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Chapter 1 Introduction 1

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 50

2.4 Differences Between Linear and Nonlinear Equations 68

2.5 Autonomous Equations and Population Dynamics 78

2.6 Exact Equations and Integrating Factors 94

2.7 Numerical Approximations: Euler's Method 101

2.8 The Existence and Uniqueness Theorem 110

2.9 First Order Difference Equations 119

Chapter 3 Second Order Linear Equations 135

3.1 Homogeneous Equations with Constant Coefficients 135

3.2 Fundamental Solutions of Linear Homogeneous Equations 143

3.3 Linear Independence and the Wronskian 153

3.4 Complex Roots of the Characteristic Equation 159

3.5 Repeated Roots; Reduction of Order 166

3.6 Nonhomogeneous Equations; Method of Undetermined Coefficients 175

3.7 Variation of Parameters 186

3.8 Mechanical and Electrical Vibrations 192

3.9 Forced Vibrations 207

Chapter 4 Higher Order Linear Equations 219

4.1 General Theory of nth Order Linear Equations 219

4.2 Homogeneous Equations with Constant Coefficients 224

4.3 The Method of Undetermined Coefficients 233

4.4 The Method of Variation of Parameters 237

Chapter 5 Series Solutions of Second Order Linear Equations 243

5.1 Review of Power Series 243

5.2 Series Solutions Near an Ordinary Point, Part Ⅰ 250

5.3 Series Solutions Near an Ordinary Point, Part Ⅱ 261

5.4 Regular Singular Points 268

5.5 Euler Equations 273

5.6 Series Solutions Near a Regular Singular Point, Part Ⅰ 279

5.7 Series Solutions Near a Regular Singular Point, Part Ⅱ 286

5.8 Bessel's Equation 294

Chapter 6 The Laplace Transform 307

6.1 Definition of the Laplace Transform 307

6.2 Solution of Initial Value Problems 314

6.3 Step Functions 325

6.4 Differential Equations with Discontinuous Forcing Functions 332

6.5 Impulse Functions 340

6.6 The Convolution Integral 346

Chapter 7 Systems of First Order Linear Equations 355

7.1 Introduction 355

7.2 Review of Matrices 364

7.3 Linear Algebraic Equations; Linear Independence, Eigenvalues,Eigenvectors 374

7.4 Basic Theory of Systems of First Order Linear Equations 385

7.5 Homogeneous Linear Systems with Constant Coefficients 390

7.6 Complex Eigenvalues 401

7.7 Fundamental Matrices 414

7.8 Repeated Eigenvalues 422

7.9 Nonhomogeneous Linear Systems 431

Chapter 8 Numerical Methods 441

8.1 The Euler or Tangent Line Method 441

8.2 Improvements on the Euler Method 452

8.3 The Runge-Kutta Method 457

8.4 Multistep Methods 462

8.5 More on Errors; Stability 468

8.6 Systems of First Order Equations 478

Chapter 9 Nonlinear Differential Equations and Stability 483

9.1 The Phase Plane: Linear Systems 483

9.2 Autonomous Systems and Stability 495

9.3 Almost Linear Systems 503

9.4 Competing Species 515

9.5 Predator-Prey Equations 528

9.6 Liapunov's Second Method 536

9.7 Periodic Solutions and Limit Cycles 547

9.8 Chaos and Strange Attractors: The Lorenz Equations 558

Chapter 10 Partial Differential Equations and Fourier Series 569

10.1 Two-Point Boundary Value Problems 569

10.2 Fourier Series 576

10.3 The Fourier Convergence Theorem 587

10.4 Even and Odd Functions 594

10.5 Separation of Variables; Heat Conduction in a Rod 603

10.6 Other Heat Conduction Problems 612

10.7 The Wave Equation: Vibrations of an Elastic String 623

10.8 Laplace's Equation 638

Appendix A Derivation of the Heat Conduction Equation 649

Appendix B Derivation of the Wave Equation 653

Chapter 11 Boundary Value Problems 657

11.1 The Occurrence of Two-Point Boundary Value Problems 657

11.2 Sturm-Liouville Boundary Value Problems 665

11.3 Nonhomogeneous Boundary Value Problems 679

11.4 Singular Sturm-Liouville Problems 695

11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 702

11.6 Series of Orthogonal Functions: Mean Convergence 709

Answers to Problems 719

Index 781