**What are you going to learn?**

**Content**

Model with first-order differential equations (DE) and identify initial value problems. Solve scalar differential equations, homogeneous and non-homogeneous, using methods including separation of variables, integrating factors, eigenvalues, LaPlace transformations. Model with systems of first-order DEs and higher-order DEs. Solve systems of linear differential equations using matrices and eigenvalues.

**Chapter 1. Introduction**

Terminology

Classification of Differential Equations

Initial Value Problem. Cauchy's Theorem of

*y' = f(x, y)*Mathematical Models and Direction of Fields

Integrating Factors

Separation of Variables

Exact Equations

First Order Linear Equations

Equations with Homogeneous Coefficients

Solution by Substitutions

Bernoulli's DE

Computer Solutions

**Chapter 2. First Order Ordinary Differential Equations (ODE)**

**Chapter 3. Second Order Ordinary Differential Equations**

The General Second Order Linear Homogeneous Equation

Linear Independence

Existence and Uniqness Theorem

The Wronskian

General Solution of a Homogeneous Equation

General Solution of a Nonhomogeneous Equation

Differential Operators

**Chapter 4. Higher Order Constant Coefficients Linear Equations**

Power Series

Series Solutions Near an Ordinary Point

Euler Equations, Regular Singular Points

Series Solutions Near an Regular Singular Point

Bessel's Equation

**Chapter 5. Series Solutions of 2d Order LDE**

Power Series

Series Solutions Near an Ordinary Point

Euler Equations, Regular Singular Points

Series Solutions Near an Regular Singular Point

Bessel's Equation

**Chapter 6. Laplace Transform**

Definition

Solution of Initial Value Probems

Step Fucntions

Impulse Functions

The Convolution Integral

**Chapter 7. Systems of First Order Differential Equations**

Review of Matrices

Systems of Linear Equations. Linear Independence

Eigenvectors, Eigenvalues

Basic Theory

Homogeneous Linear Systems with Constant Coefficients

Nonhomogeneous Linear Systems

**Chapter 8. Numerical Methods**

The Euler's Method

Runge-Kutta Method

Multi Step Methods

Systems of First order Equations

### Bibliography

Boyce, W. E., DiPrima, R.

8th ed., Wiley & Sons Inc., New York.*Elementary Differential Equations and Boundary Problems*.Zill, D.

. 10th ed., Brooks/Cole, London, 1997.*A First Course in Differential Equations with Modeling Applications*