Preface v

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 17

2 First-Order Differential Equations 26

2.1 Linear Differential Equations; Method of Integrating Factors 26

2.2 Separable Differential Equations 34

2.3 Modeling with First-Order Differential Equations 41

2.4 Differences Between Linear and Nonlinear Differential Equations 53

2.5 Autonomous Differential Equations and Population Dynamics 61

2.6 Exact Differential Equations and Integrating Factors 72

2.7 Numerical Approximations: Euler’s Method 78

2.8 The Existence and Uniqueness Theorem 86

2.9 First-Order Difference Equations 93

3 Second-Order Linear Differential Equations 106

3.1 Homogeneous Differential Equations with Constant Coefficients 106

3.2 Solutions of Linear Homogeneous Equations; the Wronskian 113

3.3 Complex Roots of the Characteristic Equation 123

3.4 Repeated Roots; Reduction of Order 130

3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients 136

3.6 Variation of Parameters 145

3.7 Mechanical and Electrical Vibrations 150

3.8 Forced Periodic Vibrations 161

4 Higher-Order Linear Differential Equations 173

4.1 General Theory of n^𝗍𝗁 Order Linear Differential Equations 173

4.2 Homogeneous Differential Equations with Constant Coefficients 178

4.3 The Method of Undetermined Coefficients 185

4.4 The Method of Variation of Parameters 189

5 Series Solutions of Second-Order Linear Equations 194

5.1 Review of Power Series 194

5.2 Series Solutions Near an Ordinary Point, Part I 200

5.3 Series Solutions Near an Ordinary Point, Part II 209

5.4 Euler Equations; Regular Singular Points 215

5.5 Series Solutions Near a Regular Singular Point, Part I 224

5.6 Series Solutions Near a Regular Singular Point, Part II 228

5.7 Bessel’s Equation 235

6 The Laplace Transform 247

6.1 Definition of the Laplace Transform 247

6.2 Solution of Initial Value Problems 254

6.3 Step Functions 263

6.4 Differential Equations with Discontinuous Forcing Functions 270

6.5 Impulse Functions 275

6.6 The Convolution Integral 280

7 Systems of First-Order Linear Equations 288

7.1 Introduction 288

7.2 Matrices 293

7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 301

7.4 Basic Theory of Systems of First-Order Linear Equations 311

7.5 Homogeneous Linear Systems with Constant Coefficients 315

7.6 Complex-Valued Eigenvalues 325

7.7 Fundamental Matrices 335

7.8 Repeated Eigenvalues 342

7.9 Nonhomogeneous Linear Systems 351

8 Numerical Methods 363

8.1 The Euler or Tangent Line Method 363

8.2 Improvements on the Euler Method 372

8.3 The Runge-Kutta Method 376

8.4 Multistep Methods 380

8.5 Systems of First-Order Equations 385

8.6 More on Errors; Stability 387

9 Nonlinear Differential Equations and Stability 400

9.1 The Phase Plane: Linear Systems 400

9.2 Autonomous Systems and Stability 410

9.3 Locally Linear Systems 419

9.4 Competing Species 429

9.5 Predator – Prey Equations 439

9.6 Liapunov’s Second Method 446

9.7 Periodic Solutions and Limit Cycles 455

9.8 Chaos and Strange Attractors: The Lorenz Equations 465

10 Partial Differential Equations and Fourier Series 476

10.1 Two-Point Boundary Value Problems 476

10.2 Fourier Series 482

10.3 The Fourier Convergence Theorem 490

10.4 Even and Odd Functions 495

10.5 Separation of Variables; Heat Conduction in a Rod 501

10.6 Other Heat Conduction Problems 508

10.7 The Wave Equation: Vibrations of an Elastic String 516

10.8 Laplace’s Equation 527

A Appendix 537

B Appendix 541

11 Boundary Value Problems and Stur-Liouville Theory 544

11.1 The Occurrence of Two-Point Boundary Value Problems 544

11.2 Sturm-Liouville Boundary Value Problems 550

11.3 Nonhomogeneous Boundary Value Problems 561

11.4 Singular Sturm-Liouville Problems 572

11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 578

11.6 Series of Orthogonal Functions: Mean Convergence 582

Answers to Problems 591

Index 624

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