Differential Equations

Preface for the Instructor xv
Preface for the Student xvii
1 What Is a Differential Equation? 1
1.1 Introductory Remarks 1
1.2 A Taste of Ordinary Differential Equations 5
1.3 The Nature of Solutions 7
1.4 Separable Equations 15
1.5 First-Order, Linear Equations 18
1.6 Exact Equations 24
1.7 Orthogonal Trajectories and Curves 30
1.8 Homogeneous Equations 36
1.9 Integrating Factors 41
1.10 Reduction of Order 46
1.10.1 Dependent Variable Missing 46
1.10.2 Independent Variable Missing 48
1.11 The Hanging Chain and Pursuit Curves 52
1.11.1 The Hanging Chain 52
1.11.2 Pursuit Curves 57
1.12 Electrical Circuits 62
1.13 The Design of a Dialysis Machine 67
Problems for Review and Discovery 72
2 Second-Order Linear Equations 77
2.1 Second-Order Linear Equations with Constant Coefficients 77
2.2 The Method of Undetermined Coefficients 85
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2.3 The Method of Variation of Parameters 90
2.4 The Use of a Known Solution to Find Another 95
2.5 Vibrations and Oscillations 100
2.5.1 Undamped Simple Harmonic Motion 100
2.5.2 Damped Vibrations 102
2.5.3 Forced Vibrations 106
2.5.4 A Few Remarks About Electricity 109
2.6 Newton's Law of Gravitation and Kepler's Laws 112
2.6.1 Kepler's Second Law 116
2.6.2 Kepler's First Law 118
2.6.3 Kepler's Third Law 121
2.7 Higher-Order Equations 128
Historical Note 135
2.8 Bessel Functions and the Vibrating Membrane 136
Problems for Review and Discovery 142
3 Power Series Solutions and Special Functions 145
3.1 Introduction and Review of Power Series 145
3.1.1 Review of Power Series 146
3.2 Series Solutions of First-Order Equations 157
3.3 Ordinary Points 163
3.4 Regular Singular Points 173
3.5 More on Regular Singular Points 180
Historical Note 190
Historical Note 192
3.6 Steady State Temperature in a Ball 193
Problems for Review and Discovery 196
4 Sturm-Liouville Problems and Boundary Value Problems 199
4.1 What is a Sturm-Liouville Problem? 199
4.2 Analyzing a Sturm-Liouville Problem 206
4.3 Applications of the Sturm-Liouville Theory 213
4.4 Singular Sturm-Liouville 220
4.5 Some Ideas from Quantum Mechanics 227
Problems for Review and Discovery 231
5 Numerical Methods 235
5.1 Introductory Remarks 236
5.2 The Method of Euler 238
5.3 The Error Term 242
5.4 An Improved Euler Method 246
5.5 The Runge-Kutta Method 252
5.6 A Constant Perturbation Method 256
Problems for Review and Discovery 260
6 Fourier Series: Basic Concepts 265
6.1 Fourier Coefficients 265
6.2 Some Remarks about Convergence 275
6.3 Even and Odd Functions: Cosine and Sine Series 282
6.4 Fourier Series on Arbitrary Intervals 289
6.5 Orthogonal Functions 293
Historical Note 299
6.6 Introduction to the Fourier Transform 300
6.6.1 Convolution and Fourier Inversion 309
6.6.2 The Inverse Fourier Transform 309
Problems for Review and Discovery 312
7 Laplace Transforms 317
7.0 Introduction 317
7.1 Applications to Differential Equations 321
7.2 Derivatives and Integrals 327
7.3 Convolutions 334
7.3.1 Abel's Mechanics Problem 337
7.4 The Unit Step and Impulse Functions 342
Historical Note 352
7.5 Flow on an Impulsively Started Flat Plate 353
Problems for Review and Discovery 356
8 Distributions 363
8.1 Schwartz Distributions 363
Problems for Review and Discovery 371
9 Wavelets 373
9.1 Localization in the Time and Space Variables 373
9.2 Building a Custom Fourier Analysis 376
9.3 The Haar Basis 379
9.4 Some Illustrative Examples 384
9.5 Construction of a Wavelet Basis 394
9.5.1 A Combinatorial Construction of the Daubechies Wavelets398
9.5.2 The Daubechies Wavelets from the Point of View of Fourier Analysis 399
9.5.3 Wavelets as an Unconditional Basis 402
9.5.4 Wavelets and Almost Diagonalizability 403
9.6 The Wavelet Transform 406
9.7 More on the Wavelet Transform 423
9.8 Decomposition and its Obverse 429
9.9 Some Applications 435
9.10 Cumulative Energy and Entropy 445
Problems for Review and Discovery 451
10 Partial Differential Equations and Boundary Value Problems455
10.1 Introduction and Historical Remarks 455
10.2 Eigenvalues and the Vibrating String 460
10.2.1 Boundary Value Problems 460
10.2.2 Derivation of the Wave Equation 461
10.2.3 Solution of the Wave Equation 463
10.3 The Heat Equation 469
10.4 The Dirichlet Problem for a Disc 478
10.4.1 The Poisson Integral 481
Historical Note 488
Historical Note 489
Problems for Review and Discovery 491
Table of Notation 495
Glossary 501
Solutions to Selected Exercises 527
Bibliography 527
Index 530