**A First Course in Differential Equations with Modeling Applications, Twelfth Edition (Metric Version)**

Dennis G. Zill

**Contents**

1 Introduction to Differential Equations 2

1.1 Definitions and Terminology 3

1.2 Initial-Value Problems 15

1.3 Differential Equations as Mathematical Models 22

Chapter 1 In Review 34

2 First-Order Differential Equations 36

2.1 Solution Curves without a Solution 37

2.1.1 Direction Fields 37

2.1.2 Autonomous First-Order DEs 39

2.2 Separable Equations 47

2.3 Linear Equations 56

2.4 Exact Equations 65

2.5 Solutions by Substitutions 73

2.6 A Numerical Method 77

Chapter 2 In Review 82

3 Modeling with First-Order Differential

Equations 84

3.1 **Linear Models** 85

3.2 Nonlinear Models 96

3.3 Modeling with Systems of First-Order DEs 107

Chapter 3 In Review 114

4 Higher-Order Differential Equations 118

4.1 Theory of Linear Equations 119

4.1.1 Initial-Value and Boundary-Value Problems 119

4.1.2 Homogeneous Equations 121

4.1.3 Nonhomogeneous Equations 127

4.2 Reduction of Order 132

4.3 Homogeneous Linear Equations with Constant Coefficients 135

4.4 Undetermined Coefficients—Superposition Approach 142

4.5 Undetermined Coefficients—Annihilator Approach 152

4.6 Variation of Parameters 159

4.7 Cauchy-Euler Equations 166

4.8 Green’s Functions 173

4.8.1 Initial-Value Problems 173

4.8.2 Boundary-Value Problems 179

4.9 Solving Systems of Linear DEs by Elimination 183

4.10 Nonlinear Differential Equations 188

Chapter 4 In Review 193

5 Modeling with Higher-Order Differential Equations 196

5.1 Linear Models: Initial-Value Problems 197

5.1.1 Spring/Mass Systems: Free Undamped Motion 197

5.1.2 Spring/Mass Systems: Free Damped Motion 202

5.1.3 Spring/Mass Systems: Driven Motion 204

5.1.4 Series Circuit Analogue 207

5.2 Linear Models: Boundary-Value Problems 215

5.3 Nonlinear Models 223

Chapter 5 In Review 233

6 Series Solutions of Linear Equations 240

6.1 Review of Power Series 241

6.2 Solutions About Ordinary Points 247

6.3 Solutions About Singular Points 256

6.4 Special Functions 266

Chapter 6 In Review 281

7 The Laplace Transform 284

7.1 Definition of the Laplace Transform 285

7.2 Inverse Transforms and Transforms of Derivatives 293

7.2.1 Inverse Transforms 293

7.2.2 Transforms of Derivatives 296

7.3 Operational Properties I 302

7.3.1 Translation on the s-Axis 302

7.3.2 Translation on the t-Axis 305

7.4 Operational Properties II 315

7.4.1 Derivatives of a Transform 315

7.4.2 Transforms of Integrals 317

7.4.3 Transform of a Periodic Function 323

7.5 The Dirac Delta Function 328

7.6 Systems of Linear Differential Equations 332

Chapter 7 In Review 339

8 Systems of Linear Differential Equations 344

8.1 Theory of Linear Systems 345

8.2 Homogeneous Linear Systems 354

8.2.1 Distinct Real Eigenvalues 355

8.2.2 Repeated Eigenvalues 358

8.2.3 Complex Eigenvalues 362

8.3 Nonhomogeneous Linear Systems 369

8.3.1 Undetermined Coefficients 369

8.3.2 Variation of Parameters 371

8.4 Matrix Exponential 376

Chapter 8 In Review 380

9 Numerical Solutions of Ordinary **Differential Equations** 382

9.1 Euler Methods and Error Analysis 383

9.2 Runge-Kutta Methods 388

9.3 Multistep Methods 392

9.4 Higher-Order Equations and Systems 395

9.5 Second-Order Boundary-Value Problems 399

Chapter 9 In Review 403

Appendices

A Integral-Defined Functions APP-3

B Matrices APP-11

C Laplace Transforms APP-29

Answers for Selected Odd-Numbered Problems ANS-1

Index I-1