Differential Equations Theory, Technique, and Practice George F. Simmons and Steven G. Krantz Higher Education Boston Burr Ridge, IL Dubuque, IA Madison, Wl New York San Francisco St. Louis Bangkok Bogota Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto
TABLE OF CONTENTS r. PREFACE. VIII CHAPTER 1 WHAT IS A DIFFERENTIAL EQUATION?.1.2.3.4.5.6.7.8.9 1.10 1.11 The Nature of Solutions Separable Equations First-Order Linear Equations Exact Equations Orthogonal Trajectories and Families of Curves Homogeneous Equations Integrating Factors Reduction of Order 1.9.1 Dependent Variable Missing 1.9.2 Independent Variable Missing The Hanging Chain and Pursuit Curves 1.10.1 The Hanging Chain 1.10.2 Pursuit Curves Electrical Circuits Anatomy of an Application: The Design of a Dialysis Machine 1 2 4 10 13 17 22 26 29 33 33 35 38 38 42 45 49 53 CHAPTER 2 SECOND-ORDER LINEAR EQUATIONS 2.1 2.2 2.3 2.4 2.5 Second-Order Linear Equations with Constant Coefficients The Method of Undetermined Coefficients The Method of Variation of Parameters The Use of a Known Solution to Find Another Vibrations and Oscillations 2.5.1 Undamped Simple Harmonic Motion 2.5.2 Damped Vibrations 2.5.3 Forced Vibrations 2.5.4 A Few Remarks "about Electricity 57 58 63 67 71 75 75 77 80 82 in
iv Table of Contents 2.6 Newton's Law of Gravitation and Kepler's Laws 84 2.6.1 Kepler's Second Law 86 2.6.2 Kepler's First Law 87 2.6.3 Kepler's Third Law 89 2.7 Higher Order Linear Equations, Coupled Harmonic Oscillators 93 Historical Note: Euler 99 Anatomy of an Application: Bessel Functions and the Vibrating Membrane 101 105 CHAPTER 3 QUALITATIVE PROPERTIES AND THEORETICAL ASPECTS 109 3.1 Review of Linear Algebra 110 3.1.1 Vector Spaces 110 3.1.2 The Concept of Linear Independence 111 3.1.3 Bases 113 3.1.4 Inner Product Spaces 114 3.1.5 Linear Transformations and Matrices 115 3.1.6 Eigenvalues and Eigenvectors 117 3.2 A Bit of Theory 119 3.3 Picard's Existence and Uniqueness Theorem 125 3.3.1 The Form of a Differential Equation 125 3.3.2 Picard's Iteration Technique 126 3.3.3 Some Illustrative Examples 127 3.3.4 Estimation of the Picard Iterates 129 3.4 Oscillations and the Sturm Separation Theorem 130 3.5 The Sturm Comparison Theorem 138 Anatomy of an Application-. The Green's Function 142 146 CHAPTER4 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 4.1 Introduction and Review of Power Series 4.1.1 Review of Power Series 4.2 Series Solutions of First-Order Differential Equations 149 150 150 159
Table of Contents v 4.3 Second-Order Linear Equations: Ordinary Points 164 4.4 Regular Singular Points 171 4.5 More on Regular Singular Points 177 4.6 Gauss's Hypergeometric Equation 184 Historical Note: Gauss 189 Historical Note: Abel 190 Anatomy of an Application: Steady-State Temperature in a Ball 192 194 CHAPTER 5 FOURIER SERIES: BASIC CONCEPTS 197 5.1 Fourier Coefficients 198 5.2 Some Remarks about Convergence 207 5.3 Even and Odd Functions: Cosine and Sine Series 211 5.4 Fourier Series on Arbitrary Intervals 218 5.5 Orthogonal Functions 221 Historical Note: Riemann 225 Anatomy of an Application: Introduction to the Fourier Transform 227 236 CHAPTER 6 PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 239 6.1 Introduction and Historical Remarks 240 6.2 Eigenvalues, Eigenfunctions, and the Vibrating String 243 6.2.1 Boundary Value Problems 243 6.2.2 Derivation of the Wave Equation 244 6.2.3 Solution of the Wave Equation 246 6.3 The Heat Equation 251 6.4 The Dirichlet Problem for a Disc 256 6.4.1 The Poisson Integral 259 6.5 Sturm-Liouville Problems 262 Historical Note: Fourier 267 Historical Note: Dirichlet 268 Anatomy of an Application: Some Ideas from Quantum Mechanics 270 273
VI Table of Contents CHAPTER LAPLACE TRANSFORMS 277 7.1 7.2 7.3 7.4 7.5 Introduction Applications to Differential Equations Derivatives and Integrals of Laplace Transforms Convolutions 7.4.1 Abel's Mechanical Problem The Unit Step and Impulse Functions Historical Note: Laplace Anatomy of an Application: Flow Initiated by an Impulsively Started Flat Plate 278 280 285 291 293 298 305 306 309 CHAPTER 8 THE CALCULUS OF VARIATIONS 315 8.1 8.2 8.3 Euler's Equation Isoperimetric Problems and the Like 8.3.1 Lagrange Multipliers 8.3.2 Integral Side Conditions 8.3.3 Finite Side Conditions Historical Note: Newton Anatomy of an Application: Hamilton's Principle and its Implications 316 319 327 328 329 333 338 340 344 CHAPTER 9 NUMERICAL METHODS 347 9.1 9.2 9.3 9.4 9.5 The Method of Euler The Error Term An Improved Euler Method The Runge-Kutta Method Anatomy of an Application: A Constant Perturbation Method for Linear, Second-Order Equations 348 349 353 357 360 365 368 CHAPTER 1 0 SYSTEMS OF FIRST-ORDER EQUATIONS 371 10.1 10.2 Linear Systems 372 374
Table of Contents vii 10.3 Homogeneous Linear Systems with Constant Coefficients 382 10.4 Nonlinear Systems: Volterra's Predator-Prey Equations 389 Anatomy of an Application: Solution of Systems with Matrices and Exponentials 395 400 CHAPTER 11 THE NONLINEAR THEORY 403 11.1 Some Motivating Examples 404 11.2 Specializing Down 404 11.3 Types of Critical Points: Stability 409 11.4 Critical Points and Stability for Linear Systems 417 11.5 Stability by Liapunov's Direct Method 427 11.6 Simple Critical Points of Nonlinear Systems 432 11.7 Nonlinear Mechanics: Conservative Systems 439 11.8 Periodic Solutions: The Poincare-Bendixson Theorem 444 Historical Note: Poincare 452 Anatomy of an Application: Mechanical Analysis of a Block on a Spring 454 457 CHAPTER 12 DYNAMICAL SYSTEMS 461 12.1 Flows 462 12.1.1 Dynamical Systems 464 12.1.2 Stable and Unstable Fixed Points 466 12.1.3 Linear Dynamics in the Plane 468 12.2 Some Ideas from Topology 475 12.2.1 Open and Closed Sets 475 12.2.2 The Idea of Connectedness 476 12.2.3 Closed Curves in the Plane 478 12.3 Planar Autonomous Systems 480 12.3.1 Ingredients of the Proof of Poincare-Bendixson 480 Anatomy of an Application: Lagrange's, Equations 489 493 BIBLIOGRAPHY 495 ANSWERS TO ODD-NUMBERED EXERCISES 497 INDEX ; 525