Ordinary Differential Equations

1. First-Order Differential Equations 1

1.1 Motivation and Overview 1

1.2 Linear First-Order Equations 11

1.3 Applications of Linear First-Order Equations 24

1.4 Nonlinear First-Order Equations That Are Separable 43

1.5 Existence and Uniqueness 50

1.6 Applications of Nonlinear First-Order Equations 59

1.7 Exact Equations and Equations That Can Be Made Exact 71

1.8 Solution by Substitution 81

1.9 Numerical Solution by Euler’s Method 87

2. Higher-Order Linear Equations 99

2.1 Linear Differential Equations of Second Order 99

2.2 Constant-Coefficient Equations 103

2.3 Complex Roots 113

2.4 Linear Independence; Existence, Uniqueness, General Solution 118

2.5 Reduction of Order 128

2.6 Cauchy-Euler Equations 134

2.7 The General Theory for Higher-Order Equations 142

2.8 Nonhomogeneous Equations 149

2.9 Particular Solution by Undetermined Coefficients 155

2.10 Particular Solution by Variation of Parameters 163

3. Applications of Higher-Order Equations 173

3.1 Introduction 173

3.2 Linear Harmonic Oscillator; Free Oscillation 174

3.3 Free Oscillation with Damping 186

3.4 Forced Oscillation 193

3.5 Steady-State Diffusion; A Boundary Value Problem 202

3.6 Introduction to the Eigenvalue Problem; Column Buckling 211

4. Systems of Linear Differential Equations 219

4.1 Introduction, and Solution by Elimination 219

4.2 Application to Coupled Oscillators 230

4.3 N-Space and Matrices 238

4.4 Linear Dependence and Independence of Vectors 247

4.5 Existence, Uniqueness, and General Solution 253

4.6 Matrix Eigenvalue Problem 261

4.7 Homogeneous Systems with Constant Coefficients 270

4.8 Dot Product and Additional Matrix Algebra 283

4.9 Explicit Solution of x’ = Ax and the Matrix Exponential Function 297

4.10 Nonhomogeneous Systems 307

5. Laplace Transform 317

5.1 Introduction 317

5.2 The Transform and Its Inverse 319

5.3 Applications to the Solution of Differential Equations 334

5.4 Discontinuous Forcing Functions; Heaviside Step Function 347

5.5 Convolution 358

5.6 Impulsive Forcing Functions; Dirac Delta Function 366

6. Series Solutions 379

6.1 Introduction 379

6.2 Power Series and Taylor Series 380

6.3 Power Series Solution About a Regular Point 387

6.4 Legendre and Bessel Equations 395

6.5 The Method of Frobenius 408

7. Systems of Nonlinear Differential Equations 423

7.1 Introduction 423

7.2 The Phase Plane 424

7.3 Linear Systems 435

7.4 Nonlinear Systems 447

7.5 Limit Cycles 463

7.6 Numerical Solution of Systems by Euler’s Method 468

Appendix A. Review of Partial Fraction Expansions 479

Appendix B. Review of Determinants 483

Appendix C. Review of Gauss Elimination 491

Appendix D. Review of Complex Numbers and the Complex Plane 497

Answers to Exercises 501 

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