Cambridge University Press Differential Equations Linear, Nonlinear, Ordinary, Partial

Cambridge University Press Differential Equations Linear, Nonlinear, Ordinary, Partial

(Parte 1 de 4)

Differential Equations Linear, Nonlinear, Ordinary, Partial

A.C. King, J. Billingham and S.R. Otto

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom


First published in print format

© Cambridge University Press 2003 2003

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Preface page ix

Part One: Linear Equations 1

1 Variable Coefficient, Second Order, Linear, Ordinary Differential Equations 3

1.1 The Method of Reduction of Order 5 1.2 The Method of Variation of Parameters 7 1.3 Solution by Power Series: The Method of Frobenius 1

2 Legendre Functions 31

2.2 The Generating Function for Pn(x) 35 2.3 Differential and Recurrence Relations Between Legendre

Polynomials 38

2.4 Rodrigues’ Formula 39 2.5 Orthogonality of the Legendre Polynomials 41 2.6 Physical Applications of the Legendre Polynomials 4 2.7 The Associated Legendre Equation 52

3 Bessel Functions 58 3.1 The Gamma Function and the Pockhammer Symbol 58 3.2 Series Solutions of Bessel’s Equation 60

3.3 The Generating Function for Jn(x), n an integer 64 3.4 Differential and Recurrence Relations Between Bessel Functions 69

3.5 Modified Bessel Functions 71 3.6 Orthogonality of the Bessel Functions 71 3.7 Inhomogeneous Terms in Bessel’s Equation 7 3.8 Solutions Expressible as Bessel Functions 79 3.9 Physical Applications of the Bessel Functions 80

4 Boundary Value Problems, Green’s Functions and Sturm–Liouville Theory 93

4.1 Inhomogeneous Linear Boundary Value Problems 96

4.2 The Solution of Boundary Value Problems by Eigenfunction Expansions 100

4.3 Sturm–Liouville Systems 107 vi CONTENTS

5 Fourier Series and the Fourier Transform 123 5.1 General Fourier Series 127 5.2 The Fourier Transform 133 5.3 Green’s Functions Revisited 141 5.4 Solution of Laplace’s Equation Using Fourier Transforms 143 5.5 Generalization to Higher Dimensions 145

6 Laplace Transforms 152 6.1 Definition and Examples 152 6.2 Properties of the Laplace Transform 154

6.3 The Solution of Ordinary Differential Equations using Laplace Transforms 157

6.4 The Inversion Formula for Laplace Transforms 162

7 Classification, Properties and Complex Variable Methods for Second Order Partial Differential Equations 175

7.1 Classification and Properties of Linear, Second Order Partial Differential Equations in Two Independent Variables 175

7.2 Complex Variable Methods for Solving Laplace’s Equation 186

Part Two: Nonlinear Equations and Advanced Techniques 201

8 Existence, Uniqueness, Continuity and Comparison of Solutions of Ordinary Differential Equations 203

8.1 Local Existence of Solutions 204 8.2 Uniqueness of Solutions 210 8.3 Dependence of the Solution on the Initial Conditions 211 8.4 Comparison Theorems 212

9 Nonlinear Ordinary Differential Equations: Phase Plane Methods 217

9.1 Introduction: The Simple Pendulum 217

9.2 First Order Autonomous Nonlinear Ordinary Differential Equations 2

9.3 Second Order Autonomous Nonlinear Ordinary Differential Equations 224

9.4 Third Order Autonomous Nonlinear Ordinary Differential Equations 249

10 Group Theoretical Methods 256 10.1 Lie Groups 257 10.2 Invariants Under Group Action 261 10.3 The Extended Group 262

10.4 Integration of a First Order Equation with a Known Group Invariant 263


10.5 Towards the Systematic Determination of Groups Under Which a First Order Equation is Invariant 265 10.6 Invariants for Second Order Differential Equations 266 10.7 Partial Differential Equations 270

1 Asymptotic Methods: Basic Ideas 274 1.1 Asymptotic Expansions 275 1.2 The Asymptotic Evaluation of Integrals 280

12 Asymptotic Methods: Differential Equations 303 12.1 An Instructive Analogy: Algebraic Equations 303 12.2 Ordinary Differential Equations 306 12.3 Partial Differential Equations 351

13 Stability, Instability and Bifurcations 372 13.1 Zero Eigenvalues and the Centre Manifold Theorem 372 13.2 Lyapunov’s Theorems 381 13.3 Bifurcation Theory 388

14 Time-Optimal Control in the Phase Plane 417 14.1 Definitions 418 14.2 First Order Equations 418 14.3 Second Order Equations 422 14.4 Examples of Second Order Control Problems 426 14.5 Properties of the Controllable Set 429 14.6 The Controllability Matrix 433 14.7 The Time-Optimal Maximum Principle (TOMP) 436

15 An Introduction to Chaotic Systems 447 15.1 Three Simple Chaotic Systems 447 15.2 Mappings 452 15.3 The Poincare Return Map 467 15.4 Homoclinic Tangles 472 15.5 Quantifying Chaos: Lyapunov Exponents and the Lyapunov Spectrum 484

Appendix 1 Linear Algebra 495 Appendix 2 Continuity and Differentiability 502 Appendix 3 Power Series 505 Appendix 4 Sequences of Functions 509 Appendix 5 Ordinary Differential Equations 511 Appendix 6 Complex Variables 517 Appendix 7 A Short Introduction to MATLAB 526

Bibliography 534 Index 536


When mathematical modelling is used to describe physical, biological or chemical phenomena, one of the most common results is either a differential equation or a system of differential equations, together with appropriate boundary and initial conditions. These differential equations may be ordinary or partial, and finding and interpreting their solution is at the heart of applied mathematics. A thorough introduction to differential equations is therefore a necessary part of the education of any applied mathematician, and this book is aimed at building up skills in this area. For similar reasons, the book should also be of use to mathematically-inclined physicists and engineers.

Although the importance of studying differential equations is not generally in question, exactly how the theory of differential equations should be taught, and what aspects should be emphasized, is more controversial. In our experience, textbooks on differential equations usually fall into one of two categories. Firstly, there is the type of textbook that emphasizes the importance of abstract mathematical results, proving each of its theorems with full mathematical rigour. Such textbooks are usually aimed at graduate students, and are inappropriate for the average undergraduate. Secondly, there is the type of textbook that shows the studenth ow to construct solutions of differential equations, with particular emphasis on algorithmic methods. These textbooks often tackle only linear equations, and have no pretension to mathematical rigour. However, they are usually well-stocked with interesting examples, and often include sections on numerical solution methods.

In this textbook, we steer a course between these two extremes, starting at the level of preparedness of a typical, but well-motivated, second year undergraduate at a British university. As such, the book begins in an unsophisticated style with the clear objective of obtaining quantitative results for a particular linear ordinary differential equation. The text is, however, written in a progressive manner, with the aim of developing a deeper understanding of ordinary and partial differential equations, including conditions for the existence and uniqueness of solutions, solutions by group theoretical and asymptotic methods, the basic ideas of control theory, and nonlinear systems, including bifurcation theory and chaos. The emphasis of the book is on analytical and asymptotic solution methods. However, where appropriate, we have supplemented the text by including numerical solutions and graphs produced using MATLAB†, version 6. We assume some knowledge of

† MATLAB is a registered trademark of The MathWorks, Inc.


MATLAB (summarized in Appendix 7), but explain any nontrivial aspects as they arise. Where mathematical rigour is required, we have presented the appropriate analysis, on the basis that the student has taken first courses in analysis and linear algebra. We have, however, avoided any functional analysis. Most of the material in the book has been taught by us in courses for undergraduates at the University of Birmingham. This has given us some insight into what students find difficult, and, as a consequence, what needs to be emphasized and re-iterated.

The book is divided into two parts. In the first of these, we tackle linear differential equations. The first three chapters are concerned with variable coefficient, linear, second order ordinary differential equations, emphasizing the methods of reduction of order and variation of parameters, and series solution by the method of Frobenius. In particular, we discuss Legendre functions (Chapter 2) and Bessel functions (Chapter 3) in detail, and motivate this by giving examples of how they arise in real modelling problems. These examples lead to partial differential equations, and we use separation of variables to obtain Legendre’s and Bessel’s equations. In Chapter 4, the emphasis is on boundary value problems, and we show how these differ from initial value problems. We introduce Sturm–Liouville theory in this chapter, and prove various results on eigenvalue problems. The next two chapters of the first part of the book are concerned with Fourier series, and Fourier and Laplace transforms. We discuss in detail the convergence of Fourier series, since the analysis involved is far more straightforward than that associated with other basis functions. Our approach to Fourier transforms involves a short introduction to the theory of generalized functions. The advantage of this approach is that a discussion of what types of function possess a Fourier transform is straightforward, since all generalized functions possess a Fourier transform. We show how Fourier transforms can be used to construct the free space Green’s function for both ordinary and partial differential equations. We also use Fourier transforms to derive the solutions of the Dirichlet and Neumann problems for Laplace’s equation. Our discussion of the Laplace transform includes an outline proof of the inversion theorem, and several examples of physical problems, for example involving diffusion, that can be solved by this method. In Chapter 7 we discuss the classification of linear, second order partial differential equations, emphasizing the reasons why the canonical examples of elliptic, parabolic and hyperbolic equations, namely Laplace’s equation, the diffusion equation and the wave equation, have the properties that they do. We also consider complex variable methods for solving Laplace’s equation, emphasizing their application to problems in fluid mechanics.

The second part of the book is concerned with nonlinear problems and more advanced techniques. Although we have used a lot of the material in Chapters 9 and 14 (phase plane techniques and control theory) in a course for second year undergraduates, the bulk of the material here is aimed at third year students. We begin in Chapter 8 with a brief introduction to the rigorous analysis of ordinary differential equations. Here the emphasis is on existence, uniqueness and comparison theorems. In Chapter 9 we introduce the phase plane and its associated techniques. This is the first of three chapters (the others being Chapters 13 and 15) that form an introduction to the theory of nonlinear ordinary differential equations,

PREFACE xi often known as dynamical systems. In Chapter 10, we show how the ideas of group theory can be used to find exact solutions of ordinary and partial differential equations. In Chapters 1 and 12 we discuss the theory and practice of asymptotic analysis. After discussing the basic ideas at the beginning of Chapter 1, we move on to study the three most important techniques for the asymptotic evaluation of integrals: Laplace’s method, the method of stationary phase and the method of steepest descents. Chapter 12 is devoted to the asymptotic solution of differential equations, and we introduce the method of matched asymptotic expansions, and the associated idea of asymptotic matching, the method of multiple scales, including Kuzmak’s method for analysing the slow damping of nonlinear oscillators, and the WKB expansion. We illustrate each of these methods with a wide variety of examples, for both nonlinear ordinary differential equations and partial differential equations. In Chapter 13 we cover the centre manifold theorem, Lyapunov functions and an introduction to bifurcation theory. Chapter 14 is about time-optimal control theory in the phase plane, and includes a discussion of the controllability matrix and the time-optimal maximum principle for second order linear systems of ordinary differential equations. Chapter 15 is on chaotic systems, and, after some illustrative examples, emphasizes the theory of homoclinic tangles and Mel’nikov theory.

There is a set of exercises at the end of each chapter. Harder exercises are marked with a star, and many chapters include a project, which is rather longer than the average exercise, and whose solution involves searches in the library or on the Internet, and deeper study. Bona fide teachers and instructors can obtain full worked solutions to many of the exercises by emailing

In order to follow many of the ideas and calculations that we describe in this book, and to fully appreciate the more advanced material, the reader may need to acquire (or refresh) some basic skills. These are covered in the appendices, and fall into six basic areas: linear algebra, continuity and differentiability, power series, sequences and series of functions, ordinary differential equations and complex variables.

We would like to thank our friends and colleagues, Adam Burbidge (Nestle Research Centre, Lausanne), Norrie Everitt (Birmingham), Chris Good (Birmingham), Ray Jones (Birmingham), John King (Nottingham), Dave Needham (Reading), Nigel Scott (East Anglia) and Warren Smith (Birmingham), who read and commented on one or more chapters of the book before it was published. Any nonsense remaining is, of course, our fault and not theirs.

ACK, JB and SRO, Birmingham 2002

Part One Linear Equations

Variable Coefficient, Second Order, Linear, Ordinary Differential Equations

Many physical, chemical and biological systems can be described using mathematical models. Once the model is formulated, we usually need to solve a differential equation in order to predict and quantify the features of the system being modelled. As a precursor to this, we consider linear, second order ordinary differential equations of the form

(Parte 1 de 4)