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Fundamentals of Differential Equations with Boundary Value Problems

by: Nagle, R. Kent; Saff, Edward B.; Snider, Arthur David

Fundamentals of Differential Equations with Boundary Value Problems

by: Nagle, R. Kent; Saff, Edward B.; Snider, Arthur David

ISBN 13:
9780321145710
ISBN 10:
0321145712
Edition: 5th
Format: Hardcover w/CD
Copyright: 01/01/2008
Publisher: Addison Wesley
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Summary

Fundamentals of Differential Equations, Sixth Edition is designed for a one-semester sophomore or junior-level course. Fundamentals of Differential Equations and Boundary Value Problems, Fourth Edition, contains enough material for a two-semester course that covers and builds on boundary-value problems. These tried-and-true texts help students understand the methods and concepts they will need to successfully complete engineering courses. The new texts retain the features that have made previous editions successful, while integrating recent advances in teaching and learning. The Fundamentals of Differential Equations and Boundary Value Problems version consists of the main text plus three additional chapters (Eigenvalue Problems and Sturm-Liouville Equations; Stability of Autonomous Systems; and Existence and Uniqueness Theory).

CHAPTER 1 INTRODUCTION

(36)

1.1 Background

(5)

1.2 Solutions and Initial Value Problems

(10)

1.3 Direction Fields

(8)

1.4 The Approximation Method of Euter

(6)

Chapter Summary

(1)

Technical Writing Exercises

(1)

Group Projects for Chapter 1

(6)

A. Taylor Series Method

(1)

B. Picard's Method

(1)

C. Magnetic Dipole

(1)

D. The Phase Line

(3)

CHAPTER 2 FIRST ORDER DIFFERENTIAL EQUATIONS

(50)

2.1 Introduction: Motion of a Falling Body

(3)

2.2 Separable Equations

(9)

2.3 Linear Equations

(9)

2.4 Exact Equations

(10)

2.5 Special Integrating Factors

(4)

2.6 Substitutions and Transformations

(8)

Chapter Summary

(1)

Review Problems

(1)

Technical Writing Exercises

(1)

Group Projects for Chapter 2

(4)

A. Torricelli's Law of Fluid Flow

(1)

B. The Snowplow Problem

(1)

C. Two Snowplows

(1)

D. Clairaut Equations and Singular Solutions

(1)

E. Asymptotic Behavior of Solutions to Linear Equations

(1)

CHAPTER 3 MATHEMATICAL MODELS AND NUMERICAL METHODS INVOLVING FIRST ORDER EQUATIONS

(67)

3.1 Mathematical Modeling

(2)

3.2 Compartmental Analysis

(12)

3.3 Heating and Cooling of Buildings

101

(7)

3.4 Newtonian Mechanics

108

(11)

3.5 Electrical Circuits

119

(4)

3.6 Improved Euler's Method

123

(11)

3.7 Higher-Order Numerical Methods: Taylor and Runge-Kutta

134

(10)

Group Projects for Chapter 3

144

(10)

A. Aquaculture

144

(1)

B. Curve of Pursuit

145

(1)

C. Aircraft Guidance in a Crosswind

146

(1)

D. Feedback and the Op Amp

147

(1)

E. Bang-Bang Controls

148

(1)

F. Price, Supply, and Demand

149

(1)

G. Stability of Numerical Methods

150

(1)

H. Period Doubling and Chaos

151

(3)

CHAPTER 4 LINEAR SECOND-ORDER EQUATIONS

4.1 Introduction: The Mass-Spring Oscillator

154

(6)

4.2 Homogeneous Linear Equations: The General Solution

160

(9)

4.3 Auxiliary Equations with Complex Roots

169

(10)

4.4 Nonhomgeneous Equations: The Method of Undetermined Coefficients

179

(7)

4.5 The Superposition Principle and Undetermined Coefficients Revisited

186

(8)

4.6 Variation of Parameters

194

(4)

4.7 Qualitative Considerations for Variable-Coefficient and Nonlinear Equations

198

(12)

4.8 A Closer Look at Free Mechanical Vibrations

210

(10)

4.9 A Closer Look at Forced Mechanical Vibrations

220

(8)

Chapter Summary

228

(2)

Review Problems

230

(1)

Technical Writing Exercises

231

(1)

Group Projects for Chapter 4

232

(9)

A. Undetermined Coefficients Using Complex Arithmetic

232

(1)

B. An Alternative to the Method of Undetermined Coefficients

233

(1)

C. Convolution Method

234

(1)

D. Linearization of Nonlinear Problems

235

(1)

E. Nonlinear Equations Solvable by First-Order Techniques

236

(1)

F. Apollo Reentry

237

(1)

G. Simple Pendulum

238

(1)

H. Asymptotic Behavior of Solutions

239

(2)

CHAPTER 5 INTRODUCTION TO SYSTEMS AND PHASE PLANE ANALYSIS

241

(76)

5.1 Interconnected Fluid Tanks

241

(2)

5.2 Elimination Method for Systems with Constant Coefficients

243

(10)

5.3 Solving Systems and Higher-Order Equations Numerically

253

(11)

5.4 Introduction to the Phase Plane

264

(15)

5.5 Coupled Mass-Spring Systems

279

(7)

5.6 Electrical Systems

286

(6)

5.7 Dynamical Systems, Poincare Maps, and Chaos

292

(11)

Chapter Summary

303

(1)

Review Problems

304

(2)

Group Projects for Chapter 5

306

(11)

A. The Growth of a Tumor

306

(2)

B. Designing a Landing System for Interplanetary Travel

308

(1)

C. Things That Bob

309

(2)

D. Periodic Solutions to Volterra-Lotka Systems

311

(1)

E. Hamiltonian Systems

312

(2)

F. Strange Behavior of Competing Species-Part I

314

(1)

G. Cleaning Up the Great Lakes

315

(2)

CHAPTER 6 THEORY OF HIGHER-ORDER LINEAR DIFFERENTIAL EQUATIONS

317

(31)

6.1 Basic Theory of Linear Differential Equations

317

(9)

6.2 Homogeneous Linear Equations with Constant Coefficients

326

(7)

6.3 Undetermined Coefficients and the Annihilator Method

333

(5)

6.4 Method of Variation of Parameters

338

(5)

Chapter Summary

343

(1)

Review Problems

344

(1)

Technical Writing Exercises

345

(1)

Group Projects for Chapter 6

346

(2)

A. Justifying the Method of Undetermined Coefficients

346

(1)

B. Transverse Vibrations of a Beam

346

(2)

CHAPTER 7 LAPLACE TRANSFORMS

348

(77)

7.1 Introduction: A Mixing Problem

348

(3)

7.2 Definition of the Laplace Transform

351

(9)

7.3 Properties of the Laplace Transform

360

(6)

7.4 Inverse Laplace Transform

366

(10)

7.5 Solving Initial Value Problems

376

(8)

7.6 Transforms of Discontinuous and Periodic Functions

384

(14)

7.7 Convolution

398

(9)

7.8 Impulses and the Dirac Delta Function

407

(7)

7.9 Solving Linear Systems with Laplace Transforms

414

(3)

Chapter Summary

417

(1)

Review Problems

418

(1)

Technical Writing Exercises

419

(2)

Group Projects for Chapter 7

421

(4)

A. Duhamel's Formulas

421

(1)

B. Frequency Response Modeling

422

(2)

C. Determining System Parameters

424

(1)

CHAPTER 8 SERIES SOLUTIONS OF DIFFERENTIAL EQUATIONS

425

(78)

8.1 Introduction: The Taylor Polynomial Approximation

425

(6)

8.2 Power Series and Analytic Functions

431

(9)

8.3 Power Series Solutions to Linear Differential Equations

440

(11)

8.4 Equations with Analytic Coefficients

451

(6)

8.5 Cauchy-Euler (Equidimensional) Equations

457

(4)

8.6 Method of Frobenius

461

(12)

8.7 Finding a Second Linearly Independent Solution

473

(11)

8.8 Special Functions

484

(12)

Chapter Summary

496

(1)

Review Problems

497

(1)

Technical Writing Exercises

498

(1)

Group Projects for Chapter 8

499

(4)

A. Spherically Symmetric Solutions to Schrodinger's Equation for the Hydrogen Atom

499

(1)

B. Airy's Equation

500

(1)

C. Buckling of a Tower

500

(1)

D. Aging Spring and Bessel Functions

501

(2)

CHAPTER 9 MATRIX METHODS FOR LINEAR SYSTEMS

503

(73)

9.1 Introduction

503

(5)

9.2 Review 1: Linear Algebraic Equations

508

(4)

9.3 Review 2: Matrices and Vectors

512

(12)

9.4 Linear Systems in Normal Form

524

(9)

9.5 Homogeneous Linear Systems with Constant Coefficients

533

(12)

9.6 Complex Eigenvalues

545

(6)

9.7 Nonhomogeneous Linear Systems

551

(7)

9.8 The Matrix Exponential Function

558

(9)

Chapter Summary

567

(3)

Review Problems

570

(1)

Technical Writing Exercises

571

(1)

Group Projects for Chapter 9

572

(4)

A. Uncoupling Normal Systems

572

(1)

B. Matrix Laplace Transform Method

572

(2)

C. Undamped Second-Order Systems

574

(1)

D. Strange Behavior of Competing Species-Part II

575

(1)

CHAPTER 10 PARTIAL DIFFERENTIAL EQUATIONS

576

(85)

10.1 Introduction: A Model for Heat Flow

576

(3)

10.2 Method of Separation of Variables

579

(10)

10.3 Fourier Series

589

(18)

10.4 Fourier Cosine and Sine Series

607

(5)

10.5 The Heat Equation

612

(13)

10.6 The Wave Equation

625

(13)

10.7 Laplace's Equation

638

(13)

Chapter Summary

651

(2)

Technical Writing Exercises

653

(1)

Group Projects for Chapter 10

654

(7)

A. Steady-State Temperature Distribution in a Circular Cylinder

654

(1)

B. A Laplace Transform Solution of the Wave Equation

655

(1)

C. Green's Function

656

(2)

D. Numerical Method for Δu = f on a Rectangle

658

(3)

CHAPTER 11 EIGENVALUE PROBLEMS AND STURM-LIOUVILLE EQUATIONS

661

(76)

11.1 Introduction: Heat Flow in a Nonuniform Wire

661

(2)

11.2 Eigenvalues and Eigenfunctions

663

(9)

11.3 Regular Sturm-Liouville Boundary Value Problems

672

(12)

11.4 Nonhomogeneous Boundary Value Problems and the Fredholm Alternative

684

(9)

11.5 Solution by Eigenfunction Expansion

693

(6)

11.6 Green's Functions

699

(9)

11.7 Singular Sturm-Liouville Boundary Value Problems

708

(9)

11.8 Oscillation and Comparison Theory

717

(9)

Chapter Summary

726

(3)

Review Problems

729

(1)

Technical Writing Exercises

730

(1)

Group Projects for Chapter 11

731

(6)

A. Hermite Polynomials and the Harmonic Oscillator

731

(1)

B. Continuous and Mixed Spectra

731

(1)

C. Picone Comparison Theorem

732

(1)

D. Shooting Method

733

(1)

E. Finite-Difference Method for Boundary Value Problems

734

(3)

CHAPTER 12 STABILITY OF AUTONOMOUS SYSTEMS

737

(69)

12.1 Introduction: Competing Species

737

(4)

12.2 Linear Systems in the Plane

741

(13)

12.3 Almost Linear Systems

754

(12)

12.4 Energy Methods

766

(9)

12.5 Lyapunov's Direct Method

775

(9)

12.6 Limit Cycles and Periodic Solutions

784

(9)

12.7 Stability of Higher-Dimensional Systems

793

(6)

Chapter Summary

799

(2)

Review Problems

801

(1)

Technical Writing Exercises

802

(1)

Group Projects for Chapter 12

803

(3)

A. Solitons and Korteweg-de Vries Equation

803

(1)

B. Burger's Equation

803

(1)

C. Computing Phase Plane Diagrams

804

(1)

D. Ecosystem on Planet GLIA-2

805

(1)

CHAPTER 13 EXISTENCE AND UNIQUENESS THEORY

806

13.1 Introduction: Successive Approximations

806

(7)

13.2 Picard's Existence and Uniqueness Theorem

813

(8)

13.3 Existence of Solutions of Linear Equations

821

(6)

13.4 Continuous Dependence of Solutions

827

(7)

Chapter Summary

834

(1)

Review Problems

835

(1)

Technical Writing Exercises

835

APPENDICES

A-1

A. Newton's Method

A-1

B. Simpson's Rule

A-3

C. Cramer's Rule

A-5

D. Method of Least Squares

A-6

E. Runge-Kutta Procedure for n Equations

A-9

ANSWERS TO ODD-NUMBERED PROBLEMS

B-1

INDEX

I-1

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Fundamentals of Differential Equations with Boundary Value Problems

Fundamentals of Differential Equations with Boundary Value Problems

9780321145710

0321145712

Summary

Table of Contents

Supplemental Materials

Write a Review

Amazon no longer offers textbook rentals. We do!

Fundamentals of Differential Equations with Boundary Value Problems

Fundamentals of Differential Equations with Boundary Value Problems

9780321145710

0321145712

Summary

Table of Contents

Supplemental Materials

Write a Review

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