Elementary Differential Equations and Boundary Value Problems 11e, like its predecessors, is written from the viewpoint of the applied mathematician, whose interest in differential equations may sometimes be quite theoretical, sometimes intensely practical, and often somewhere in between. The authors have sought to combine a sound and accurate (but not abstract) exposition of the elementary theory of differential equations with considerable material on methods of solution, analysis, and approximation that have proved useful in a wide variety of applications. While the general structure of the book remains unchanged, some notable changes have been made to improve the clarity and readability of basic material about differential equations and their applications. In addition to expanded explanations, the 11^th edition includes new problems, updated figures and examples to help motivate students.

The program is primarily intended for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for engaging with the program is a working knowledge of calculus, gained from a normal two or three semester course sequence or its equivalent. Some familiarity with matrices will also be helpful in the chapters on systems of differential equations.

Dr. William E. Boyce received his B.A. degree in Mathematics from Rhodes College, and his M.S. and Ph.D. degrees in Mathematics from Carnegie-Mellon University. He is a member of the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. He is currently the Edward P. Hamilton Distinguished Professor Emeritus of Science Education (Department of Mathematical Sciences) at Rensselaer. He is the author of numerous technical papers in boundary value problems and random differential equations and their applications. He is the author of several textbooks including two differential equations texts. In 1991 he received the William H. Wiley Distinguished Faculty Award given by Rensselaer.

Preface vii

1 Introduction 1

1.1 Some Basic Mathematical Models; Direction Fields 1

1.2 Solutions of Some Differential Equations 9

1.3 Classification of Differential Equations 16

2 First-Order Differential Equations 24

2.1 Linear Differential Equations; Method of Integrating Factors 24

2.2 Separable Differential Equations 33

2.3 Modeling with First-Order Differential Equations 39

2.4 Differences Between Linear and Nonlinear Differential Equations 51

2.5 Autonomous Differential Equations and Population Dynamics 58

2.6 Exact Differential Equations and Integrating Factors 70

2.7 Numerical Approximations: Euler’s Method 76

2.8 The Existence and Uniqueness Theorem 83

2.9 First-Order Difference Equations 91

3 Second-Order Linear Differential Equations 103

3.1 Homogeneous Differential Equations with Constant Coefficients 103

3.2 Solutions of Linear Homogeneous Equations; the Wronskian 110

3.3 Complex Roots of the Characteristic Equation 120

3.4 Repeated Roots; Reduction of Order 127

3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients 133

3.6 Variation of Parameters 142

3.7 Mechanical and Electrical Vibrations 147

3.8 Forced Periodic Vibrations 159

4 Higher-Order Linear Differential Equations 169

4.1 General Theory of n^th Order Linear Differential Equations 169

4.2 Homogeneous Differential Equations with Constant Coefficients 174

4.3 The Method of Undetermined Coefficients 181

4.4 The Method of Variation of Parameters 185

5 Series Solutions of Second-Order Linear Equations 189

5.1 Review of Power Series 189

5.2 Series Solutions Near an Ordinary Point, Part I 195

5.3 Series Solutions Near an Ordinary Point, Part II 205

5.4 Euler Equations; Regular Singular Points 211

5.5 Series Solutions Near a Regular Singular Point, Part I 219

5.6 Series Solutions Near a Regular Singular Point, Part II 224

5.7 Bessel’s Equation 230

6 The Laplace Transform 241

6.1 Definition of the Laplace Transform 241

6.2 Solution of Initial Value Problems 248

6.3 Step Functions 257

6.4 Differential Equations with Discontinuous Forcing Functions 264

6.5 Impulse Functions 270

6.6 The Convolution Integral 275

7 Systems of First-Order Linear Equations 281

7.1 Introduction 281

7.2 Matrices 286

7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 295

7.4 Basic Theory of Systems of First-Order Linear Equations 304

7.5 Homogeneous Linear Systems with Constant Coefficients 309

7.6 Complex-Valued Eigenvalues 319

7.7 Fundamental Matrices 329

7.8 Repeated Eigenvalues 337

7.9 Nonhomogeneous Linear Systems 345

8 Numerical Methods 354

8.1 The Euler or Tangent Line Method 354

8.2 Improvements on the Euler Method 363

8.3 The Runge-Kutta Method 367

8.4 Multistep Methods 371

8.5 Systems of First-Order Equations 376

8.6 More on Errors; Stability 378

9 Nonlinear Differential Equations and Stability 388

9.1 The Phase Plane: Linear Systems 388

9.2 Autonomous Systems and Stability 398

9.3 Locally Linear Systems 407

9.4 Competing Species 417

9.5 Predator-Prey Equations 428

9.6 Liapunov’s Second Method 435

9.7 Periodic Solutions and Limit Cycles 444

9.8 Chaos and Strange Attractors: The Lorenz Equations 454

10 Partial Differential Equations and Fourier Series 463

10.1 Two-Point Boundary Value Problems 463

10.2 Fourier Series 469

10.3 The Fourier Convergence Theorem 477

10.4 Even and Odd Functions 482

10.5 Separation of Variables; Heat Conduction in a Rod 488

10.6 Other Heat Conduction Problems 496

10.7 The Wave Equation: Vibrations of an Elastic String 504

10.8 Laplace's Equation 514

11 Boundary Value Problems and Sturm-Liouville Theory 529

11.1 The Occurrence of Two-Point Boundary Value Problems 529

11.2 Sturm-Liouville Boundary Value Problems 535

11.3 Nonhomogeneous Boundary Value Problems 545

11.4 Singular Sturm-Liouville Problems 556

11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 562

11.6 Series of Orthogonal Functions: Mean Convergence 566

Answers to Problems 573

Index 608