9780130673374

This practical book reflects the new technological emphasis that permeates differential equations, including the wide availability of scientific computing environments likeMaple, Mathematica, and MATLAB; it does not concentrate on traditional manual methods but rather on new computer-based methods that lead to a wider range of more realistic applications. The book starts and ends with discussions of mathematical modeling of real-world phenomena, evident in figures, examples, problems, and applications throughout the book. For mathematicians and those in the field of computer science.

Application Modules

vii

Preface

ix

First-Order Differential Equations

1

(76)

Differential Equations and Mathematical Models

1

(9)

Integrals as General and Particular Solutions

10

(8)

Slope Fields and Solution Curves

18

(13)

Separable Equations and Applications

31

(15)

Linear First-Order Equations

46

(12)

Substitution Methods and Exact Equations

58

(19)

Mathematical Models and Numerical Methods

77

(67)

Population Models

77

(13)

Equilibrium Solutions and Stability

90

(8)

Acceleration-Velocity Models

98

(12)

Numerical Approximation: Euler's Method

110

(12)

A Closer Look at the Euler Method

122

(10)

The Runge-Kutta Method

132

(12)

Linear Equations of Higher Order

144

(98)

Introduction: Second-Order Linear Equations

144

(14)

General Solutions of Linear Equations

158

(12)

Homogeneous Equations with Constant Coefficients

170

(12)

Mechanical Vibrations

182

(13)

Nonhomogeneous Equations and Undetermined Coefficients

195

(14)

Forced Oscillations and Resonance

209

(13)

Electrical Circuits

222

(7)

Endpoint Problems and Eigenvalues

229

(13)

Introduction to Systems of Differential Equations

242

(39)

First-Order Systems and Applications

242

(12)

The Method of Elimination

254

(11)

Numerical Methods for Systems

265

(16)

Linear Systems of Differential Equations

281

(85)

Matrices and Linear Systems

281

(19)

The Eigenvalue Method for Homogeneous Systems

300

(15)

Second-Order Systems and Mechanical Applications

315

(13)

Multiple Eigenvalue Solutions

328

(16)

Matrix Exponentials and Linear Systems

344

(14)

Nonhomogeneous Linear Systems

358

(8)

Nonlinear Systems and Phenomena

366

(69)

Stability and the Phase Plane

366

(12)

Linear and Almost Linear Systems

378

(15)

Ecological Models: Predators and Competitors

393

(13)

Nonlinear Mechanical Systems

406

(17)

Chaos in Dynamical Systems

423

(12)

Laplace Transform Methods

435

(62)

Laplace Transforms and Inverse Transforms

435

(11)

Transformation of Initial Value Problems

446

(11)

Translation and Partial Fractions

457

(10)

Derivatives, Integrals, and Products of Transforms

467

(8)

Periodic and Piecewise Continuous Input Functions

475

(11)

Impulses and Delta Functions

486

(11)

References for Further Study

497

(3)

Appendix: Existence and Uniqueness of Solutions

500

(15)

Answers to Selected Problems

515

Index

1

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Differential Equations : Computing and Modeling

Differential Equations : Computing and Modeling

0130673374

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