Differential Equations With Applications

Coherent, balanced introductory text focuses on initial- and boundary-value problems, general properties of linear equations, and the differences between linear and nonlinear systems. Includes large number of illustrative examples worked out in detail and extensive sets of problems. Answers or hints to most problems appear at end.

Preface

vii

Basic Concepts

1

(35)

Introduction

1

(1)

Classifications and Examples of Differential Equations

1

(3)

The Motion of a Particle

4

(5)

The Solution of a Differential Equation

9

(6)

Initial- and Boundary-value Problems

15

(5)

The Differential Equation y' = f(x)

20

(3)

Integrals as Functions of Parameters

23

(4)

Elementary and Non-elementary Functions

27

(1)

The Gamma Function

28

(8)

Appendix

31

(5)

Special Methods for First-order Equations

36

(31)

Introduction

36

(1)

Separation of Variables

36

(4)

The First-order Linear Differential Equation

40

(7)

Exact Differential Equations

47

(5)

Integrating Factors

52

(5)

Use of Substitutions

57

(5)

Second-order Equations Reducible to First-order

62

(2)

Summary

64

(3)

Applications of First-order Equations

67

(28)

Introduction

67

(1)

Falling Bodies with Air Resistance

67

(4)

Motion on a Given Curve

71

(4)

Linear Motion with Variable Mass

75

(2)

Newton's Law of Cooling

77

(3)

Dilution Problems

80

(1)

Chemical Reactions

81

(2)

Population Growth

83

(2)

A Simple Electrical Circuit

85

(6)

Families of Curves and Orthogonal Trajectories

91

(4)

Existence and Uniqueness and Methods of Approximation

95

(24)

Introduction

95

(1)

The Direction Field

95

(5)

Existence and Uniqueness of Solutions

100

(3)

The Picard Method

103

(4)

The Cauchy-Euler Method

107

(2)

Taylor Series

109

(3)

Existence and Uniqueness Theorems for Systems of Equations and Higher-order Equations

112

(2)

Existence and Uniqueness Theorems for Linear Equations

114

(5)

Appendix: Proof of existence and uniqueness theorems

115

(4)

Linear Differential Equations

119

(37)

Introduction

119

(1)

Fundamental Theory of Second-order Linear Equations

119

(7)

Complex-valued Solutions

126

(3)

Homogeneous Linear Equations with Constant Coefficients

129

(6)

Undetermined Coefficients

135

(6)

Variation of Parameters

141

(3)

Euler's Equation

144

(3)

Formulas of Lagrange and Abel

147

(2)

Linear Equations of the nth Order

149

(7)

Applications of Second-order Linear Differential Equations

156

(29)

Introduction

156

(1)

Free Vibrations

157

(9)

Forced Vibrations

166

(6)

Electrical Circuits

172

(8)

The Equations of Planetary Motion

180

(5)

Linear Differential Equations with Variable Coefficients

185

(39)

Introduction

185

(1)

Solution by Power Series

186

(10)

Solution near a Singular Point

196

(8)

Bessel's Equation

204

(12)

Hypergeometric Equation

216

(2)

Legendre's Equation

218

(6)

Systems of Linear Differential Equations

224

(43)

Introduction

224

(1)

Some Illustrative Examples

224

(5)

A Two-degree-of-freedom Vibration

229

(4)

Vectors and Matrices

233

(6)

Theory of Systems of Linear Differential Equations

239

(8)

Homogeneous Linear Systems with Constant Coefficients

247

(4)

Solution by Matrix Methods

251

(16)

The Laplace Transform

267

(41)

Introduction

267

(1)

Improper Integrals

267

(1)

The Laplace Transform

268

(5)

Properties of the Laplace Transform

273

(6)

Solution of Linear Equations with Constant Coefficients

279

(6)

Product of Transform Functions; Convolutions

285

(4)

Discontinuous Functions

289

(4)

Linear Systems Analysis

293

(15)

Appendix

303

(5)

Nonlinear Differential Equations

308

(27)

Introduction

308

(1)

The Pendulum

309

(6)

Singularities and the Phase Plane

315

(9)

Van der Pol's Equation

324

(3)

Piecewise Linear System

327

(3)

Liapunov's Second Method

330

(5)

Linear Difference Equations

335

(38)

Introduction

335

(1)

First-order Linear Difference Equations

336

(4)

Second-order Linear Difference Equations

340

(4)

Homogeneous Linear Difference Equations with Constant Coefficients

344

(4)

The Nonhomogeneous Equation

348

(7)

The Vector Space EN

355

(8)

A Boundary-value Problem

363

(4)

N Beads on a Tightly Stretched String

367

(6)

Numerical Methods

373

(26)

Introduction

373

(1)

The Euler Method

374

(3)

Error Analysis

377

(5)

Parasitic Solutions and Stability

382

(3)

A Second-order Predictor-Corrector Method

385

(4)

Fourth-order Predictor-Corrector Methods

389

(4)

Starting Methods and Runge-Kutta Methods

393

(3)

Higher-order Equations and Systems of Equations

396

(3)

Boundary-value Problems

399

(45)

Introduction

399

(1)

Homogeneous Boundary-value Problems

399

(5)

Eigenvalue Problems

404

(5)

Orthogonal Functions

409

(5)

Generalized Fourier Series

414

(6)

Weight Functions

420

(1)

The Sturm-Liouville Problem

421

(3)

Theorems on Eigenvalues and Eigenfunctions

424

(2)

Ordinary Fourier Series

426

(7)

Fourier-Bessel Series

433

(2)

Fourier-Legendre Series

435

(3)

Nonhomogeneous Boundary-value Problems

438

(6)

Partial Differential Equations of Mathematical Physics

444

(32)

Introduction

444

(1)

The Vibrating String

445

(14)

Heat Conduction

459

(6)

Laplace's Equation

465

(6)

Theory of Second-order Equations

471

(5)

Further Applications of Partial Differential Equations

476

(17)

Introduction

476

(1)

Laplace's Equation in Three Dimensions

476

(8)

Temperature in an Infinite Cylinder

484

(2)

Vibrating Membranes

486

(7)

Appendix A Infinite Series

493

(5)

Appendix B Functions of a Complex Variable

498

(5)

References

503

(2)

Answers and Hints

505

(32)

Index

537

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Differential Equations With Applications

Summary

Table of Contents

Supplemental Materials