1 INTRODUCTION TO DIFFERENTIAL EQUATIONS

1.1 Definitions and Terminology

1.2 Initial-Value Problems

1.3 Differential Equations as Mathematical Models

Chapter 1 in Review

2 FIRST-ORDER DIFFERENTIAL EQUATIONS

2.1 Solution Curves Without the Solution

2.2 Separable Variables

2.3 Linear Equations

2.4 Exact Equations

2.5 Solutions by Substitutions

2.6 A Numerical Solution

Chapter 2 in Review

3 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS

3.1 Linear Equations

3.2 Nonlinear Equations

3.3 Systems of Linear and Nonlinear Differential Equations

Chapter 3 in Review

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4 HIGHER-ORDER DIFFERENTIAL EQUATIONS

4.1 Preliminary Theory:Linear Equations

4.1.1 Initial-Value and Boundary-Value Problems

4.1.2 Homogeneous Equations

4.1.3 Nonhomogeneous Equations

4.2 Reduction of Order

4.3 Homogeneous Linear Equations with Constant Coefficients

4.4 Undetermined Coefficients—Superposition Approach

4.5 Undetermined Coefficients—Annihilator Approach

4.6 Variation of Parameters

4.7 Cauchy-Euler Equation

4.8 Solving Systems of Linear Equations by Elimination

4.9 Nonlinear Equations

Chapter 4 in Review

5 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS

5.1 Linear Equations:Initial-Value Problems

5.1.1 Spring/Mass Systems:Free Undamped Motion

5.1.2 Spring/Mass Systems:Free Damped Motion

5.1.3 Spring/Mass Systems:Driven Motion

5.1.4 Series Circuit Analogue

5.2 Linear Equations:Boundary-Value Problems

5.3 Nonlinear Equations

Chapter 5 in Review

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6 SERIES SOLUTIONS OF LINEAR EQUATIONS

6.1 Solutions About Ordinary Points

6.1.1 Review of Power Series

6.1.2 Power Series Solutions

6.2 Solutions About Singular Points

6.3 Two Special Equations

Chapter 6 in Review

7 THE LAPLACE TRANSFORM

7.1 Definition of the Laplace Transform

7.2 Inverse Transform and Transforms of Derivatives

7.3 Translation Theorems

7.3.1 Translation on the s-Axis

7.3.2 Translation on the t-Axis

7.4 Additional Operational Properties

7.5 Dirac Delta Function

7.6 Systems of Linear Equations

Chapter 7 in Review

8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS

8.1 Preliminary Theory

8.2 Homogeneous Linear Systems with Constant Coefficients

8.2.1 Distinct Real Eigenvalues

8.2.2 Repeated Eigenvalues

8.2.3 Complex Eigenvalues

8.3 Variation of Parameters

8.4 Matrix Exponential

Chapter 8 in Review

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9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

9.1 Euler Methods and Error Analysis

9.2 Runge-Kutta Methods

9.3 Multistep Methods

9.4 Higher-Order Equations and Systems

9.5 Second-Order Boundary-Value Problems

Chapter 9 in Review