After a brief review of first-order differential equations, this book focuses on second-order equations with constant coefficients that derive their general solution using only results described previously. Higher-order equations are provided since the patterns are more readily grasped by students. Stability and fourth order equations are also discussed since these topics typically appear in further study for engineering and science majors. In addition to applications to engineering systems, applications from the biological and life sciences are emphasized. Ecology and population dynamics are featured since they involve both linear and nonlinear equations, and these topics form one application thread that weaves through the chapters. Diffusion of material, heat, and mechanical and electrical oscillators are also important in biological and engineering systems and are discussed throughout. A complete Instructor Solution Manual is available upon request and contains solutions to all exercises as well as Maple code. While the book is not dependent on the use of one specific software, some of the exercises do call on the use of such systems to solve certain differential equations or to plot the results. A Student Solutions Manual is available to supplement the book, and while the first manual will feature Maple, the author is also preparing versions using Mathematica® and MATLAB® to accommodate instructor preferences. Chapter coverage includes First-Order Differential Equations; Higher-Order Linear Equations; Applications of Higher-Order Linear Equations; Systems of Linear Differential Equations; Laplace Transform; Series Solution; Systems of Nonlinear Differential Equations; and Appendices on Partial Fraction Expansions, Determinants, Gauss Elimination, and Complex Numbers and the Complex Plane.

Preface viii

**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**