Ordinary Differential Equations and Calculus of Variations Tilts Page Is Intentionally Left Blank ORDINARY DIFFERENTIAL EQUATIONS and CALCULUS of VARIATIONS

w ORDINARY i DIFFERENTIAL EQUATIONS AND CALCULUS OF VARIATIONS

M. V. Makarets V. Yu. Reshetnyak

World Scientific Ordinary Differential Equations and Calculus of Variations Tilts page is intentionally left blank ORDINARY DIFFERENTIAL EQUATIONS AND CALCULUS OF VARIATIONS

Book of Problems

M. V. Makarets Kiev T. Shevchenko University, Ukraine V. Yu. Reshetnyak Institute of Surface Chemistry, Ukraine

World Scientific Vh Singapore • New Jersey • London • Hong Kong Published by World Scientific Publishing Co. Pie. Lid. PO Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, Rivet Edge, NJ 07661 UK office: 57 Shelton Street, Coven! Garden. London WC2H 9HE

ORDINARY DIFFERENTIAL EQUATIONS AND CALCULUS OF VARIATIONS Copyright© 1995 by World Scientific Publishing Co. Pte. Ud. All rights reserved. This book, or parrs thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-G2-2191-6

This book is printed on acid-free paper

Printed in Singapore by Uto-Prinl Contents

PREFACE ix

1 FIRST ORDER DIFFERENTIAL EQUATIONS 1 1.1 Separable equations . ... 1 1.2 Homogeneous equations 9 1.2.1 Quasi homogeneous Equations . ... 16 1.3 Exact equations . 19 1.3.1 Integrating Factors .25 1.1 Linear equations ... 33 1.4.1 Bernoulli's Equation .41 1.4.2 Darboux's Equation . . 44 1.4.3 Riccati's Equation . 46 1.4.4 Bool's Equation . ... 50 1.5 Nonlinear equations ...... 52 1.5.1 Solvable Equations. General Solution . . 53 1.5.2 Solvable Equations. Singular Solution ... 59 1.5.3 Unsolvable Equations ...... 62 1.6 Applications in physics . . . . 64 1.6.1 Mechanics ...... 64 1.6.2 Hydrodynamics . 67 1.6.3 Electrical Networks . 68 1.6.4 Kinetic Theory . 69 1.6.5 Nuclear Physics .72 1.6.6 Optics ... .72 1.7 Miscellaneous problems . ... 74

2 N-th ORDER DIFFERENTIAL EQUATIONS 77 2.1 Reduction of order ... . . 77 2.1.1 Simple Cases . . 78 2.1.2 Homogeneous Equations . . 79 2.1.3 Exact Equations . 80 2.1.4 Linear Equations 82 2.1.5 The Initial Value Problem 83 2.2 Linear homogeneous equations ...... 87 vi CONTENTS

2.2.1 Exponential Solution ... 89 2.2.2 Power Solution .... 90 2.2.3 Transformations of Equation . . 92 2.2.4 The Initial Value Problem 94 2.3 Linear nonhomogeneous equations . 97 2.3.1 Method of Variation of Parameters . . 98 2.3.2 Method of Undetermined Coefficients 100 2.3.3 The Influence Function . 102 2.3.4 The Initial Value Problem . 103 2.4 Linear equation with constant coefficients . 107 2.4.1 The Homogeneous Equation with Constant Coefficients. 107 2.4.2 The Complete Equation with Constant Coefficients. Method of Undetermined Coefficients. . . . 112 2.4.3 The Method of Variatiou of Parameters . . .. 120 2.4.4 Symbolic Methods . . 123 2.4.5 Laplace Transform . . . 131 2.5 Equations with polynomial coefficients 140 2.5.1 Changes of Variable . 141 2.5.2 Substitutions . .143 2.5.3 Substitutions and Changes of Variable 145 2.5.4 Series Solutions 146

3 LINEAR SECOND ORDER EQUATIONS 153 3.1 Series solutions . 153 3.1.1 Ordinary Point . . 153 3.1.2 Regular Singular Point 157 3.1.3 Irregular Singular Point 166 3.2 Linear boundary value problem . 172 3.2.1 Homogeneous Problem 173 3.2.2 Nonhomogeneous Problem 175 3.2.3 Green's Function . 178 3.3 Eigenvalues and eigenf unctions 182 3.3.1 Self-adjoint Problems 184 3.3.2 The Sturm-Llouville Problem 186 3.3.3 Nonhomogeneous Problem 188

4 SYSTEMS OF DIFFERENTIAL EQUATIONS 191 4.1 Linear systems with constant coefficients 191 4.1.1 Homogeneous Systems . 191 4.1.2 Homogeneous Systems. Euler's Method 192 4.1.3 Euler's Method. Different Eigenvalues 192 4.1.4 Euler's Method. Repeated Eigenvalues 193 4.1.5 Repeated Eigenvalues. Method of Associated Vectors 194 CONTENTS vii

4.1.6 Repeated Eigenvalues. Method of Undetermined Coefficients . 198 4.1.7 Homogeneous Systems. Matrix Method . • • • • 199 4.1.8 Nonhomogeneous Systems 203 4.1.9 Method of Variation of Parameters . ... 203 4.1.10 Method of Undetermined Coefficients . 204 4.1.11 Matrix Method .... 205 4.1.12 Initial Value Problem ... 206 4.1.13 Laplace Transform ... 207 4.1.14 Systems of Higher Order Equations 208 4.2 Linear systems . 216 4.2.1 Solution by Eliminations . 216 4.2.2 Matrix Method 219 4.2.3 Nonhomogeneous Linear Systems .... . 219 4.2.4 Initial Value Problem 221 4.3 Nonlinear systems ...... 224 4.3.1 Method of Eliminations 225 4.3.2 Method of Integrable Combinations. 228 4.3.3 Systems of Bernoulli's Form 230 4.3.4 Method of Complex Variable . .... 231 4.3.5 Systems of Canonical Form ...... 232

5 PARTIAL EQUATIONS OF THE FIRST ORDER 237 5.1 Linear partial equations .... 237 5.2 Pfaffian equation . .... 244 5.2.1 Mayer's Method...... 246 5.3 Nonlinear partial equations ... . 248 5.3.1 Lagrange - Charpit's Method . . 250

6 NONLINEAR EQUATIONS AND STABILITY 255 6.1 Phase plane. Linear systems .... . 257 6.2 Almost linear systems ...... 266 6.3 Liapunov's second method . . 273

7 CALCULUS OF VARIATIONS 279 7.1 Euler's equation 279 7.2 Conditional extremum 284 7.2.1 Isoperimetric Problem ...... 288 7.3 Movable end points 292 7.4 Bolza problem . . 299 7.5 Euler-Poisson equation ...... 301 7.6 Ostrogradsky equation ...... 303 viii CONTENTS

8 ANSWERS TO PROBLEMS 307 8.1 Separable equations . 307 8.2 Homogeneous equations • 308 8.3 Exact equations ... • - 310 8.4 Linear equations 312 8.5 Nonlinear equations 315 8.6 Applications in physics . . . 318 8.7 Miscellaneous problems . . . 321 8.8 Reduction o( order 323 8.9 Linear homogeneous equations . • • 326 8.10 Linear nonhomogeneous equations . ... - 327 8.11 Linear equation with constant coefficients. . 330 8.12 Equations with polynomial coefficients 336 8.13 Series solutions . 338 8.14 Linear boundary value problems 342 8.15 Eigenvalues and eigenfunctions 344 8.16 Systems with constant coefficients . 346 8.17 Linear systems ...... 350 8.18 Nonlinear systems . ... 351 8.19 Linear partial equations . . 354 8.20 Pfaffian equation . 355 8.21 Nonlinear partial equations. . . 355 8.22 Phase plane. Linear systems . . . 357 8.23 Almost linear systems . 358 8.24 Liapunov's second method ...... 359 8.25 Euler's equation ...... 359 8.26 Conditional extremum . . 360 8.27 Isoperimetric problem . . 361 8.28 Movable end points . . 361 8.29 Bolza problem . . ... 362 8.30 Euler-Poissou equation ... 362 8.31 Ostrogradsky equation. . ... 363

BIBLIOGRAPHY 365

INDEX 369 Preface

This problem book contains exercises for courses in differential equations and cal• culus of variations at universities and technical institutes. It is designed for no n-mat hematics students and also for scientists and practicing engineers who feel a need to refresh their knowledge of such an important area of higher mathematics as differential equations and calculus of variations. Each section of the text begins with a summary of basic facts. This is followed by detailed solutions of examples and problems. The book contains more than 260 examples and about 1400 problems to be solved by the students, a considerable part of which have been composed by the authors themselves. Numerous references are given at the end of the book. These furnish sources for detailed theoretical approaches, and expanded treatment of applications. In preparing this book for publication, Mr. Y.-S. Kim rendered a great help to us.

ix Tilts page is intentionally left blank Chapter 1

FIRST ORDER DIFFERENTIAL EQUATIONS

1.1 Separable equations

A differential equation which can be written in the form

M(x)dx + N(y)dy = 0, (1) where M is a function of X alone and N is a function of y alone, is said to be separable. The solution is j M{x)dx + j N(y)dy = C, (2) where C is an arbitrary constant. The problem is then reduced to the problem of evaluating the two integrals in (2). In Eq.(l) we say that the variables are separated. Example 1. Find the solution of the equation

y' = e'*> which is such that y — 0 when x = 0. The equation may be written as

y' = eV, from which it is seen that the separated form is

e~*dy — e'dx.

Integrating now gives the general solution

-e~> = tz + C, and we have to find the value of the constant C such that x and y vanish simultane• ously. On putting t = y — 0, we have — 1 = 1 + C whence C = —2. The appropriate solution is given by

e"* = 2 - e'

1 2 CHAPTER 1. FIRST ORDER DIFFERENTIAL EQUATIONS

Example 2. Solve the equation

xydx+(x+l)dy = 0. (3)

If y j£ 0 and x + 1 ^ 0, we can divide by tj and i + I and put the equation in the form

dy xdx = 0.

Integrating,

J y J x + 1

Mjrl + i-ln \x + l\ = C. Taking exponential of both sides yields

jz-d^ + lje"1, C, = In |C|.

Equation (3) has also solutions y = 0 and x = — 1 The first one can be obtained from the general solution when arbitrary constant C\ — 0 and therefore JJ = 0 is the particular solution. The second solution x — — 1 can't be obtained from the genera! solution and therefore x = — 1 is the singular solution. Then the solution of the problem (3) is y = C,(x+ \)e-* if x jt -1; also I = -1. Example 3. Solve the initial value problem

y,coti + y = 2; y(j] = 0. (4)

KJf 2 and cot x ^ 0 the differential equation can be written as

dy • + tan xdx — 0.

Integrating, f dy l sin xdx _ ^

I y — 2 J cos x '

ln|s-2|-ln|eosi| = C. Whence

B = 2+&eosjf) (5) where C, = ]a\G\ is an arbitrary constant. To determine the particular solution satisfying the prescribed initial condition we substitute x = TT/3 and y = 0 into 1.1. SEPARABLE EQUATIONS 3

Eq,(5)> obtaining Ci — —4. Hence the desired particular solution is given explicitly by y - 2 - icosx. Example 4. Solve the equation

y2 (r3 + l) dx + (x3 - 5x2 + 6x) dy = 0.

If y ji 0 and x ^ 0, 2, 3 then the separated form is

dy (x3 + l)dx 2 3 2 If x -Sx + 6x or Sx2 - 6x + 1 x3-^ + 6x } " Using partial fractions we can write

5ia - 6i + 1 _ 5ia - 6x + 1 _ A B C i3 - 5i2 + 6i ~ s.(g - 2)(x - 3} ~ x+ x-2 ' 1-3'

Multiplying this by i we find

5I2 - fa + 1 Cx (i - 2)(z - 3) = 4+ z-2 +••- g-3

Then letting i —< 0, we have

5i2 - 6i + 1 [x - 2)(x - 3)

Similarly, multiplying by I - 2 and letting x — 2 yields

5x2 - 6x + 1 B = *(* - 3) and multiplying by x - 3 and letting i — 3 yields C = 28/3, Putting these values, we get 28 1 dx j/5 I 6i 2i- 2 + 3 i-3 which, on integration, gives ill 28 i = + ilnii|-^lQ|x-2| + -ln |i-3| + C, y r 6 2 J where C is an arbitrary constant. The given equation has also the singular solutions y = 0, (a; # 0, x * 2, a; # 3), s = 0 (y ^ 0), x = 2 (J ji 0) and x = 3 (y # 0). 4 CHAPTER I. FIRST ORDER DIFFERENTIAL EQUATIONS

Example 5. Solve the equation

(x3 + \)dy - ydx = 0.

If i £ -1 and g/Owt can write this equation in the form

dy dx 7 ~ *3 + l'

Since i3 + 1 = (i + l)(iJ — i + 1), we have using partial fractions

1 A Bx + C + i x+1 + xa - x+r where A, S, C are undetermined constants. Multiplying by (x + 1 )(x2 - i + 1) we obtain

1 = A {x* - x + l) + {Bx + C)(x + 1) = x'(-4 + B) + x{-A + B + C) + [A + C).

Since this is an identity we have on equating coefficients of like powers of x,

A + B = 0t

-A-i-B + C = 0,

A + C = I. Solving these we find A = 1/3, B - -1/3, C - 2/3. On substituting these values and integrating we have

f dy _ 1 / _rfx_ _ 1 / x-2 / S 3/x + l 3/ + l

= I|n|I+ll_I f ~'/2)d(x - 1/2) 1 r dx 3 ' 1 3/ (x-l/2)J + 3/4 +2J (x~ 1/2)'+ 3/4

= ^ In |x + 11 - i In (x! - x + 1) + -L arctan + fj, or , I | 1 . (x +1)5 1 2x - 1 _ in ji = - In — — + -= arctan —7=- + C, 6 x1 - x + 1 ,/3 where C is an arbitrary constant. The original equation has also the singular solutions x = -l (y?!0) andV = 0(x^-l). If a differential equation can be written in the form

y' = f[ax + 63,) (6]