Elementary Differential Equations

, by Rainville, Earl D.; Bedient, Phillip E.; Bedient, Richard E.

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ISBN: 9780135080115 | 0135080118
Cover: Paperback
Copyright: 10/23/1996

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A clear, concise book that emphasizes finding solutions to differential equations where applications play an important role. Each chapter includes many illustrative examples to assist the reader.The book emphasizes methods for finding solutions to differential equations. It provides many abundant exercises, applications, and solved examples with careful attention given to readability.Elementary Differential Equationsincludes a thorough treatment of power series techniques. In addition, the book presents a classical treatment of several physical problems to show how Fourier series become involved in the solution of those problems. The eighth edition ofElementary Differential Equationshas been revised to include a new supplement in many chapters that provides suggestions and exercises for using a computer to assist in the understanding of the material in the chapter. It also now provides an introduction to the phase plane and to different types of phase portraits.A valuable reference book for readers interested in exploring the technological and other applications of differential equations.

Preface

xiii

1 Definitions; Families of Curves

(17)

1.1 Examples of Differential Equations

(1)

1.2 Definitions

(3)

1.3 Families of Solutions

(5)

1.4 Geometric Interpretation

(2)

1.5 The Isoclines of an Equation

(2)

1.6 An Existence Theorem

(1)

1.7 Computer Supplement

(3)

2 Equations of Order One

(27)

2.1 Separation of Variables

(6)

2.2 Homogeneous Functions

(1)

2.3 Equations with Homogeneous Cofficients

(4)

2.4 Exact Equations

(6)

2.5 The Linear Equation of Order One

(3)

2.6 The General Solution of a Linear Equation

(5)

2.7 Computer Supplement

(2)

3 Numerical Methods

(17)

3.1 General Remarks

(1)

3.2 Euler's Method

(3)

3.3 A Modification of Euler's Method

(1)

3.4 A Method of Successive Approximation

(2)

3.5 An Improvement on the Method of Successive Approximation

(1)

3.6 The Use of Taylor's Theorem

(2)

3.7 The Runge-Kutta Method

(4)

3.8 A Continuing Method

(2)

3.9 Computer Supplement

(2)

Elementary Applications

(13)

4.1 Velocity of Escape from the Earth

(2)

4.2 Newton's Law of Cooling

(1)

4.3 Simple Chemical Conversion

(4)

4.4 Logistic Growth and the Price of Commodities

(4)

4.5 Computer Supplement

(2)

5 Additional Topics on Equations of Order One

(24)

5.1 Integrating Factors Found by Inspection

(4)

5.2 The Determination of Integrating Factors

(4)

5.3 Substitution Suggested by the Equation

(3)

5.4 Bernoulli's Equation

(3)

5.5 Coefficients Linear in the Two Variables

(5)

5.6 Solutions Involving Nonelementary Integrals

(3)

5.7 Computer Supplement

(2)

6 Linear Differential Equations

(18)

6.1 The General Linear Equation

(1)

6.2 An Existence and Uniqueness Theorem

100

(2)

6.3 Linear Independence

102

(1)

6.4 The Wronskian

103

(3)

6.5 General Solution of a Homogeneous Equation

106

(1)

6.6 General Solution of a Nonhomogeneous Equation

107

(2)

6.7 Differential Operators

109

(1)

6.8 The Fundamental Laws of Operation

111

(2)

6.9 Some Properties of Differential Operators

113

(2)

6.10 Computer Supplement

115

(2)

7 Linear Equations with Constant Coefficients

117

(17)

7.1 Introduction

117

(1)

7.2 The Auxiliary Equation: Distinct Roots

117

(3)

7.3 The Auxiliary Equation: Repeated Roots

120

(3)

7.4 A Definition of exp Z for Imaginary Z

123

(2)

7.5 The Auxiliary Equation: Imaginary Roots

125

(2)

7.6 A Note on Hyperbolic Functions

127

(5)

7.7 Computer Supplement

132

(2)

8 Nonhomogeneous Equations: Undetermined Coefficients

134

(18)

8.1 Construction of a Homogeneous Equation from a Specific Solution

134

(3)

8.2 Solution of a Nonhomogeneous Equation

137

(2)

8.3 The Method of Undetermined Coefficients

139

(5)

8.4 Solution by Inspection

144

(6)

8.5 Computer Supplement

150

(2)

9 Variation of Parameters

152

(13)

9.1 Introduction

152

(1)

9.2 Reduction of Order

152

(4)

9.3 Variation of Parameters

156

(5)

9.4 Solution of y"+y=f(x)

161

(3)

9.5 Computer Supplement

164

(1)

Applications

165

(21)

10.1 Vibration of a Spring

165

(2)

10.2 Undamped Vibrations

167

(2)

10.3 Resonance

169

(3)

10.4 Damped Vibrations

172

(5)

10.5 The Simple Pendulum

177

(1)

10.6 Newton's Laws and Planetary Motion

178

(1)

10.7 Central Force and Kepler's Second Law

179

(1)

10.8 Kepler's First Law

180

(2)

10.9 Kepler's Third Law

182

(2)

10.10 Computer Supplement

184

(2)

11 Linear Systems of Equations

186

(38)

11.1 Introduction

186

(1)

11.2 First-Order Systems with Constant Coefficients

186

(1)

11.3 Solution of a First-Order System

187

(2)

11.4 Some Matrix Algebra

189

(6)

11.5 First-Order Systems Revisited

195

(9)

11.6 Complex Eigenvalues

204

(4)

11.7 Repeated Eigenvalues

208

(8)

11.8 The Phase Plane

216

(6)

11.9 Computer Supplement

222

(2)

12 Nonhomogeneous Systems of Equations

224

(19)

12.1 Nonhomogeneous Systems

224

(4)

12.2 Arms Races

228

(4)

12.3 Electric Circuits

232

(3)

12.4 Simple Networks

235

(8)

13 The Existence and Uniqueness of Solutions

243

(9)

13.1 Preliminary Remarks

243

(1)

13.2 An Existence and Uniqueness Theorem

243

(3)

13.3 A Lipschitz Condition

246

(1)

13.4 A Proof of the Existence Theorem

246

(4)

13.5 A Proof of the Uniqueness Theorem

250

(1)

13.6 Other Existence Theorems

251

(1)

14 The Laplace Transform

252

(22)

14.1 The Transform Concept

252

(1)

14.2 Definition of the Laplace Transform

253

(1)

14.3 Transforms of Elementary Function

253

(4)

14.4 Sectionally Continuous Functions

257

(1)

14.5 Functions of Exponential Order

258

(3)

14.6 Functions of Class A

261

(2)

14.7 Transforms of Derivatives

263

(3)

14.8 Derivatives of Transforms

266

(1)

14.9 The Gamma Function

267

(2)

14.10 Periodic Functions

269

(5)

15 Inverse Transforms

274

(46)

15.1 Definition of an Inverse Transform

274

(3)

15.2 Partial Fractions

277

(3)

15.3 Initial Value Problems

280

(6)

15.4 A Step Function

286

(8)

15.5 A Convolution Theorem

294

(4)

15.6 Special Integral Equations

298

(5)

15.7 Transform Methods and the Vibration of Springs

303

(4)

15.8 The Deflection of Beams

307

(3)

15.9 Systems of Equations

310

(6)

15.10 Computer Supplement

316

(4)

16 Nonlinear Equations

320

(22)

16.1 Preliminary Remarks

320

(1)

16.2 Factoring the Left Member

320

(3)

16.3 Singular Solutions

323

(2)

16.4 The c-Discriminant Equation

325

(1)

16.5 The p-Discriminant Equation

326

(2)

16.6 Eliminating the Dependent Variable

328

(2)

16.7 Clairaut's Equation

330

(4)

16.8 Dependent Variable Missing

334

(1)

16.9 Independent Variable Missing

335

(3)

16.10 The Catenary

338

(4)

17 Power Series Solutions

342

(16)

17.1 Linear Equations and Power Series

342

(1)

17.2 Convergence of Power Series

343

(2)

17.3 Ordinary Points and Singular Points

345

(2)

17.4 Validity of the Solutions Near an Ordianry Point

347

(1)

17.5 Solutions Near an Ordinary Point

347

(9)

17.6 Computer Supplement

356

(2)

18 Solutions Near Regular Singular Points

358

(38)

18.1 Regular Singular Points

358

(2)

18.2 The Indicial Equation

360

(2)

18.3 Form and Validity of the Solutions Near a Regular Singular Point

362

(1)

18.4 Indicial Equation with Difference of Roots Nonintegral

363

(4)

18.5 Differentiation of a Product of Functions

367

(1)

18.6 Indicial Equation with Equal Roots

368

(6)

18.7 Indicial Equation with Equal Roots: An Alternative

374

(3)

18.8 Indicial Equation with Difference of Roots a Positive Integer: Nonlogarithmic Case

377

(4)

18.9 Indicial Equation with Difference Roots a Positive Integer: Logarithmic Case

381

(4)

18.10 Solution for Large x

385

(3)

18.11 Many-Term Recurrence Relations

388

(4)

18.12 Summary

392

(4)

19 Equations of Hypergeometric Type

396

(8)

19.1 Equations to Be Treated in This Chapter

396

(1)

19.2 The Factorial Function

396

(1)

19.3 The Hypergeometric Function

397

(2)

19.4 Laguerre Polynomials

399

(1)

19.5 Bessel's Equation with Index Not an Integer

400

(1)

19.6 Bessel's Equation with Index an Integer

401

(1)

19.7 Hermite Polynomials

402

(1)

19.8 Legendre Polynomials

403

(1)

20 Partial Differential Equations

404

(14)

20.1 Remarks on Partial Differential Equations

404

(1)

20.2 Some Partial Differential Equations of Applied Mathematics

404

(2)

20.3 Method of Separation of Variables

406

(5)

20.4 A Problem on the Conduction of Heat in a Slab

411

(5)

20.5 Computer Supplement

416

(2)

21 Orthogonal Sets of Functions

418

(7)

21.1 Orthogonality

418

(1)

21.2 Simple Sets of Polynomials

419

(1)

21.3 Orthogonal Polynomials

419

(2)

21.4 Zeros of Orthogonal Polynomials

421

(1)

21.5 Orthogonality of Legendre Polynomials

422

(2)

21.6 Other Orthogonal Sets

424

(1)

22 Fourier Series

425

(22)

22.1 Orthogonality of a Set of Sines and Cosines

425

(2)

22.2 Fourier Series: An Expansion Theorem

427

(4)

22.3 Numerical Examples of Fourier Series

431

(7)

22.4 Fourier Sine Series

438

(2)

22.5 Fourier Cosine Series

441

(2)

22.6 Numerical Fourier Analysis

443

(1)

22.7 Improvement in Rapidity of Convergence

444

(1)

22.8 Computer Supplement

445

(2)

23 Boundary Value Problems

447

(20)

23.1 The One-Dimensional Heat Equation

447

(6)

23.2 Experimental Verification of the Validity of the Heat Equation

453

(2)

23.3 Surface Temperature Varying with Time

455

(2)

23.4 Heat Conduction in a Sphere

457

(1)

23.5 The Simple Wave Equation

458

(3)

23.6 Laplace's Equation in Two Dimensions

461

(3)

23.7 Computer Supplement

464

(3)

24 Additional Properties of the Laplace Transform

467

(14)

24.1 Power Series and Inverse Transforms

467

(4)

24.2 The Error Function

471

(7)

24.3 Bessel Functions

478

(2)

24.4 Differential Equations with Variable Coefficients

480

(1)

25 Partial Differential Equations Transform Methods

481

(19)

25.1 Boundary Value Problems

481

(4)

25.2 The Wave Equation

485

(3)

25.3 Diffusion in a Semi-Infinite Solid

488

(3)

25.4 Canonical Variables

491

(2)

25.5 Diffusion in a Slab of Finite Width

493

(3)

25.6 Diffusion in a Quarter-Infinite Solid

496

(4)

Answers to Odd-numbered Exercises

500

(27)

Index

527

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Elementary Differential Equations

Summary

Table of Contents

Supplemental Materials