MT 284 - Introduction to Differential Equations

• demonstrate an ability to classify differential equations into ordinary and partial, linear and nonlinear, and to tell the order of a given equation;
• solve and, where applicable, to graph the solutions of first order equations including linear and separable equations. Also, to apply this skill to problems in population dynamics, compound interest, and some problems in mechanics;
• solve second and possibly higher order homogeneous equations with constant coefficients by constructing fundamental sets of solutions and to use the methods of undetermined coefficients and possibly variation of parameters to solve corresponding non-homogeneous equations. The ability to solve such differential equations must extend to the case where the characteristic equation has real or complex roots;
• create and where appropriate, solve equations that model physical problems in vibratory motion and some LRC electrical circuits, and to graph and interpret solutions obtained therefrom.
• solve certain second order linear equations with variable coefficients by the use of infinite series. This skill is to extend to solutions near ordinary points, regular singular points, and to Euler equations;
• solve second order initial value problems using Laplace transform methods and to apply these methods to problems involving step functions, discontinuous forcing functions, and impulse functions;
• solve systems of first order linear equations with the aid of matrix methods, beginning with homogeneous systems with constant coefficients and extending to non-homogeneous systems. The student is expected to demonstrate skill in finding eigenvalues and eigenvectors and in constructing graphs and interpreting solutions of linear systems; and
• use the Euler method for finding numerical solutions of first order initial value problems.

Laboratory Objectives: At the end of the course the student should be able to use the computer algebra system “Maple” for performing various mathematical procedures. These procedures include, but are not limited to the following:

• introduction to the calculus and graphing capabilities of the math software package Maple;
• plot direction fields and solve first order differential equations;
• solve and graph solutions of second order ordinary differential equations;
• solve and graph solutions to a spring-mass system;
• solve and plot solutions to systems of damped/undamped electrical vibrations;
• plot direction fields and solve systems of differential equations; and
• find approximate solutions to differential equations using numerical methods (e.g., Euler)