# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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2.9 First Order Difference Equations 115

Chapter 3 Second Order Linear Equations 129

3.1 Homogeneous Equations with Constant Coefficients 129

3.2 Fundamental Solutions of Linear Homogeneous Equations 137

3.3 Linear Independence and the Wronskian 147

3.4 Complex Roots of the Characteristic Equation 153

3.5 Repeated Roots; Reduction of Order 160

3.6 Nonhomogeneous Equations; Method of Undetermined Coefficients 169

3.7 Variation of Parameters 179

3.8 Mechanical and Electrical Vibrations 186

3.9 Forced Vibrations 200

Chapter 4 Higher Order Linear Equations 209

4.1 General Theory of nth Order Linear Equations 209

4.2 Homogeneous Equations with Constant Coeffients 214

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Contents

xv

4.3 The Method of Undetermined Coefficients 222

4.4 The Method of Variation of Parameters 226

Chapter 5 Series Solutions of Second Order Linear Equations 231

5.1 Review of Power Series 231

5.2 Series Solutions near an Ordinary Point, Part I 238

5.3 Series Solutions near an Ordinary Point, Part II 249

5.4 Regular Singular Points 255

5.5 Euler Equations 260

5.6 Series Solutions near a Regular Singular Point, Part I 267

5.7 Series Solutions near a Regular Singular Point, Part II 272

5.8 Bessel’s Equation 280

Chapter 6 The Laplace Transform 293

6.1 Definition of the Laplace Transform 293

6.2 Solution of Initial Value Problems 299

6.3 Step Functions 310

6.4 Differential Equations with Discontinuous Forcing Functions 317

6.5 Impulse Functions 324

6.6 The Convolution Integral 330

Chapter 7 Systems of First Order Linear Equations 339

7.1 Introduction 339

7.2 Review of Matrices 348

7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 357

7.4 Basic Theory of Systems of First Order Linear Equations 368

7.5 Homogeneous Linear Systems with Constant Coefficients 373

7.6 Complex Eigenvalues 384

7.7 Fundamental Matrices 393

7.8 Repeated Eigenvalues 401

7.9 Nonhomogeneous Linear Systems 411

Chapter 8 Numerical Methods 419

8.1 The Euler or Tangent Line Method 419

8.2 Improvements on the Euler Method 430

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Contents

8.3 The Runge-Kutta Method 435

8.4 Multistep Methods 439

8.5 More on Errors; Stability 445

8.6 Systems of First Order Equations 455

Chapter 9 Nonlinear Differential Equations and Stability 459

9.1 The Phase Plane; Linear Systems 459

9.2 Autonomous Systems and Stability 471

9.3 Almost Linear Systems 479

9.4 Competing Species 491

9.5 Predator-Prey Equations 503

9.6 Liapunov’s Second Method 511

9.7 Periodic Solutions and Limit Cycles 521

9.8 Chaos and Strange Attractors; the Lorenz Equations 532

Chapter 10 Partial Differential Equations and Fourier Series 541

10.1 Two-Point Boundary Valve Problems 541

10.2 Fourier Series 547

10.3 The Fourier Convergence Theorem 558

10.4 Even and Odd Functions 564

10.5 Separation of Variables; Heat Conduction in a Rod 573

10.6 Other Heat Conduction Problems 581

10.7 The Wave Equation; Vibrations of an Elastic String 591

10.8 Laplace’s Equation 604

Appendix A. Derivation of the Heat Conduction Equation 614

Appendix B. Derivation of the Wave Equation 617

Chapter 11 Boundary Value Problems and Sturm-Liouville Theory 621

11.1 The Occurrence of Two Point Boundary Value Problems 621

11.2 Sturm-Liouville Boundary Value Problems 629

11.3 Nonhomogeneous Boundary Value Problems 641

11.4 Singular Sturm-Liouville Problems 656

11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 663

11.6 Series of Orthogonal Functions: Mean Convergence 669 Answers to Problems 679

Index 737

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CHAPTER

1

Introduction

In this chapter we try in several different ways to give perspective to your study of differential equations. First, we use two problems to illustrate some of the basic ideas that we will return to and elaborate upon frequently throughout the remainder of the book. Later, we indicate several ways of classifying equations, in order to provide organizational structure for the book. Finally, we outline some of the major figures and trends in the historical development of the subject. The study of differential equations has attracted the attention of many of the world’s greatest mathematicians during the past three centuries. Nevertheless, it remains a dynamic field of inquiry today, with many interesting open questions.

1.1 Some Basic Mathematical Models; Direction Fields

Before embarking on a serious study of differential equations (for example, by reading this book or major portions of it) you should have some idea of the possible benefits to be gained by doing so. For some students the intrinsic interest of the subject itself is enough motivation, but for most it is the likelihood of important applications to other fields that makes the undertaking worthwhile.

Many of the principles, or laws, underlying the behavior of the natural world are statements or relations involving rates at which things happen. When expressed in mathematical terms the relations are equations and the rates are derivatives. Equations containing derivatives are differential equations. Therefore, to understand and to investigate problems involving the motion of fluids, the flow of current in electric

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