Master differential equations and succeed in your course DIFFERENTIAL EQUATIONS WITH BOUNDARY-VALUE PROBLEMS with accompanying CD-ROM and technology! Straightfoward and readable, this mathematics text provides you with tools such as examples, explanations, definitions, and applications designed to help you succeed. The accompanying DE Tools CD-ROM makes helps you master difficult concepts through twenty-one demonstration tools such as Project Tools and Text Tools. Studying is made easy with iLrn Tutorial, a text-specific, interactive tutorial software program that gives the practice you need to succeed.

Preface

xi

Acknowledgments

xv

Introduction to Differential Equations

1

(29)

Definitions and Terminology

2

(9)

Initial-Value Problems

11

(6)

Differential Equations as Mathematical Models

17

(13)

Review Exercises

28

(2)

First-Order Differential Equations

30

(30)

Separable Variables

31

(6)

Exact Equations

37

(7)

Linear Equations

44

(10)

Solutions by Substitutions

54

(6)

Review Exercises

59

(1)

Modeling with First-Order Differential Equations

60

(33)

Linear Equations

61

(11)

Nonlinear Equations

72

(10)

Systems of Linear and Nonlinear Equations

82

(11)

Review Exercises

91

(2)

Differential Equations of Higher Order

93

(76)

Preliminary Theory: Linear Equations

94

(15)

Initial-Value and Boundary Value Problems

94

(2)

Homogeneous Equations

96

(7)

Nonhomogeneous Equations

103

(6)

Reduction of Order

109

(4)

Homogeneous Linear Equations with Constant Coefficients

113

(8)

Undetermined Coefficients---Superposition Approach

121

(11)

Undetermined Coefficients---Annihilator Approach

132

(9)

Variation of Parameters

141

(6)

Cauchy-Euler Equation

147

(7)

Systems of Linear Equations

154

(7)

Nonlinear Equations

161

(8)

Review Exercises

167

(2)

Modeling with Higher-Order Differential Equations

169

(44)

Linear Equations: Initial-Value Problems

170

(22)

Spring/Mass Systems: Free Undamped Motion

170

(5)

Spring/Mass Systems: Free Damped Motion

175

(3)

Spring/Mass Systems: Driven Motion

178

(3)

Analogous Systems

181

(11)

Linear Equations: Boundary-Value Problems

192

(8)

Nonlinear Equations

200

(13)

Review Exercises

210

(3)

Series Solutions of Linear Equations

213

(45)

Review of Power Series; Power Series Solutions

214

(8)

Solutions About Ordinary Points

222

(8)

Solutions About Singular Points

230

(13)

Two Special Equations

243

(15)

Review Exercises

257

(1)

Laplace Transform

258

(61)

Definition of the Laplace Transform

259

(8)

Inverse Transform

267

(6)

Translation Theorems and Derivatives of a Transform

273

(11)

Transforms of Derivatives, Integrals, and Periodic Functions

284

(8)

Applications

292

(14)

Dirac Delta Function

306

(4)

Systems of Linear Equations

310

(9)

Review Exercises

316

(3)

System of Linear First-Order Differential Equations

319

(34)

Preliminary Theory

320

(10)

Homogeneous Linear Systems with Constant Coefficients

330

(14)

Distinct Real Eigenvalues

330

(3)

Repeated Eigenvalues

333

(5)

Complex Eigenvalues

338

(6)

Variation of Parameters

344

(5)

Matrix Exponential

349

(4)

Review Exercises

352

(1)

Numerical Methods for Ordinary Differential Equations

353

(35)

Direction Fields

354

(3)

Euler Methods

357

(9)

Runge-Kutta Methods

366

(7)

Multistep Methods

373

(3)

Higher-Order Equations and Systems

376

(5)

Second-Order Boundary-Value Problems

381

(7)

Review Exercises

386

(2)

Plane Autonomous Systems and Stability

388

(48)

Autonomous Systems, Critical Points, and Periodic Solutions

389

(8)

Stability of Linear Systems

397

(12)

Linearization and Local Stability

409

(13)

Modeling Using Autonomous Systems

422

(14)

Review Exercises

433

(3)

Orthogonal Functions and Fourier Series

436

(38)

Orthogonal Functions

437

(6)

Fourier Series

443

(5)

Fourier Cosine and Sine Series

448

(10)

Sturm-Liouville Problem

458

(7)

Bessel and Legendre Series

465

(9)

Fourier-Bessel Series

465

(4)

Fourier-Legendre Series

469

(3)

Review Exercises

472

(2)

Partial Differential Equations and Boundary-Value Problems in Rectangular Coordinates

474

(38)

Separable Partial Differential Equations

475

(5)

Classical Equations and Boundary-Value Problems

480

(6)

Heat Equation

486

(3)

Wave Equation

489

(6)

Laplace's Equation

495

(4)

Nonhomogeneous Equations and Boundary Conditions

499

(3)

Use of Generalized Fourier Series

502

(5)

Boundary-Value Problems Involving Fourier Series in Two Variables

507

(5)

Review Exercises

510

(2)

Boundary-Value Problems in Other Coordinate Systems

512

(19)

Problems Involving Laplace's Equation in Polar Coordinates

513

(5)

Problems in Polar and Cylindrical Coordinates: Bessel functions

518

(7)

Problems in Spherical Coordinates: Legendre Polynomials

525

(6)

Review Exercises

528

(3)

Integral Transform Method

531

(28)

Error Function

532

(2)

Applications of the Laplace Transform

534

(9)

Fourier Integral

543

(6)

Fourier Transforms

549

(10)

Review Exercises

556

(3)

Numerical Methods for Partial Differential Equations

559

Elliptic Equations

560

(7)

Parabolic Equations

567

(8)

Hyperbolic Equations

575

Review Exercises

581

Appendix I Gamma Function

APP-I

Appendix II Introduction to Matrices

APP-3

Appendix III Laplace Transforms

APP-23

Answers to Odd-Numbered Problems

A-1

Index

1-1

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