MATH 251

COURSE DESCRIPTION: Ordinary and Partial Differential Equations (4:4:0)
First- and second-order equations; numerical methods; special functions;
Laplace transform solutions; higher order equations; Fourier series,
partial differential equations. Students who have passed Math 250 may
only take a one credit section of this course.

PREREQUISITE: Math 141

TOPICS

INTRODUCTION
Classification of Differential Equations

FIRST ORDER DIFFERENTIAL EQUATIONS
Linear Equations
Further Discussion of Linear Equations
Separable Equations
Applications of First Order Linear Equations
Population Dynamics and Some Related Problems
Problems in Mechanics
Exact Equations and Integrating Factors

SECOND ORDER LINEAR EQUATIONS
Homogeneous Equations with Constant Coefficients
Fundamental Solutions of Linear Homogeneous Equations
Linear Independence and the Wronskian
Complex Roots of the Characteristic Equations
Repeated Roots; Reduction of Order
Nonhomogeneous Equations; Method of Undetermined Coefficients
Variation of Parameters
Mechanical and Electrical Vibrations
Forced Vibrations

HIGHER ORDER LINEAR EQUATIONS
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients

SERIES SOLUTIONS OF SECOND ORDER LINEAR EQUATIONS
Series Solutions near an Ordinary Point

THE LAPLACE TRANSFORM
Definition of the Laplace Transform
Solution of Initial Value Problems
Step Functions
Differential Equations with Discontinuous Forcing Functions
Impulse Functions

NUMERICAL METHODS
The Euler or Tangent Line Method
Errors in Numerical Procedures
Improvements on the Euler Method
The Runge-Kutta Method

PARTIAL DIFFERENTIAL EQUATIONS AND FOURIER SERIES
Separation of Variables
Fourier Series
The Fourier Theorem
Even and Odd Functions
Solutions of Heat Conduction Problems
The Wave Equation: Vibrations of an Elastic String
Laplace's Equation

AMK 6/21/95