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: a C- or better in 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
