficients of Equations Are Quadratic in x, y, and z .................... 6.2.3. Coefficients of Equations Contain Other Powers of x, y, and z .............. 6.2.4. Coefficients of Equations Contain Arbitrary Powers of x, y, and z ...........

143 143 147 150 151

127 127 128 129 129 130 j 30 131 132 133 133 133 134 134 135 135 137 138

6.3. Equations Containing Exponential Functions .................................. 154 6.3.1. Coefficients of Equations Contain Exponential Functions .................. 154 6.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions. . . .. 155 6.4. Equations Containing Hyperbolic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.4.1. Coefficients of Equations Contain Hyperbolic Sine ....................... 6.4.2. Coefficients of Equations Contain Hyperbolic Cosine .......... . . . . . . . . . .. 6.4.3. Coefficients of Equations Contain Hyperbolic Tangent .................... 6.4.4. Coefficients of Equations Contain Hyperbolic Cotangent .................. 6.4.5. Coefficients of Equations Contain Different Hyperbolic Functions ...........

157 157 158 159 159 160

CONTENTS

ix

6.5. Equations Containing Logarithmic Functions .................................. 161 6.5.1. Coefficients of Equations Contain Logarithmic Functions. . . . . . . . . . . . . . . . .. 161 6.5.2. Coefficients of Equations Contain Logarithmic and Power-Law Functions. . . .. 161 6.6. Equations Containing Trigonometric Functions ................................ 6.6.1. Coefficients of Equations Contain Sine ................................ 6.6.2. Coefficients of Equations Contain Cosine .............................. 6.6.3. Coefficients of Equations Contain Tangent ............................. 6.6.4. Coefficients of Equations Contain Cotangent . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.6.5. Coefficients of Equations Contain Different Trigonometric Functions ........

162 162 162 163 163 164

6.7. Equations Containing Inverse Trigonometric Functions .......................... 6.7.1. Coefficients of Equations Contain Arcsine. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.7.2. Coefficients of Equations Contain Arccosine ............................ 6.7.3. Coefficients of Equations Contain Arctangent ........................... 6.7.4. Coefficients of Equations Contain Arccotangent .........................

164 164 165 166 166

6.8. Equations Containing Arbitrary Functions ....... ,............................ 6,8.1. Coefficients of Equations Contain Arbitrary Functions of x ................ 6.8.2. Coefficients of Equations Contain Arbitrary Functions of Different Variables .. 6.8.3. Coefficients of Equations Contain Arbitrary Functions of Two Variables

167 167 169 170

7. Linear Equations of the Form It ~= + 12 ~~ + 13 ~': = 14, In = In{x,y, z) ...... 7. I. Preliminary Remarks ..................................................... 7.1.1. Solution Methods ................................................. 7.1.2. Examples.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

173 173 173 174

7.2. Equations Containing Power-Law Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.2.1. Coefficients of Equations Are Linear in x, y, and z ....................... 7.2.2. Coefficients of Equations Are Quadratic in x, y, and z .................... 7.2.3. Coefficients of Equations Contain Other Powers in x, y, and z .............. 7.2.4. Coefficients of Equations Contain Arbitrary Powers of x, y. and z ...........

175 175 176 177 178

7.3. Equations Containing Exponential Functions .................................. 180 7.3.1. Coefficients of Equations Contain Exponential Functions. . . . . . . . . . . . . . . . .. 180 7.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions .. . .. 181 7.4. Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.4.1. Coefficients of Equations Contain Hyperbolic Sine ....................... 7.4.2. Coefficients of Equations Contain Hyperbolic Cosine ............... ..... 7.4.3. Coefficients of Equations Contain Hyperbolic Tangent .................... 70404. Coefficients of Equations Contain Hyperbolic Cotangent .................. 704.5. Coefficients of Equations Contain Different Hyperbolic Functions . . . . . . . . . ..

182 182 183 184 185 186

7.5. Equations Containing Logarithmic Functions .................................. 186 7.5.1. Coefficients of Equations Contain Logarithmic Functions . . . . . . . . . . . . . . . . .. 186 7.5.2. Coefficients of Equations Contain Logarithmic and Power-Law Functions. . . .. 187 7.6. Equations Containing Trigonometric Functjons ................................ 7.6.1. Coefficients of Equations Contain Sine ................................ 7.6.2. Coefficients of Equations Contain Cosine .............................. 7.6.3. Coefficients of Equations Contain Tangent ....... -.......... ;". . . . . . . . . .. 7.6.4. Coefficients of Equations Contain Cotangent ..................... , . . . . .. 7.6.5. Coefficients of Equations Contain Different Trigonometric Functions ........

187 187 188 189 190 190

7.7. Equations Containing Inverse Trigonometric Functions .......................... 191 7.7.1. Coefficients of Equations Contain Arcsine . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 191 7.7.2. Coefficients of Equations Contain Arccosine . . . . . . . . . . . . . . . . . . . . . . . . . . .. 192

x

CONTENTS

7.7.3. Coefficients of Equations Contain Arctangent .... " . . . . .. .. . . . . . . . . .. . .. 192 7.704. Coefficients of Equations Contain Arccotangent ......................... 193 7.8. Equations Containing Arbitrary Functions .................................... 7.8.1. Coefficients of Equations Contain Arbitrary Functions of x ................ 7.8.2. Coefficients of Equations Contain Arbitrary Functions of Different Variables .. 7.8.3. Coefficients of Equations Contain Arbitrary Functions of Two Variables ......

8. Linear Equations of the Form il ~~ + i2 ~; + i3 ~~ = f4 W , in

= fn(x, y, z)

193 193 195 196

.... 199

8.1. Preliminary Remarks ..................................................... 199 8.1.1. Solution Methods ................................................. 199 8.1.2. Examples .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 200 8.2. Equations Containing Power-Law Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8.2. I. Coefficients of Equations Are Linear in x, y, and z ....................... 8.2.2. Coefficients of Equations Are Quadratic in x, y, and z .................... 8.2.3. Coefficients of Equations Contain Other Powers of x, y, and z .............. 8.204. Coefficients of Equations Contain Arbitrary Powers of x. y, and z ...........

201 201 202 203 204

8.3. Equations Containing Exponential Functions .................................. 206 8.3.1. Coefficients of Equations Contain Exponential Functions .................. 206 8.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions ..... 207 8.4. Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 804.1. Coefficients of Equations Contain Hyperbolic Sine .................... . .. 8.4.2. Coefficients of Equations Contain Hyperbolic Cosine ..................... 804.3. Coefficients of Equations Contain Hyperbolic Tangent .................... 8.4.4. Coefficients of Equations Contain Hyperbolic Cotangent .................. 8.4.5. Coefficients of Equations Contain Different Hyperbolic Functions. , . . . . . . . ..

208 208 209 2 J0 211 211

8.5. Equations Containing Logarithmic Functions .................................. 212 8.5.1. Coefficients of Equations Contain Logarithmic Functions . . . . . . . . . . . . . . . . .. 212 8.5.2. Coefficients of Equations Contain Logarithmic and Power-Law Functions. . . .. 213 8.6. Equations Containing Trigonometric Functions ... , ........... , ...... , ......... 8.6.1. Coefficients of Equations Contain Sine ................................ 8.6.2. Coefficients of Equations Contain Cosine .............................. 8.6.3. Coefficients of Equations Contain Tangent ............................. 8.604. Coefficients of Equations Contain Cotangent. . . . . . . . . . . . . . . . . . . . . . . . . . .. 8.6.5. Coefficients of Equations Contain Different Trigonometric Functions ........

213 213 214 215 215 216

8.7. Equations Containing Inverse Trigonometric Functions .......................... 8.7.1. Coefficients of Equations Contain Arcsine. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8.7.2. Coefficients of Equations Contain Arccosine ..... . . . . . . . . . . . . . . . . . . . . . .. 8.7.3. Coefficients of Equations Contain Arctangent , .......................... 8.704. Coefficients of Equations Contain Arccotangent .........................

217 217 2 17 218 218

8.8. Equations Containing Arbitrary Functions ..... ,.............................. 8.8.1. Coefficients of Equations Contain Arbitrary Functions of x ................ 8.8.2. Coefficients of Equations Contain Arbitrary Functions of Different Variables .. 8.8.3. Coefficients of Equations Contain Arbitrary Functions of Two Variables ......

219 2I9 221 222

9. Linear Equations of the Form it ~: +!2 ~~ + i3 ~'; = !4 W + is, in = in(x, y, z)

225

9.1. Preliminary Remarks ..................................................... 225 9.1.1. Solution Methods ................................................. 225 9.1.2. Examples ......................... , , . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 226

CONTENTS

9.2. Equations Containing Power-Law Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.2.1. Coefficients of Equations Are Linear in x, y, and z ....................... 9.2.2. Coefficients of Equations Are Quadratic in x, y, and z .................... 9.2.3. Coefficients of Equations Contain Other Powers of x, y. and z ......... . . . .. 9.2.4. Coefficients of Equations Contain Arbitrary Powers of x, y. and z ...........

xi

226 226 228 228 229

9.3. Equations Containing Exponential Functions .................................. 231 9.3.1. Coefficients of Equations Contain Exponential Functions ............ . . . . .. 231 9.3.2. Coefficients of Equations Contain Exponential and Power-Law Functions .. . .. 232 9.4. Equations Containing Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.4.1. Coefficients of Equations Contain Hyperbolic Sine ....................... 9.4.2. Coefficients of Equations Contain Hyperbolic Cosine ..................... 9.4.3. Coefficients of Equations Contain Hyperbolic Tangent .................... 9.4.4. Coefficients of Equations Contain Hyperbolic Cotangent .................. 9.4.5. Coefficients of Equations Contain Different Hyperbolic Functions . . . . . . . . . ..

233 233 234 235 235 236

9.5. Equations Containing Logarithmic Functions ................. . . . . . . . . . . . . . . . .. 237 9.5.1. Coefficients of Equations Contain Logarithmic Functions . . . . . . . . . . . . . . . . .. 237 9.5.2. Coefficients of Equations Contain Logarithmic and Power-Law Functions. . . .. 237. 9.6. Equations Containing Trigonometric Functions ................................ 9.6.1. Coefficients of Equations Contain Sine ................................ 9.6.2. Coefficients of Equations Contain Cosine .............................. 9.6.3. Coefficients of Equations Contain Tangent ............................. 9.6.4. Coefficients of Equations Contain Cotangent .......................... " 9.6.5. Coefficients of Equations Contain Different Trigonometric Functions ........

238 238 239 240 240 241

9.7. Equations Containing Inverse Trigonometric Functions .......................... 9.7.1. Coefficients of Equations Contain Arcsine. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.7.2. Coefficients of Equations Contain Arccosine. . . . . . . . . . . . . . . . . . . . . . . . . . .. 9.7.3. Coefficients of Equations Contain Arctangent ........................... 9.7.4. Coefficients of Equations Contain Arccotangent .........................

242 242 242 243 243

9.8. Equations Containing Arbitrary Functions .................................... 9.8.1. Coefficients of Equations Contain Arbitrary Functions of x ................ 9.8.2. Coefficients of Equations Contain Arbitrary Functions of Different Variables .. 9.8.3. Coefficients of Equations Contain Arbitrary Functions of Two Variables ......

244 244 245 246

10. Linear Equations With Four or More Independent Variables ................. "

249

10.1. Preliminary Remarks .................................................... 249 10.1.1. Linear Homogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 249 10.1.2. Linear Nonhomogeneous Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 250 10.2. Specific Equations ...................................................... 10.2.1. Equations Containing Power-Law Functions ........................... 10.2.2. Other Equations Containing Arbitrary Parameters ................ . . . . . .. 10.2.3. Equations Containing Arbitrary Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . ..

251 251 254 256

Part III. Nonlinear Equations

259

11. Quasilinear Equations of the Form f(x, y) 88 w:z: + g(x, y) 8aWy = hex, y, w) .. . . . . . .. 261 11.1. Preliminary Remarks .................................................... 261 11.1.1. Solution Methods ................................................ 261 11.1.2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 262

CONTENTS

xii

11.2. Equations Containing Arbitrary Parameters .................................. 11.2.1. Coefficients of Equations Contain Power-Law Functions ................. 11.2.2. Coefficients of Equations Contain Exponential Functions ................. 11.2.3. Coefficients of Equations Contain Hyperbolic Functions ................. 11.2.4. Coefficients of Equations Contain Logarithmic Functions . . . . . . . . . . . . . . . .. 11.2.5. Coefficients of Equations Contain Trigonometric Functions ...............

263 263 264 266 267 268

11.3. Equations Containing Arbitrary Functions ................................... 268 11.3.1. Equations Contain Arbitrary Functions of One Variable .................. 268 11.3.2. Equations Contain Arbitrary Functions of Two Variables ................. 271

=

12. Quasilinear Equations of the Form f(x, y, w) ~~ + g(x, y, w) ~~ hex, y, w) ... 12.1. Preliminary Remarks .................................................... 12.1.1. Solution Methods ................................................ 12.1.2. Cauchy Problem. Existence and Uniqueness Theorem ................... 12.1.3. Equation ~~ + few) ~~ = O. Qualitative Features and Discontinuous Solutions 12.1.4. Generalized Solutions of Quasilinear Equations ........................

273 273 273 274 276 286

12.2. Equations Containing Power-Law Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12.2.1. Coefficients of Equations Are Linear in w ............................. 12.2.2. Coefficients of Equations Are Quadratic in w .......................... 12.2.3. Coefficients of Equations Contain Other Powers of w ....................

290 290 293 295

12.3. Other Equations Containing Arbitrary Parameters ............................. 12.3.1. Coefficients of Equations Contain Exponential Functions ................. 12.3.2. Coefficients of Equations Contain Hyperbolic Functions ................. 12.3.3. Coefficients of Equations Contain Logarithmic Functions. . . . . . . . . . . . . . . .. 12.3.4. Coefficients of Equations Contain Trigonometric Functions ...............

297 297 299 302 304

12.4. Equations Containing Arbitrary Functions ................................... 12.4.1. Equations Contain Arbitrary Functions of Independent Variables ........... 12.4.2. Equations Contain Arbitrary Functions of the Unknown Variable .. . . . . . . . .. 12.4.3. Equations Contain Arbitrary Functions of Two Variables .................

306 306 310 314

13. Equations With 1\vo Independent Variables Quadratic in Derivatives . . . . . . . . . . .. 317 13.1. Preliminary Remarks .................................................... 317 13.2. Equations Containing Arbitrary Parameters .................................. 317 13.2.1. Equations of the Form ~~ ~~ = f(x, y, w) ............................ 317 13.2.2. Equations of the Form lex, y, w) ~~ ~~ + g(x, y, w) ~~ = hex, y, w) ........ 319 13.2.3. Equations of the Form f(x, y, w) ~~ ~~ + g(x, y, W) ~~ + hex, y, w) ~; sex, y, w)

=

...................................................... 321

13.2.4. Equations of the Form ~~ + lex, y, w)( ~;)2

= g(x, y, w)

................ 324

13.2.5. Equations of the Form ~~ +1(x,y,w)(~~)2+g(x,y,w)~~ =h(x,y,w) ... 331 13.2.6. Equations of the Form f(x,y.w)(~~)2 + g(x,y.w)(~~)2 = h(x,V,w) ...... 335 13.2.7. Equations of the Form lex, y)( ~~)2 + g(x, y) ~~ ~~ = hex, y, w) .......... 340 13.2.8. Other Equations ................................................. 343 13.3. Equations Containing Arbitrary Functions ................................... 347 13.3.1. Equations of the Form ~~ ~; = f(x, y, w) ............................ 347

= hex, y, w) .... . . . . . . . .. Equations of the Form lex, y) ~~ + g(x, y, w)( ~~)2 = hex, y, w) ........... Equations of the Form ~~ + j(x, y, w) (~~)2 + g(x, y, w) ~~ = hex, y, 'LV) ... Equations of the Form f(x,y,w)(~~)2 + g(x,y,w)(~~)2 = h(x,y,w) ......

13.3.2. Equations of the Form f(x, y) ~~ ~; + g(x, y) ~~

349

13.3.3.

351

13.3.4. 13.3.5.

356 359

CONTENTS

xiii

13.3.6. EquationsoftheFonn (~~)2+f(x,y,w)~~ ~~ =g(x,y,w) ...... , ..... 361 13.3.7. Other Equations ............................................. _ ... 364

14. Nonlinear Equations With Two Independent Variables of General Form ......... 14.1. Preliminary Remarks ................................................... 14.1.1. Solution Methods ................................................ 14.1.2. Cauchy Problem. Existence and Uniqueness Theorem ................... 14.1.3. Generalized Viscosity Solutions and Their Applications ....... _. . . . . . . . .. 14.2. Equations Containing Cubic Nonlinearities With Respect to Derivatives. . . . . . . . . . .. 14.2.1. Equations of the Fonn ~~ (~~)2 = f(x, v, w) .......................... 14.2.2. Equations of the Fonn f(x , V' w)( ~~)3 + g(x, Y, w) ~; = h(:J;, y, w) ........ 14.2.3. Equations of the Fonn f(x, y, w)( ~~)3 + g(x, y, w)( ~~) 2 = hex, y, w) . . . . .. 14.2.4. Equations of the Fonn f(x, y, w)( ~~)3 + g(x, y, W) ~~ ~; = h(x, y, w) ..... 14.2.5. Other Equations ................................................. 14.3. Nonlinear Equations Containing Arbitrary Parameters .......................... 14.3.1. Equations Contain the Fourth Powers of Derivatives ..................... 14.3.2. Equations Contain Derivatives in Radicands ........................... 14.3.3. Equations Contain Arbitrary Powers of Derivatives ...................... 14.3.4. More Complicated Equations ....................................... 14.4. Equations Containing Arbitrary Functions ofIndependent Variables. . . . . . . . . . . . . .. 14.4.1. Equations Contain One Arbitrary Power of Derivative ................... 14.4.2. Equations Contain Two or Three Arbitrary Powers of Derivatives .......... 14.5. Equations Containing Arbitrary Functions of Derivatives . . . . . . . . . . . . . . . . . . . . . . .. 14.5.1. Equations Contain Arbitrary Functions of One Variable .................. 14.5.2. Equations Contain Arbitrary Functions of Two Variables ................. 14.5.3. Equations Contain Arbitrary Functions of Three Variables ................ 14.5.4. Equations Contain Arbitrary Functions of Four Variables .................

367 367 367 371 374 378 378 379 381 382 383 385 385 387 387 390 392 392 395 397 397 400 404 406

15. Nonlinear Equations With Three or More Independent Variables ............... 15. 1. Preliminary Remarks .................................................... 15.1.1. Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15.1.2. Nonlinear Equations .............................................. 15.1.3. Generalized Viscosity Solutions .............. ...................... 15.2. QuasilinearEquations ................................................... 15.2.1. Equations \Vith Three Variables ..................................... 15.2.2. Equations \Vith Arbitrary Number of Variables ......................... 15.3. Nonlinear Equations With Three Variables Quadratic in Derivatives ............... 15.3.1. Equations Contain Squares of One or Two Derivatives ................... 15.3.2. Equations Contain Squares of Three Derivatives ........................ 15.3.3. Equations Contain Products of Derivatives With Respect to Different Variables 15.3.4. Equations Contain Squares and Products of Derivatives .................. 15.4. Other Nonlinear Equations With Three Variables Containing Parameters ........... 15.4.1. Equations Cubic in Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15.4.2. Equations Contain RooLs and Moduli of Derivatives .................... 15.4.3. Equations Contain Arbitrary Powers of Derivatives ......... ;' ........... , 15.5. Nonlinear Equations With Three Variables Containing Arbitrary Functions ......... 15.5.1. Equations Quadratic in Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15.5.2. Equations With Power Nonlinearity in Derivatives ...................... 15.5.3. Equations With Arbitrary Dependence on Derivatives. . . . . . . . . . . . . . . . . . .. 15.5.4. Nonlinear Equations of General Fonn ................................

407 407 407 409 416 418 418 422 424 424 429 430 432 432 432 433 434 437 437 443 445 446

xiv

CONTENTS

15.6. Nonlinear Equations With Four Independent Variables. . . . . . . . . . . . . . . . . . . . . . . . .. 450 15.6.1. Equations Quadratic in Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 450 15.6.2. Equations Contain Power-Law Functions of Derivatives .................. 452 15.7. Nonlinear Equations With Arbitrary Number of Variables Containing Arbitrary Parameters ............................................................ 15.7.1. Equations Quadratic in Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15.7.2. Equations With Power-Law Nonlinearity in Derivatives .................. 15.8. Nonlinear Equations With Arbitrary Number of Variables Containing Arbitrary Functions ............................................................. 15.8.1. Equations Quadratic in Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15.8.2. Equations With Power-Law Nonlinearity in Derivatives .................. 15.8.3. Equations Contain Arbitrary Functions of Two Variables ................. 15.804. Nonlinear Equations of General FOnTI • . . • . . • . . . • . . . . . • . . • . . . . . . . . . . . .

454 454 456 457 457 462 463 464

Supplement. Solution of Differential Equations Through the CONVODE Software . . .. S.l. Introduction ............................................................ S.I.I. Preliminary Remarks .............................................. S.1.2. Reduce Notation Used in CONVODE ................................. S.I.3. How CONVODE Solves Equations ................................... S.2. Examples of Solving Ordinary Differential Equations ........................... S.2.1. Riccati Equation (Example 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. S.2.2. Riccati Equation (Example 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. S.2.3. A Nonlinear Equation Quadratic in the Derivative. . . . . . . . . . . . . . . . . . . . . . .. S.3. Examples of Solving Partial Differential Equations ............................. S.3.l. A First Order Linear Equation (Example 1) ............................. S.3.2. A First Order Linear Equation (Example 2) ............ . . . . . . . . . . . . . . . .. S.3.3. A Second Order Nonlinear Equation .................................. SA. How to Use CONVODE .................................................. SA.1. Arguments ofthe CONVODE procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. SA.2. Global variables .................................................. SA.3. CONVODE via e-mail .............................................

469 469 469 469 470 471 471 475 479 481 481 483 486 490 490 491 492

References ............................................................... "

493

Index ..................................................................... 497

PREFACE First order partial differential equations are encountered in various fields of science and numerous applications (differential geometry, analytical mechanics, solid mechanics, gas dynamics, geometric optics, wave theory, heat and mass transfer, multi phase flows, control theory, differential games, calculus of variations, dynamic programming, chemical engineering sciences, etc.). Exact (closed-fonn) 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. They can be used to verify the consistency and estimate errors of various numerical, asymptotic, and approximate methods. The book contains about 3000 first order partial differential equations with solutions. Many new exact solutions to linear and nonlinear equations are included (a large portion of these solutions was constructed by "recalculating" the corresponding results obtained by the authors over the last decade in the field of ordinary differential equations). Special attention is paid to equations of general form which depend on arbitrary functions. Other equations contain one or more fr~e parameters (lhe book actually deals with families of differential equations); it is the reader's option to fix these parameters. A number of differential equations are considered which are encountered in various fields of applied mathematics, mechanics, physics, control theory, and engineering sciences. Totally, the number of equations described is several times greater than in any other book available. The handbook consists of chapters, sections, and subsections. The equations within a subsection are arranged in the increasing order of complexity. An extensive table of contents provides rapid access to the desired equations. Each chapter opens with a "Preliminary Remarks" section, which briefly outlines basic analytical methods for solving the corresponding types of differential equations and presents specific examples. Both classical (smooth) and generalized (nonsmooth, discontinuous) solutions of the Cauchy problem for nonlinear equations are considered. To meet the demands of a wider readership with diverse mathematical backgrounds, the authors tried to avoid the use of special terminology wherever possible. Therefore, some of the methods are outlined in a schematic and somewhat simplified manner, with necessary references made to books where these methods are considered in more detail. The main material is followed by a supplement which presents CONVODE, a specialized software package for solving ordinary differential equations and first order partial differential equations analytically. The reader can get access to CONVODE via e-mail. We would like to express our deep gratitude to Dr. Alexei Zhurov for useful remarks and invaluable help in preparing the camera-ready copy of the book and to Prof. Arik Melikyan for fruitful discussion of Sections 14.1 and 15.1. We also thank Richard Mairesse of the FUNDP computing center for helpful comments, maintaning Reduce, and more. The authors hope that the handbook will prove helpful for a wide readership of researchers, college and university teachers, engineers, and students in various fields of applied mathematics, mechanics, physics, optimal control, differential garnes, and engineering sciences.

Andrei D. Polyanill Valentin F. Zaitsev Alain Moussiat/x

AUTHORS Andrei D. Polyanin, D.Se., Ph.D., is a noted scientist of broad interests (ordinary differential, partial differential, and integral equations, mathematical physics, engineering mathematics, nonlinear mechanics, heat and mass transfer. chemical hydrodynamics, and others). Andrei Polyanin graduated from the Department of Mechanics and Mathematics of the Moscow State University in 1974. He received his Ph.D. degree in 1981 and D.Se. degree in 1986 at the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1975, Andrei Polyanin has been a member of the staff of the Institute for Problems in Mechanics of the Russian Academy of Sciences. Professor Polyanin is an author of 21 books in English. Russian, German, and Bulgarian. His publications also include more than 120 research papers and three patents, His most significant books are A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for OrdiTla 1}' Differential Equati01ls, CRC Press, 1995 and A. D. Polyanin and A. V. Manzhirov, Handbook of 11llegral Eqllations, CRC Press, 1998. In 1991, Andrei Polyanin was awarded a Chaplygin Prize of the USSR Academy of Sciences for his research in mechanics. Address: Institute for Problems in Mechanics. RAS. 101 Vcmadsky Avenue, Building I, 117526 Moscow, Russia E-mail: polyanin@ipmncLru

Valentin F. Zaitsev, Ph.D., D.Se., is a noted scientist in the fields of ordinary differential equations, mathematical physics, and nonlinear mechanics. Valentin Zaitsev graduated from the Radio Electronics Faculty of the Leningrad Poly technical Institute (now Saint-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 received his Doctor of Sciences degree; his D.Sc. thesis was dedicated to the discrete-group analysis of ordinary differential equations. In 1971-1996, Valentin Zaitsev worked in the Research Institute for Computational Mathematics and Control Processes of the SL 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 110 scientific publications. including 15 books and one patent. Address: Russian Slale Pedagogical University. 48 NuberezhnllYu reki Moiki. 191186 Sainl-Petersburg. Russia E-mail: z:[email protected]

AJain Moussiaux, Ph.D., D.Se., is a prominent scientist in the fields of computer algebra, general relativity models. and differential equations. Alain Moussiaux was born in 1943 in Belgium. He graduated from the University of Liege as a physicist in 1965 and received his Ph.D. degree in 1972 at the University of Namm (Belgium). His thesis was devoted to the application of the Riemann method for sLUdying progressive waves in various star models. Since 1968. Doctor Moussiaux has been a member of the staff of the Physical Department at the University of Namur. Alain Moussiaux came to computer algebra from general relativity models and mathematical physics a software models. One of his most significant achievements is the development of the CONVODE for analytical solution of ordinary and partial differentia] equations [see A. Moussiaux (l996)]. Doctor Moussiaux is an author of various books and publications in the fields of computer algebra, mathematical physics, and general relativity models. In 1982, Alain Moussiaux became a laureate of the Belgium Academy of Sciences for his work in computer algebra. Address: Facu1tes Universitaires de Namur, Depanement de Physique, 61 rue de Bruxelles. 5000 Namur. Belgique E-mail: [email protected]

ANNOTATION The book contains about 3000 first order partial differential equations with solutions. Many new exact solutions to linear and nonlinear equations are included. Special attention is paid to equations of general form, which depend on arbitrary functions. Other equations contain one or more free parameters (it is the reader's option to fix these parameters). In total, the number of equations described is several times greater than in any other book available. At the beginning of each section, basic solution methods for the corresponding types of differential equations are outlined and specific examples are considered. A number of differential equations are presented which are encountered in various applications (differential geometry, nonlinear mechanics, gas dynamics, geometric optics, wave theory, control theory, differential games, etc.). The handbook is intended for a wide readership of researchers, university teachers, engineers, and students in various fields of applied mathematics, mechanics, physics, control theory, and engineering sciences.

SOME NOTA-rION AND REMARKS by

1. In the original first order partial differential equations, the independent variables are denoted y, and z (or XI, ... , x n ), and the unknown function by w.

X,

2. The arbitrary constants in solutions (integrals) of differential equations are denoted bye and

e l , ... , en.

3. In some cases we use the following brief notation for partial derivatives:

8w Wx

8w

w y :;::: By'

8x'

8w W;:

8z'

8w

Pk

= 8 X k'

4. For a function of one variable, f

:;: : I(x), we denote the ordinary derivative dl/dx by I~. 5. All equations contain arbitrary functions f = I(x), 9 =g(y), h;;::; h(w), F(x, Wy), etc. andlor

arbitrary free parameters a, b, c, etc. As regards these functions and parameters, the following assumptions are usually adopted: (a) f:;::: I(x), g =g(y), h of real arguments;

= h(w), F(x, u), etc. are continuously differentiable real-valued functions

(b) the free parameters a, b, c, etc. may assume any real values for which the expressions occurring in the equation and the solution make sense (for example, if a solution contains a factor I~(l' then it is implied that a :;t:. 1; as a rule, this is not specified in the text).

6. If the solution contains an expression like k~l xk+1 (and the other terms of the solution make sense for k -1), then this expression can be extended by In Ixl at Ii, = -1. Such an extension is related to the computation of the integral

!

xk

dx = {

Ikn'~llxxi k+l

occurring in the solution. For brevity, the solution for k

if k:;c -1, if k =-1

=-1 is omitted.

7. If the solution contains an expression like teAl: (and the other terms of the solution make sense for A :;::: 0), then this expression can be extended by x at A = O. Such an extension is related to the computation of the integral

!

dx

= {xteAx

if,\ =/:. 0, if A=O occurring in the solution. For brevity, the solution for A 0 is omitted. eAX

=

8. The general solutions of many linear and quasilinear equations are described by an arbitrary function of one or several arguments. The function is assumed to be continuously differentiable with respect to its arguments. As a rule, the arbitrariness and smoothness of the function are not pointed out in the text. 9. In Chapters 12-15, singular solutions are not given. To find a singular solution is easy and does require solving differential equations (see Subsections 14.1.1 and 15.l.2). 10. Equations are numbered separately in each subsection. When referencing a particular equation, we use a notation like 3.2.4.5, which implies equation 5 from Subsection 3.2.4. 11. For calculating indefinite integrals encountered in this book in solutions of differential equations. it is useful to consult the handbooks by H. B. Dwight (1961), 1. S. Gradshteyn and L M. Ryzhik (1980). and A. P. Prudnikov, Yu. A. Brychkov, and O. 1. Marichev (1986, 1988).

Part I

ar E u tions With Ind pe dent ria les

Chapter 1

quations Containing One Derivative The equations with two independent variables which contain only one partial derivative may be treated as ordinary differential equations for the function w(x, V), where y (or x) plays the role of a parameter. Solutions of such linear equations are given below.

t> Notation: (I>

1.


8w

- : : : f(x,y).

8x

General solution: w = / I(x, y) dx + q,(y). In the integration, y is considered a parameter.

2.

8w

-

8y

= f(x, y).

General solution: w::: / I(x, y) dy +
3.

8w

-

8x

=f(x,y}w.

General solution: w

= if!(y) exp [ / f (x. y) dX].

In the integration, y is considered a param-

eter.

4.

8w

8y

= f{x,y)w.

General solution:

1/1

= if!(x) exp [ / f.(x, y) dY], In the integration, x is considered a param-

eter. 5.

8w

-

8x

::: f(x, y)w + g(x, y).

General solution:

w ::: E(x, y) [
g(x, Y)

/

E

(x,y)

dx

j

I

E(x, y)

=exp [ / f(x. y) dx

j.

In the integration, y is considered a parameter.

6.

8w

-

8y

= j(x, y)w + g(x, y}.

General solution:

W ::: E(x, y) [cI>(X) + /

~(x, Y) dyj, (x,y)

In the integration, x is considered a parameter.

E(x.y)

exp[j f(x,Y)dyj.

Chapter 2

Linear Equations of the Form

=0

f(x,y)~~ +g(x,y)~; 2.1. Preliminary Remarks I 2.1.1.

Solution Method

I

1. The characteristic equatioll. Consider a first order linear homogeneous partial differential equation with two independent variables of the form

8w

8w

x

y

j(x, Y)-8 + g(:1;, 1))-8 = o.

(1)

The first order ordinary differential equation

dx

_

dy

f(x, y) - g(x, U)

(2)

is called the characteristic equation corresponding to the partial differential equation (1). The integral curves of equation (2) are called characteristics.

Remark 1. Suppose the variables x and y belong to a domain G. Let the functions f(x, y) and g(x, y) be continuously differentiable with respect to both x and yin G and let f2(x, y)+ gl(x, y);i: 0 in G. Then only one characteristic passes through each point of G. 2. Formula/or tile general solution. Let the general solution of the characteristic equation (2)

be given by

3(x, y) = C,

(3)

where C is an arbitrary constant. Then the general solution of equation (I) has the form

w


(4)

where (P =
Remark 2. For brevity, only principal integrals (3) will often be presented in Chapler 2. The general solution of the equation is given by Eq. (4). 3. Physical itlterpretation. Equation (1) describes a steady-state distribution of the concentration of a substance in a plane flow (without regard for diffusion). Moreover, it is assumed that the fluid velocity components along the x- and y-axes are specified by the functions f and g.

o

References for Subsection 2.1.1: E. Kamke (1965). I. G. Petrovskii (970), H. Rhec. R. Ans. and N. R, Amundson (1986).

6

LINEAR EQUATIONS OF THE FORM f(x, y)

I 2.1.2.

Cauchy Problem (Initial Value Problem)

~~ + g(x, y) ~~

=0

I

1. Classical Cauchy problem. Find a solution w = w(x, y) of equation (1) satisfying the condition

w

= hey)

x

at

=XQ,

(5)

where h(y) is a given function. The solution of the Cauchy problem (also called the initial value problem) can be obtained from the general solution (4). Substituting the initial data (5) into Eq. (4), we have

hey) =

This relation serves to determine the function
2. Physical i1lterpretation of the Cauchy problem. Let x and y be spatial coordinates and let w be the concentration of a substance. It is assumed that the concentration distribution is governed by the steady-state transfer equation (1) and the concentration profile (5) is specified at the input cross-sec lion I = Xo. The concentration w w(x, y) is to be determined in the flow after the input cross-section (for x ;::: xo), Another, non-steady-state interpretation of the Cauchy problem is possible. Let x be time, y the spatial coordinate, and w the concentration (f == I). It is assumed that the concentration distribution is governed by the non-steady-state transfer equation (1) and that the concentration profile (5) is prescribed at the initial instant x = Xo. The concentration w = w(x, y) is to be determined for all subsequent instants of time (x ;::: IO). 3. Generalized Cauchy problem. Find a solution w

= w(x, y) of equation (I) satisfying the

initial conditions

(6) where ~ is a parameter (n ::;; E::;; /3), hi = hI (~) and h'1 = h2«() are given functions, and Ih~ I + Ih~l:;: o. A geometric interpretation of this problem is: find an integral surface of equation (1) passing through the line (6) defined parametrically. The solution of the generalized Cauchy problem can be obtained from the general solution (4) by substituting the initial data (6) in it.

Remark 1. In the formulation of the Cauchy problem (1), (6), it is assumed that the plane curve x =hI (~). Y = h2(~) is not tangent to any characteristic at any point, that 1S, the following inequality holds: f(h l , hz)hi g(hl, hz)h~ ;c O. Remark 2. If the curve x = hi (E), y = h2«() is a characteristic, then: (a) the Cauchy problem has no solution if h 3(O :;:E const or (b) the Cauchy problem has infinitely many solutions if h)(E) == const.

o References for Subsection

2.1.2: E. Kamke (1965). I. G, Pctrovskii (1970), H, Rhce, R. Aris, and N. R. Amundson

(1986).

I 2.1.3. Examples Example 1. Consider the equation

8w

aw o. ay

a y - +bx- =

ax

The general solution of the corresponding characteristic equation

dx ay

bx

(7)

2.2, EQUATIONS CONTAINING POWER-LAW FUNCTIONS

7

has the fonn bx 2 ay"!. = C. Thus. the general solution of the original partial differential equation can be expressed via an arbitrary composite function

Example 2. Consider the equation

8w ae :r 8w + b 8y

O.

The general solution of the characteristic equation

dx b C. Hence, the general solution of the original partial differential equation is given by ae X

has the fonn ay + be-X:

Example 3. Find a solution of the Cauchy problem for equation (7) with the initial condition w::: y2

at

x

=L

(9)

Substituting the initial data (9) inlO the general solution (8) yields y2

This allows us to conclude that (Nu) problem in the fonn

= q,(b ay2),

b-u = --. Substituting this expression into Eq. (8), we find the solution of the Cauchy a b(l - x 2) + ay2 w=------a

Example 4. Find a solution of equation (7) satisfying the following initial condition in parametric fonn:

x ={.

y

~,

w

e.

(0)

The general solution of the equation in question is given by Eq. (8). Taking into account the initial condition (10), we obtain

c = b-a. Hence, we find that ~(u) in the fonn

= (u/c)2

Substituting this expression inlo Eq. (8), we arrive at the solution of the Cauchy problem

w= (bx2_a

y2 )2

b-a

=

This solution is valid if a ':t b. For a b, the sl.raightline y = x ::: ~ (where the initial data are specified) is a characteristic; hence, the Cauchy problem in question has no solutions.

2.2. Equations Containing Power-law Functions

I 2.2.1.

Coefficients of Equations Are Linear in x and y

1.

aw ax

I

aw ay

a-+b-:::O.

General solution: w::: (bx - ay). where.p is an arbitrary function.

o 2.

Reference: E. Kamke (1965).

aw ax

Principal integral: 3.

aW ay

a - + (bx + c ) - ::: O.

aw ax

-

:=:::: -ibx2 + ex -

ay.

aw ay

+ (ax + by + c ) - ::: O.

Principal integral: ::::::: (abx + b2 y + a + bc)e- bx ,

8

4.

aw

aw

ax

ay

=0

OF THE FORM

LINEAR

ax-+by-=O. For a

= b, this is a conoid equation. Principal integral:

o Reference: E. Kamke (1965). 5.

8w ax

aw ay

:::: = [xlb[yr a •

=O.

ay- + bx-

Principal integral:::::

bx z ay2.

o Reference: E. Kamke (1965). 6.

aw 8w + (y + a ) ax ay

y-

Principal integral:::::

7.

x - y + a In Iy + al.

8w ax

aw 8y

(ay+bx+c)--(by+kx+s)-=O. Principal

8.

=O.

(alx

=ayZ + kx 2 + 2(bxy + cy + 5X), aw 8w S

+ bty +

8x + (a2 x + b2y + C2) ay

=O.

The principal integral is detennined by solutions of the following auxiliary system of algebraic equations for the parameters s, A. Jl, a, p. and ,: (a1 -

s)(b 2 - s) = a2 b(,

(l)

+ a2/-L = sA, btA + bZIL = SIL. qa + czp - s, = CIA + C2IL. (al - s)a + a2p = As, bl a + (b2 - s)p

(2)

al/\

(3)

=ILS.

(4)

Case 1: (al b2)2 + 4a2bt ;: O. Equation (1) has two different roots 51 and 52. To these roots there correspond two sets of solutions, AJ, JlI and A2. IL2, of system (2). 1.1. If al b2 alb!:;!:: 0, then 5, :;!:: 0 and 52 =F O. Hence the principal integral has the fonn + JlI CZl 52 '-' - IS2(A2 X + /-L2Y) + A2q + Jl2 cz1 5 [ __ lSI (AI X + ILlY) + Al Ct

•

1.2. If a,b 2 -a2bt = a, then 51 = s = al + b2 and 52 = O. Principal integral for AlCI + ILZC2 ;: 0: A.., x + ..... - A -lnlst(AIX + ILlY) + "'\JCI + ILt C21· ICI + IL2 C2 Principal integral for "'\2C) + IL2C2 = 0:

B=

AlX

+ /-LzY.

=

Case 2: (al-b 2 )2 +4U2bl = O. Equation (1) has the double root S t(al +b2). System (2) gives ...\ and Jl not equal to zero simultaneously. 2.1. If 5 0, then we find, from (3) and take nonzero 0: and 13 that satisfy relations (4). This leads to the principal integral

"*

_ .=

2.2. If S

s(a.x + py + ,)

=In IS(Ax + ILY) + Ct ...\ + c2ILI- 5 (A X+ILY ) +CI A +C2/-L

0, then b2 ;;:; -al. We have ~ ;;:; a2x2 -- 2alxy - bIY 2 + 2C2X - 2cI y.

o Reference: E. Kamke ( 1965).