#### Description

This course is an introduction to ordinary differential equations. The course introduces the basic techniques for solving and/or analyzing first and second order differential equations, both linear and nonlinear, and systems of differential equations. Emphasis is placed on qualitative and numerical methods, as well as on formula solutions. The use of mathematical software system is an integral and substantial part of the course. Beginning Spring 2003, all sections of the course will use the software system MATLAB. Credit will be granted for only one of the following: MATH 246 or MATH 341.

#### Prerequisites

Required: MATH 141. Recommended: MATH240 or ENES102 or PHYS161 or PHYS171.

**Topics**

**Introduction to and Classification of Differential Equations**

**First Order Equations**

Linear, separable and exact equations

Introduction to symbolic solutions using a MSS

Existence and uniqueness of solutions

Properties of nonlinear vs. linear equations

Qualitative methods for autonomous equations

Plotting direction fields using a MSS

Models and applications

**Numerical Methods**

Introduction to a numerical solver in a MSS

Elementary numerical methods: Euler, Improved Euler, Runge-Kutta

Local and global error, reliability of numerical methods

**Second Order Equations**

Theory of linear equations

Homogeneous linear equations with constant coefficients

Reduction of order

Methods of undetermined coefficients and variation of parameters for non-homogeneous equations

Symbolic and numerical solutions using a MSS

Mechanical and electrical vibrations

**Laplace Transforms**

Definition and calculation of transforms

Applications to differential equations with discontinuous forcing functions

**Systems of First Order Linear Equations**

General theory

Eigenvalue-eigenvector method for systems with constant coefficients

Finding eigenpairs and solving linear systems with a MSS

The phase plane and parametric plotting with a MSS

**Nonlinear Systems and Stability**

Autonomous systems and critical points

Stability and phase plane analysis of almost linear systems

Linearized stability analysis and plotting vector fields using a MSS

Numerical solutions and phase portraits of nonlinear systems using a MSS

Models and applications