Differential Equations with Boundary Value

Ideal for one or two term first courses in differential equations for engineering, math, biology, and finance majors. This text strikes a balance between the traditional and the modern. It combines the traditional material with a modern systems emphasis. It offers flexibility of use that will allow faculty at a variety of institutions to use the book with extensive problem sets.

Preface

ix

Introduction to Differential Equations

1

(17)

Differential Equation Models

2

(5)

The Derivative

7

(3)

Integration

10

(8)

First-Order Equations

18

(105)

Differential Equations and Solutions

18

(13)

Solutions to Separable Equations

31

(13)

Models of Motion

44

(10)

Linear Equations

54

(10)

Mixing Problems

64

(9)

Exact Differential Equations

73

(17)

Existence and Uniqueness of Solutions

90

(12)

Dependence of Solutions on Initial Conditions

102

(5)

Autonomous Equations and Stability

107

(16)

Project 2.10 The Daredevil Skydiver

120

(3)

Modeling and Applications

123

(40)

Modeling Population Growth

124

(14)

Models and the Real World

138

(5)

Personal Finance

143

(9)

Electrical Circuits

152

(11)

Project 3.5 The Spruce Budworm

158

(3)

Project 3.6 Social Security, Now or Later

161

(2)

Second-Order Equations

163

(68)

Definitions and Examples

163

(11)

Second-Order Equations and Systems

174

(5)

Linear, Homogeneous Equations with Constant Coefficients

179

(11)

Harmonic Motion

190

(9)

Inhomogeneous Equations; the Method of Undetermined Coefficients

199

(10)

Variation of Parameters

209

(6)

Forced Harmonic Motion

215

(16)

Project 4.8 Nonlinear Oscillators

228

(3)

The Laplace Transform

231

(70)

The Definition of the Laplace Transform

232

(9)

Basic Properties of the Laplace Transform

241

(7)

The Inverse Laplace Transform

248

(8)

Using the Laplace Transform to Solve Differential Equations

256

(10)

Discontinuous Forcing Terms

266

(14)

The Delta Function

280

(7)

Convolutions

287

(11)

Summary

298

(3)

Project 5.9 Forced Harmonic Oscillators

299

(2)

Numerical Methods

301

(34)

Euler's Method

302

(12)

Runge-Kutta Methods

314

(8)

Numerical Error Comparisons

322

(5)

Practical Use of Solvers

327

(5)

A Cautionary Tale

332

(3)

Project 6.6 Numerical Error Comparison

334

(1)

Matrix Algebra

335

(61)

Vectors and Matrices

335

(13)

The Geometry of Systems of Linear Equations

348

(5)

Solving Systems of Equations

353

(9)

Properties of Solution Sets

362

(9)

Subspaces

371

(13)

Determinants

384

(12)

An Introduction to Systems

396

(48)

Definitions and Examples

396

(9)

Geometric Interpretation of Solutions

405

(12)

Qualitative Analysis

417

(8)

Linear Systems

425

(19)

Project 8.5 Long-Term Behavior of Solutions

441

(3)

Linear Systems with Constant Coefficents

444

(101)

Overview of the Technique

444

(8)

Planar Systems

452

(14)

Phase Plane Portraits

466

(18)

Higher Dimensional Systems

484

(8)

The Exponential of a Matrix

492

(18)

Qualitative Analysis of Lienar Systems

510

(5)

Higher-Order Linear Equations

515

(13)

Inhomogeneous Linear Systems

528

(17)

Project 9.9 Phase Plane Portraits

538

(1)

Project 9.10 Oscillations of Linear Molecules

539

(6)

Nonlinear Systems

545

(92)

The Linearization of a Nonlinear System

545

(14)

Long-Term Behavior of Solutions

559

(7)

Invariant Sets and the Use of Nullclines

566

(8)

Long-Term Behavior of Solutions to Planar Systems

574

(12)

Conserved Quantities

586

(6)

Nonlinear Mechanics

592

(17)

The Method of Lyapunov

609

(10)

Predator-Prey Systems

619

(18)

Project 10.9 Human Immune Response to Infectious Disease

631

(3)

Project 10.10 Analysis of Competing Species

634

(3)

Series Solutions to Differential Equations

637

(75)

Review of Power Series

638

(12)

Series Solutions Near Ordinary Points

650

(12)

Legendre's Equation

662

(6)

Types of Singular Points---Euler's Equation

668

(9)

Series Solutions Near Regular Singular Points

677

(13)

Solutions in the Exceptional Cases

690

(11)

Bessel's Equation and Bessel Functions

701

(11)

Fourier Series

712

(38)

Computation of Fourier Series

713

(11)

Convergence of Fourier Series

724

(9)

Fourier Cosine and Sine Series

733

(5)

The Complex Form of a Fourier Series

738

(3)

The Discrete Fourier Transform and the FFT

741

(9)

Partial Differential Equations

750

Derivation of the Heat Equation

750

Separation of Variables for the Heat Equation

756

The Wave Equation

768

Laplace's Equation

778

Laplace's Equation on a Disk

785

Sturm Liouville Problems

791

Orthogonality and Generalized Fourier Series

801

Temperature in a Ball---Legendre Polynomials

810

Time Dependent PDEs in Higher Dimension

814

Domains with Circular Symmetry---Bessel Functions

821

Answers to Odd-Numbered Problems

A-1

Index

I-1

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Differential Equations with Boundary Value Problems

Summary

Table of Contents

Supplemental Materials