Math 111.01/02 (Applied Mathematical Analysis I)

Fall 1998

Plan for Week 6

Our objective this week is to determine how to solve differential equations of the form

y'' + p(t) y' + q(t) y = g(t),

that is, second-order, linear, nonhomogeneous equations. We may think of the left-hand side of this equation as a description of a physical system, such as a spring-mass-dashpot system or an electrical circuit. The right-hand side represents a "driving" or "forcing" component, such as an external push on the moving mass or an AC generator in the electrical circuit.

We first see that we can treat this problem in two parts, first finding the general solution

yh = c1y1 + c2y2

of the homogeneous equation

y'' + p(t) y' + q(t) y = 0,

then finding one particular solution yp of the nonhomogeneous equation. The general solution of the nonhomogeneous equation turns out to be

y = c1y1 + c2y2 + yp.

If initial values y(0) and y'(0) are specified, we can use that information to set up two linear (algebraic) equations that determine c1 and c2. We will see that those equations always have a unique solution.

As we know already, the first part of this two-step strategy is completely solved if the coefficient functions p(t) and q(t) happen to be constants -- an important special case that includes damped spring-mass systems and RLC circuits. We will find that what we learned about characteristic roots also helps us find yp in many important cases. In particular, if g(t) is itself the solution of a linear equation with constant coefficients, we can guess the form of yp and then calculate its "undetermined coefficients."

Next we consider what to do when we can't guess the form of the solution. We will see that -- in theory -- there is a technique that assures a solution whenever g(t) is continuous. However, like the other "general" techniques we have seen for finding symbolic solutions, the practicality of this one depends on whether we can evaluate certain integrals in closed form.

Our Week 5 lab was an abstract look at the variety of functions that can be solutions of second-order linear homogeneous differential equations with constant coefficients. This week's lab explores the same family of equations as models for a spring-mass system, starting with actual data from such a system, both without and with damping.

Here is the syllabus for Week 6:

Week 6 Date Topic Reading Activity
M 10/5 Undetermined coefficients 3.6
W 10/7 Variation of parameters 3.7
F 10/9 Spring motion 3.8 Lab: Spring Motion

Notes

  1. Your instructor will be out of town on Friday this week. A substitute will supervise the lab.
  2. Because of Fall Break next week, submission dates for this week's homework and lab are moved back two days. You may want to consider having homework finished before you leave for Fall Break. However, you will have time to finish the lab after Break, if necessary.
  3. Your next homework papers will be turned in on Wednesday, October 14. Those papers should include solutions to all problems in the assignment below. The assignment dates are start dates.
  4. In general, no homework problem will be given full credit unless you have written an explanation of why you know it is correct. For example, an acceptable explanation for a solution of a differential equation is that you have substituted the proposed solution into the original equation and found that it satisfies the equation -- but you have to show your work. Two examples of unacceptable explanations: (a) It matches the answer in the back of the book. (b) I did the work again and it came out the same.
  5. Submit your Week 6 lab report (the Maple file) via e-mail by the end of the day Friday, October 16.
  6. Remember to submit your e-mail journal entry on Friday, October 9.

Assignments

David A. Smith <das@math.duke.edu>

Last modified: July 14, 1998