## Table of contents

**1. Introduction**

1.1 Some Basic Mathematical Models; Direction Fields

1.2 Solutions of Some Differential Equations

1.3 Classification of Differential Equations

**2. First-Order Differential Equations**

2.1 Linear Differential Equations; Method of Integrating Factors

2.2 Separable Differential Equations

2.3 Modeling with First-Order Differential Equations

2.4 Differences Between Linear and Nonlinear Differential Equations

2.5 Autonomous Differential Equations and Integrating Factors

2.6 Exact Differential Equations and Integrating Factors

2.7 Numerical Approximations: Euler's Method

2.8 The Existence and Uniqueness Theorem

2.9 First-Order Difference Equations

**3. Second-Order Linear Differential Equations**

3.1 Homogeneous Differential Equations with Constant Coefficients

3.2 Solutions of Linear Homogeneous Equations; the Wronskian

3.3 Complex Roots of the Characteristic Equation

3.4 Repeated Roots; Reduction of Order

3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients

3.6 Variation of Parameters

3.7 Mechanical and Electrical Vibrations

3.8 Forced Periodic Vibrations

**4. Higher-Order Linear Differential Equations**

4.1 General Theory of

*n*Order Linear Differential Equations

^{th}4.2 Homogeneous Differential Equations with Constant Coefficients

4.3 The Method of Undetermined Coefficients

4.4 The Method of Variation of Parameters

**5. Series Solutions of Second-Order Linear Equations**

5.1 Review of Power Series

5.2 Series Solutions Near an Ordinary Point, Part I

5.3 Series Solutions Near an Ordinary Point, Part II

5.4 Euler Equations; Regular Singular Points

5.5 Series Solutions Near a Regular Singular Point, Part I

5.6 Series Solutions Near a Regular Singular Point, Part II

5.7 Bessel's Equation

**6. The Laplace Transform**

6.1 Definition of the Laplace Transform

6.2 Solution of Initial Value Problems

6.3 Step Functions

6.4 Differential Equations with Discontinuous Forcing Functions

6.5 Impulse Functions

6.6 The Convolution Integral

**7. Systems of First-Order Linear Equations**

7.1 Introduction

7.2 Matrices

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

7.4 Basic Theory of Systems of First-Order Linear Equations

7.5 Homogeneous Linear Systems with Constant Coefficients

7.6 Complex-Valued Eigenvalues

7.7 Fundamental Matrices

7.8 Repeated Eigenvalues

7.9 Nonhomogeneous Linear Systems

**8. Numerical Methods**

8.1 The Euler or Tangent Line Method

8.2 Improvements on the Euler Method

8.3 The Runge-Kutta Method

8.4 Multistep Methods

8.5 Systems of First-Order Equations

8.6 More on Errors; Stability

**9. Nonlinear Differential Equations and Stability**

9.1 The Phase Plane: Linear Systems

9.2 Autonomous Systems and Stability

9.3 Locally Linear Systems

9.4 Competing Species

9.5 Predator-Prey Equations

9.6 Liapunov's Second Method

9.7 Periodic Solutions and Limit Cycles

9.8 Chaos and Strange Attractors: The Lorenz Equations