Rent 9780534955748 | First Course in Differential Equations with Modeling Applications

Table of Contents

Preface

(4)

Acknowledgments

xiii

1 Introduction to Differential Equations

(29)

1.1 Definitions and Terminology

... MORE

(9)

1.2 Initial-Value Problems

(6)

1.3 Differential Equations as Mathematical Models

(11)

Chapter 1 Review Exercises

(2)

2 First-Order Differential Equations

(30)

2.1 Separable Variables

(6)

2.2 Exact Equations

(7)

2.3 Linear Equations

(10)

2.4 Solutions by Substitutions

(5)

Chapter 2 Review Exercises

(1)

3 Modeling with First-Order Differential Equations

(33)

3.1 Linear Equations

(11)

3.2 Nonlinear Equations

(10)

3.3 Systems of Linear and Nonlinear Equations

(9)

Chapter 3 Review Exercises

A Modeling Application AZT and Survivability with AIDS

After pages 64

Ivan Kramer

A Modeling Application Wolf Population Dynamics

C. J. Knickerbocker

4 Differential Equations of Higher Order

(76)

4.1 Preliminary Theory: Linear Equations

(15)

4.1.1 Initial-Value and Boundary-Value Problems

(2)

4.1.2 Homogeneous Equations

(7)

4.1.3 Nonhomogeneous Equations

103

(6)

4.2 Reduction of Order

109

(4)

4.3 Homogeneous Linear Equations with Constant Coefficients

113

(8)

4.4 Undetermined Coefficients -- Superposition Approach

121

(11)

4.5 Undetermined Coefficients -- Annihilator Approach

132

(9)

4.6 Variation of Parameters

141

(6)

4.7 Cauchy-Euler Equation

147

(7)

4.8 Systems of Linear Equations

154

(7)

4.9 Nonlinear Equations

161

(6)

Chapter 4 Review Exercises

167

(2)

5 Modeling with Higher-Order Differential Equations

169

(44)

5.1 Linear Equations: Initial-Value Problems

170

(22)

5.1.1 Spring/Mass Systems: Free Undamped Motion

170

(5)

5.1.2 Spring/Mass Systems: Free Damped Motion

175

(3)

5.1.3 Spring/Mass Systems: Driven Motion

178

(3)

5.1.4 Analogous Systems

181

(11)

5.2 Linear Equations: Boundary-Value Problems

192

(8)

5.3 Nonlinear Equations

200

(10)

Chapter 5 Review Exercises

210

A Modeling Application Decay of Satellite Orbits

After page 192

John Ellison

A Modeling Application The Collapse of the Tacoma Narrows Suspension Bridge

Gilbert N. Lewis

6 Series Solutions of Linear Equations

213

(45)

6.1 Review of Power Series; Power Series Solutions

214

(8)

6.2 Solutions About Ordinary Points

222

(8)

6.3 Solutions About Singular Points

230

(13)

6.4 Two Special Equations

243

(14)

Chapter 6 Review Exercises

257

(1)

7 Laplace Transform

258

(61)

7.1 Definition of the Laplace Transform

259

(8)

7.2 Inverse Transform

267

(6)

7.3 Translation Theorems and Derivatives of a Transform

273

(11)

7.4 Transforms of Derivatives, Integrals, and Periodic Functions

284

(8)

7.5 Applications

292

(14)

7.6 Dirac Delta Function

306

(4)

7.7 Systems of Linear Equations

310

(6)

Chapter 7 Review Exercises

316

(3)

8 Systems of Linear First-Order Differential Equations

319

(34)

8.1 Preliminary Theory

320

(10)

8.2 Homogeneous Linear Systems with Constant Coefficients

330

(14)

8.2.1 Distinct Real Eigenvalues

330

(3)

8.2.2 Repeated Eigenvalues

333

(5)

8.2.3 Complex Eigenvalues

338

(6)

8.3 Variation of Parameters

344

(5)

8.4 Matrix Exponential

349

(3)

Chapter 8 Review Exercises

352

A Modeling Application Modeling an Arms Race

After page 352

Michael Olinick

9 Numerical Methods for Ordinary Differential Equations

353

9.1 Direction Fields

354

(3)

9.2 Euler Methods

357

(9)

9.3 Runge-Kutta Methods

366

(7)

9.4 Multistep Methods

373

(3)

9.5 Higher-Order Equations and Systems

376

(5)

9.6 Second-Order Boundary-Value Problems

381

(5)

Chapter 9 Review Exercises

386

Appendix I Gamma Function

APP-1

Appendix II Introduction to Matrices

APP-3

Appendix III Laplace Transforms

APP-23

Answers to Odd-Numbered Problems

A-1

Index

I-1

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First Course in Differential Equations with Modeling Applications

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