Syllabus: Math 365 - Differential Equations

Department of Mathematics

Millersville University

Description

The study of:

1. first order differential equations
2. linear 1st and 2nd order initial value problems
3. power series solutions and method of Frobenius

In addition, at least one of the following topics: special functions of mathematical physics, Laplace transforms or systems of 1st order equations. Heavy emphasis on applications. (3 credits)

Prerequisite

Math 261.

Rationale

This course provides an introduction to ordinary differential equations and their applications. This serves as the prerequisite for MATH 467, Partial Differential Equations.

A knowledge of differential equations is essential for students majoring in science, especially those in physics and meteorology and for those who wish to pursue an engineering degree. This is also an important course for mathematics majors, as it provides students with significant and meaningful applications of the calculus to the problems of classical physics.

Objectives

Upon completion of this course, the student will:

1. be able to solve a variety of ordinary differential equations
2. appreciate the theory underlying the techniques of solution
3. be conversant with methods of applying ordinary differential equations in various applications

Course Outline

1. First order ordinary differential equations and their applications
   1. separable equations
   2. First order linear equations
   3. exact differential equations
   4. existence and uniqueness theory (linear and nonlinear equations)
   5. equations reducible to first order
2. Linear differential equations of second order
   1. linear independence and Wronskians
   2. existence and uniqueness theory
   3. homogeneous equations with constant coefficients
   4. reduction of order
   5. method of undetermined coefficients
   6. variation of parameters
   7. Euler equations
3. Series solutions
   1. power series solutions about an ordinary point
   2. power series solutions about a regular singular point
4. Laplace transforms (optional)
   1. definition and properties of the Laplace transform
   2. applications to initial value problems
   3. applications to systems of differential equations
   4. the unit step function and the Dirac delta function
   5. the convolution theorem
5. Systems of first order equations (optional)
   1. Matrix algebra
   2. solution of homogeneous systems with constant coefficients
   3. nonhomogeneous systems by variation of parameters
6. Special functions of mathematical physics (optional)
   1. series solutions about a regular singular point where roots of the indicial equation are repeated or differ by an integer
   2. Bessel equations
   3. Bessel functions of 1st and 2nd kind of integral order

Suggested Text:

Boyce, W.E. and DiPrima, R.C., Elementary Differential Equations and Boundary Value Problems, 6th edition, John Wiley, 1997.

Prepared February 18, 1999 by R.T. Smith