Course Outline

I. Introduction to Differential Equations

1. Review of Some Elementary Differential Equations
2. Solutions of Differential Equations (Families of Solutions, and Direction Fields)
3. Fundamental Theory: Uniqueness and Existence
4. Solving Differential Equations Numerically: Euler's Method

II. First-Order Equations

1. Separation of Variables
2. Equations with Homogeneous Function Coefficients
3. Exact Differential Equations
4. The Linear Equation of Order One and its General Solution
5. Elementary Applications
6. Other Methods in Solving First-Order Equations

III. Linear Differential Equations

1. The General Linear Equations
2. An Existence and Uniqueness Theorem
3. Linear Independence and the Wronskian.
4. General Solution of a Homogeneous Equation
5. General Solution of a Non-Homogeneous Equation
6. Differential Operators and the Fundamental Laws of Operation
7. Some Properties of Differential Operators

IV. Linear DE with Constant Coefficients

1. Homogeneous DE
2. Non-Homogeneous DE

V. Non-Homogeneous DE

1. Variation of Parameters
2. Applications

VI. The Laplace Transform

1. Definition of the Laplace Transform
2. Transforms of Elementary Functions
3. Sectionally Continuous Functions
4. Functions of Exponential Order
5. Functions of Class A
6. Transforms of Derivatives
7. Derivatives of Transforms
8. The Gamma Function
9. Periodic Functions

VII. Inverse Transforms

1. Definition of an Inverse Transform
2. Partial Fractions
3. Solution of Initial Value Problems
4. A Step Function
5. A Convolution Theorem
6. Special Integral Equation
7. Transform Methods and the Vibration of Springs
8. The Deflection of Beams

VII. Linear Systems of Equations

1. Elementary Elimination Calculus
2. First-Order Systems with Constant Coefficient
3. Solution of a First-Order System
4. Some Matrix Algebra
5. First-Order Systems Revisited
6. Complex Eigenvalues
7. Repeated Eigenvalues
8. Non-Homogeneous Systems
9. Arms Races
10. The Laplace Transform
11. Application on Electric Circuits and Simple Networks