Elementary Differential Equations and Boundary Value Problems

1.1 Some Basic Mathematical Models; Direction Fields 1

1.2 Solutions of Some Differential Equations 10

1.3 Classification of Differential Equations 19

1.4 Historical Remarks 26

Chapter 2 First Order Differential Equations 31

2.1 Linear Equations; Method of Integrating Factors 31

2.2 Separable Equations 42

2.3 Modeling with First Order Equations 51

2.4 Differences Between Linear and Nonlinear Equations 68

2.5 Autonomous Equations and Population Dynamics 78

2.6 Exact Equations and Integrating Factors 95

2.7 Numerical Approximations: Euler’s Method 102

2.8 The Existence and Uniqueness Theorem 112

2.9 First Order Difference Equations 122

Chapter 3 Second Order Linear Equations 137

3.1 Homogeneous Equations with Constant Coefficients 137

3.2 Solutions of Linear Homogeneous Equations; the Wronskian 145

3.3 Complex Roots of the Characteristic Equation 158

3.4 Repeated Roots; Reduction of Order 167

3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients 175

3.6 Variation of Parameters 186

3.7 Mechanical and Electrical Vibrations 192

3.8 Forced Vibrations 207

Chapter 4 Higher Order Linear Equations 221

4.1 General Theory of nth Order Linear Equations 221

4.2 Homogeneous Equations with Constant Coefficients 228

4.3 The Method of Undetermined Coefficients 236

4.4 The Method of Variation of Parameters 241

Chapter 5 Series Solutions of Second Order Linear Equations 247

5.1 Review of Power Series 247

5.2 Series Solutions Near an Ordinary Point, Part I 254

5.3 Series Solutions Near an Ordinary Point, Part II 265

5.4 Euler Equations; Regular Singular Points 272

5.5 Series Solutions Near a Regular Singular Point, Part I 282

5.6 Series Solutions Near a Regular Singular Point, Part II 288

5.7 Bessel’s Equation 296

Chapter 6 The Laplace Transform 309

6.1 Definition of the Laplace Transform 309

6.2 Solution of Initial Value Problems 317

6.3 Step Functions 327

6.4 Differential Equations with Discontinuous Forcing Functions 336

6.5 Impulse Functions 343

6.6 The Convolution Integral 350

Chapter 7 Systems of First Order Linear Equations 359

7.1 Introduction 359

7.2 Review of Matrices 368

7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 378

7.4 Basic Theory of Systems of First Order Linear Equations 390

7.5 Homogeneous Linear Systems with Constant Coefficients 396

7.6 Complex Eigenvalues 408

7.7 Fundamental Matrices 421

7.8 Repeated Eigenvalues 429

7.9 Nonhomogeneous Linear Systems 440

Chapter 8 Numerical Methods 451

8.1 The Euler or Tangent Line Method 451

8.2 Improvements on the Euler Method 462

8.3 The Runge–Kutta Method 468

8.4 Multistep Methods 472

8.5 Systems of First Order Equations 478

8.6 More on Errors; Stability 482

Chapter 9 Nonlinear Differential Equations and Stability 495

9.1 The Phase Plane: Linear Systems 495

9.2 Autonomous Systems and Stability 508

9.3 Locally Linear Systems 519

9.4 Competing Species 531

9.5 Predator–Prey Equations 544

9.6 Liapunov’s Second Method 554

9.7 Periodic Solutions and Limit Cycles 565

9.8 Chaos and Strange Attractors: The Lorenz Equations 577

Chapter 10 Partial Differential Equations and Fourier Series 589

10.1 Two-Point Boundary Value Problems 589

10.2 Fourier Series 596

10.3 The Fourier Convergence Theorem 607

10.4 Even and Odd Functions 614

10.5 Separation of Variables; Heat Conduction in a Rod 623

10.6 Other Heat Conduction Problems 632

10.7 TheWave Equation: Vibrations of an Elastic String 643

10.8 Laplace’s Equation 658

AppendixA Derivation of the Heat Conduction Equation 669

Appendix B Derivation of theWave Equation 673

Chapter 11 Boundary Value Problems and Sturm–Liouville Theory 677

11.1 The Occurrence of Two-Point Boundary Value Problems 677

11.2 Sturm–Liouville Boundary Value Problems 685

11.3 Nonhomogeneous Boundary Value Problems 699

11.4 Singular Sturm–Liouville Problems 714

11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 721

11.6 Series of Orthogonal Functions: Mean Convergence 728

Answers to Problems 739

Index 799