Differential Equations:

An Introduction With Applications

LOTHAR COLLATZ

University of Hamburg Federal Republic of Germany

Translated by

E. R. DAWSON,

University of Dundee UK

A Wiley-Interscience Publication

JOHN WILEY & SONS Chichester • New York • Brisbane • Toronto • Singapore

Contents

Pref ace xi

Classification of differential equations 1 1 Definitions and notation 1 2 Examples of differential equations arising in physics 2

I Ordinary differential equations of the first order

§1 The direction field and the simplest integrable types 5 3 Solution curves in the direction field 5 4 Separation of the variables 7 7 5 The similarity differential equation 10 6 Simple cases reducible to the similarity differential equation .. 11

§2 The linear differential equation of first order 13 7 Homogeneous and inhomogeneous equations; the trivial

Solution 13 8 Solution of a homogeneous equation 14 9 Solution of the inhomogeneous equation 14

§3 The Bernoulli differential equation 17 10 Reduction to a linear differential equation 17 11 The Riccati differential equation 18

§4 Integrating factors 19 12 Exact differential equations 19 13 Integrating factors 20

§5 Preliminaries to the questions of existence and uniqueness 22 14 Single-valued and multi-valued direction fields 22 15 Non-uniqueness of the Solution 23 16 The Lipschitz condition; its stronger and weaker forms 24 17 The method of successive approximations 26

§6 The general existence and uniqueness theorem 29 18 The existence theorem 29 19 Proof of uniqueness 32

v

•J

vi

20 Systems of differential equations and a differential equation of nth order 33

21 Some fundamental concepts in functional analysis 36 22 Banach's fixed-point theorem and the existence theorem for

ordinary differential equations 42 §7 Singular line-elements 45

23 Regulär and singular line-elements. Defmitions and examples . 45 24 Isolated singular points 49 25 On the theory of isolated singular points 52 26 Clairaut's and d'Alembert's differential equations 55 27 Oscillations with one degree of freedom. Phase curves 58 28 Examples of oscillations and phase curves 60 29 Periodic oscillations of an autonomous undamped System with

one degree of freedom 66 §8 Miscellaneous problems and Solutions 71

30 Problems 71 31 Solutions 73

II Ordinary differential equations of higher order

§1 Some types of non-linear differential equations 77 1 The dependent variable y does not appear explicitly 77 2 The equation y" = f(y) and the energy integral 78 3 The general differential equation in which the independent

variable x does not occur explicitly 79 4 The differential equation contains only the ratios y(v)/y 81

§2 Fundamental theorems on linear differential equations 82 5 Notation 82 6 The superposition theorem 83 7 Reduction of the order of a linear differential equation 84

§3 The fundamental Systems of a linear differential equation 85 8 Linear dependence of functions 85 9 The Wronskian criterion for linear independence of functions 88

10 The general Solution of a linear differential equation 90 §4 Linear differential equations with constant coefficients 92

11 A trial Substitution for the Solution; the characteristic equation 92 12 Multiple zeros of the characterisitc equation 95 13 A criterion for stability 99 14 The equation for forced oscillations 99 15 Solution of the homogeneous equation for oscillations 101

§5 Determination of a particular Solution of the inhomogeneous linear differential equation 103

16 The method of Variation of the constants 104 17 The rule-of-thumb method 105 18 Introduction of a complex differential equation 107 19 The case of resonance 109

vii

§6 The Euler differential equation 109 20 The Substitution for the Solution; the characteristic equation 109 21 Examples 111

§7 Systems of linear differenetial equations 112 22 Example: vibrations of a motorcar; (types of coupling) 112 23 The fundamental System of Solutions 116 24 Solution of the inhomogeneous System by the variation-of-the-

constants method 118 25 Matrix A constant; characteristic roots of a matrix 118 26 The three main classes in the theory of square matrices 119 27 Application to the theory of oscillations 121 28 Example of a physical System which leads to a non-

normalizable matrix 122 29 Transformation of a normal or normalizable matrix to

diagonal form 124 §8 Linear differential equations with periodic coefficents 130

30 Engineering examples leading to differential equations with periodic coefficients 130

31 Periodic Solutions of the homogeneous System 132 32 Stability 133 33 Periodic Solutions for the inhomogeneous System 135 34 An example for the stability theory 136

§9 The Laplace transformation 138 35 Definition of the Laplace transformation 138 36 Differentiation and integration of the original element 141 37 Using the Laplace transformation to solve initial-value prob­

lems for ordinary differential equations 143 38 Transients and periodic Solutions 146 39 The convolution theorem and integral equations of the first

kind 152 40 The inverse Laplace transformation and tables 156

III Boundary-value problems and, in particular, eigenvalue problems

1 Initial-value problems and boundary-value problems 159 §1 Examples of linear boundary-value problems 160

2 A beam. The several fields of the differential equation 160 3 The number of Solutions in linear boundary-value problems . . 163

§2 Examples of non-linear boundary-value problems 164 4 The differential equation of a catenary 164 5 The differential equation y" = y2 167 6 A countable infinity of Solutions of a boundary-value problem

with the equation y" = - _y3 171 §3 The alternatives in linear boundary-value problems with ordinary

differential equations 173

V l l l

7 Semi-homogeneous and fully homogeneous boundary-value problems 173

8 The general alternative 174 §4 Solving boundary-value problems by means of the Green's

function 176 9 Some very simple examples of Green's functions 176

10 The Green's function as an influence function 178 11 General defmition of the Green's function 180 12 The Solution formula for the boundary-value problem 183

§5 Examples of eigenvalue problems 184 13 The completely homogeneous boundary-value problem 184 14 The corresponding non-linear boundary-value problem 188 15 Partial differential equations 189 16 The Bernoulli Substitution for natural vibrations 191

§6 Eigenvalue problems and orthonormal Systems 194 17 Self-adjoint and positive-definite eigenvalue problems 194 18 Orthogonality of the eigenfunctions 197 19 Orthonormal Systems 199 20 Approximation in the mean 204 21 On the expansion theorem 205 22 The theorem on inclusion of an eigenvalue between two

quotients 207 §7 Connections with the calculus of variations 210

23 Some simple examples 210 24 The Euler equation in the calculus of variations in the simplest

case 213 25 Free boundary-value problems and the calculus of variations . 216

§8 Branching problems 218 26 Branching problems with ordinary differential equations of the

second order 218 27 Non-linear eigenvalue problems and branching problems 218 28 Example. The branching diagram for a Urysohn integral

equation 219 §9 Miscellaneous problems and Solutions on chapters II and III . . . 221

29 Problems 221 30 Solutions 224

IV Particular differential equations

§1 Spherical harmonics 231 1 Solutions of Laplace's equation 231 2 The generating function 233 3 Legendre functions of the second kind 237 4 Another explicit representation of the Legendre polynomials . . 238 5 Orthogonality 239

ix

§2 Besscl functions 240 6 The partial differential equation for vibrations of a membrane 240 7 The Bernoulli Substitution for vibrations of a membrane 241 8 The generating function 243 9 Deductions from the generating function 244

10 An integral representation 246 11 An example from astronomy: the Kepler equation 247 12 Bessel functions of the second kind 249 13 More general differential equations giving Bessel functions . . . 249 14 Vibration modes of a circular membrane 254

§3 Series expansions; the hypergeometric function 254 15 The series Substitution; the indicial equation 255 16 The roots of the indicial equation 256 17 Examples; the hypergeometric function 257 18 The method of perturbations and Singular points 260 19 An example of the series-expansion technique 262

V An excursus into partial differential equations

§1 General Solutions of linear partial differential equations with constant coefficients 265

1 Simple linear partial differential equations 265 2 The wave equation and Laplace's equation 267 3 Non-linear differential equations. The breaking of a wave . . . . 270

§2 Initial-value problems and boundary-value problems 275 4 The three fundamental types of a quasi-linear partial

differential equation of the second order 275 5 Range of influence and domain of continuation 278 6 Solution of a boundary-value problem for a circular domain . 280 7 Example: a temperature distribution 282

§3 The boundary-maximum theorem and monotonicity 286 8 The boundary-maximum theorem in potential theory in the

plane and in three-dimensional space 286 9 Continuous dependence of the Solution of the boundary-value

problem on the data 288 10 Monotonicity theorems. Optimization and approximation . . . . 290 11 A numerical example; a torsion problem 292

§4 Well-posed and free boundary-value problems 294 12 Well-posed and not well-posed problems 294 13 Further examples of problems which are not well-posed 296 14 Examples of free boundary-value problems 297

§5 Relations with the calculus of variations and the finite-element method 299

15 The variational problem for the boundary-value problem of the first kind in potential theory 299

16 The Ritz method 301

X

17 The method of finite elements 302 18 An example 305

§6 The Laplace transformation and Fourier transformation with partial differential equations 306

19 A parabolic equation (the equation of heat conduction) 306 20 The Laplace transformation with the wave equation 308 21 The reciprocity formulae for the Fourier transformation 309 22 The Fourier transformation with the heat-conduction equation 313

VI Appendix. Some methods of approximation and further examples for practice

§1 Some methods of approximation to the Solution of ordinary differential equations 315

1 Preliminary remarks and some rough methods of approximation 315

2 The Runge and Kutta method of approximate numerical integration 316

3 The method of central differences 318 4 The method of difference quotients 321 5 Multi-point methods 323 6 The Ritz method 324

§2 Further problems and their Solutions 326 7 Miscellaneous problems on Chapter I 326 8 Solutions 328 9 Miscellaneous problems on Chapters II and III 332

10 Solutions 336 11 Miscellaneous problems on Chapters IV and V 347 12 Solutions 352

§3 Some biographical data 363 13 The dates of some mathematicians 363

Bibliography 364

Index 367