MATHS 374: Differential Equations (3)

Syllabus

1.            Prerequisite:  MATHS 162 or 166 or permission of the department chairperson.

2.            Catalog Description:  Introduction to nth-order ordinary differential equations, equations of order one, elementary applications, linear equations with constant coefficients, nonhomogeneous equations, undetermined coefficients, variation of parameters, linear systems of equations, and the Laplace transform.  Use of standard computer software.

3.            Course Objectives:  This course is an introduction to the study of differential equations.  After this course, a student will be able to solve standard ordinary differential equations using techniques learned in integral calculus. Students will also learn how to interpret the behavior of solutions of differential equations. In addition, the student will learn how to utilize a computer algebra package such as Mathematica in problem solving—a crucial skill for today’s applied mathematician.

4.            Course Rationale:  One of the most important branches of mathematics is the subject of differential equations.  Models that involve rates of change of a function can often be written in terms of differential equations.   Areas where differential equations are used include physics, engineering, chemistry, biology, and finance.  In order to work with such models, it is important to know how to solve and interpret the behavior of solutions of differential equations. 

5.            Course Content:  The core of the course should include the following topics:

a.       Definitions; Families of Curves:  examples of differential equations, definitions, families of solutions, method of isoclines, existence of solutions.

b.       Equations of Order One:  separation of variables, exact equations, linear equations of order one.

c.       Applications:  simple applications of differential equations of order one such as escape velocity, exponential growth/decay, Newton’s law of cooling, terminal velocity, logistic growth/decay.

d.       Linear Differential Equations:  general linear equation, existence and uniqueness of solutions, linear independence, the Wronskian, general solution of a homogeneous equation, general solution of a nonhomogeneous equation.

e.       Linear Equations with Constant Coefficients:  auxiliary equation, distinct roots, repeated roots, complex roots.

f.        Nonhomogeneous Solution Techniques:  method of undetermined coefficients, reduction of order, variation of parameters.

g.       Systems of Equations:  matrix operations, vector functions, first-order systems, characteristic equation, distinct eigenvalues, complex eigenvectors, repeated eigenvectors, homogeneous and nonhomogeneous systems.

h.       Laplace Transform:  definition, transforms of elementary functions, inverse transform, applications to solving differential equations.

 It may be necessary to include a brief review of integration techniques.  Illustrations of how computer algebra systems, such as Mathematica or Maple, can be applied to solve or explore the core ideas should be included in the course.  Additional topics are at the discretion of the instructor.  For a more applied flavor, more time can be spent on the Laplace transform and additional applications such as the harmonic oscillator, Kepler’s laws, arms races, and simple networks.  For a more theoretical flavor, more time can be spent on simple numerical methods such as Euler’s method, power series solutions, and the theory of existence and uniqueness of solutions of differential equations.   

6.            Course Format:  This course is usually taught in a lecture or lecture/discussion format.  Although not required, computer labs, group discussion, and projects may be used to help illustrate ideas.

7.            Methods of Evaluating Student Performance:  Course grades are determined primarily by student performance on tests, quizzes, and projects, as well as possibly homework, labs, or class participation. The evaluation and weight of these various components are at the discretion of the individual instructor.

8.            Evaluation of the Course:  The instruction of the course is evaluated by departmental student evaluations and peer evaluation.  The course is reviewed and revised periodically by the Departmental Undergraduate Programs Committee.

9.      Addendum:  Textbook for the course:  Boyce and Diprima-Elementary Differential Equations and Boundary Value Problems (8th), Wiley.

Revised Fall 2007, M. Karls