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1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 
1.1 Definitions and Terminology 
1.2 Initial-Value Problems 
1.3 Differential Equations as Mathematical Models 

2 FIRST-ORDER DIFFERENTIAL EQUATIONS 
2.1 Solution Curves Without a Solution 
2.1.1 Direction Fields 
2.1.2 Autonomous First-Order DEs 
2.2 Separable Equations
2.3 Linear Equations 
2.4 Exact Equations
2.5 Solutions by Substitutions
2.6 A Numerical Method 

3 MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 
3.1 Linear Models 
3.2 Nonlinear Models 
3.3 Modeling with Systems of First-Order DEs 

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 Coefficient
4.4 Undetermined Coefficients—Superposition Approach 
4.5 Undetermined Coefficients—Annihilator Approach
4.6 Variation of Parameters 
4.7 Cauchy-Euler Equation
4.8 Green’s Functions 
4.8.1 Initial-Value Problems 
4.8.2 Boundary-Value Problems 
4.9 Solving Systems of Linear DEs by Elimination 
4.10 Nonlinear Differential Equations

5 MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 
5.1 Linear Models: 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 Models: Boundary-Value Problems
5.3 Nonlinear Models

6 SERIES SOLUTIONS OF LINEAR EQUATIONS 
6.1 Review of Power Series 
6.2 Solutions About Ordinary Points
6.3 Solutions About Singular Points
6.4 Special Functions

7 THE LAPLACE TRANSFORM 
7.1 Definition of the Laplace Transform
7.2 Inverse Transforms and Transforms of Derivatives
7.2.1 Inverse Transforms
7.2.2 Transforms of Derivatives 
7.3 Operational Properties I 
7.3.1 Translation on the s-Axis 
7.3.2 Translation on the t-Axis 
7.4 Operational Properties II 
7.4.1 Derivatives of a Transform 
7.4.2 Transforms of Integrals 
7.4.3 Transform of a Periodic Function
7.5 The Dirac Delta Function 
7.6 Systems of Linear Differential Equations

8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 
8.1 Preliminary Theory—Linear Systems 
8.2 Homogeneous Linear Systems
8.2.1 Distinct Real Eigenvalues 
8.2.2 Repeated Eigenvalues 
8.2.3 Complex Eigenvalues
8.3 Nonhomogeneous Linear Systems
8.3.1 Undetermined Coefficient
8.3.2 Variation of Parameters 
8.4 Matrix Exponential