CONTENTS Preface xi 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1 1.1 Definitions and Terminology 2 1.2 Initial-Value Problems 13 1.3 Differential Equations as Mathematical Models 19 CHAPTER 1 IN REVIEW 32 2 FIRST-ORDER DIFFERENTIAL EQUATIONS 34 2.1 Solution Curves Without a Solution 35 2.1.1 Direction Fields 35 2.1.2 Autonomous First-Order DEs 37 2.2 Separable Variables 44 2.3 Linear Equations 53 2.4 Exact Equations 62 2.5 Solutions by Substitutions 70 2.6 A Numerical Method 75 CHAPTER 2 IN REVIEW 80 3 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 82 3.1 Linear Models 83 3.2 Nonlinear Models 94 3.3 Modeling with Systems of First-Order DEs 105 CHAPTER 3 IN REVIEW 1133 v CONTENTS 1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1 Preface xi 1.1 Definitions and Terminology 2 1.2 Initial-Value Problems 13 1.3 Differential Equations as Mathematical Models 19 CHAPTER 1 IN REVIEW 32 2 FIRST-ORDER DIFFERENTIAL EQUATIONS 34 2.1 Solution Curves Without a Solution 35 2.1.1 Direction Fields 35 2.1.2 Autonomous First-Order DEs 37 2.2 Separable Variables 44 2.3 Linear Equations 53 2.4 Exact Equations 62 2.5 Solutions by Substitutions 70 2.6 A Numerical Method 75 CHAPTER 2 IN REVIEW 80 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 82 3.1 Linear Models 83 3.2 Nonlinear Models 94 3.3 Modeling with Systems of First-Order DEs 105 CHAPTER 3 IN REVIEW 113