# Math 230 - Chapters 5-6 Study Guide

1. A differential equation is given that corresponds to a spring/mass system. Identify whether the system described is 1) free or driven and 2) undamped or damped. Three parts.
2. A differential equation is given. Determine whether the system is overdamped, underdamped, or critically damped. Three parts.
3. A spring/mass system is described. Find the equation of motion.
4. Find the eigenvalues and eigenfunctions for the given boundary-value problem. Two parts. Look at problems 5.2.9-18.
5. Without actually solving the given differential equation. Find a lower bound for the radius of convergence of a power series solution about the ordinary point x=0. Two parts. Look at problems 6.1.15-16.
6. Rewrite the given expression as a single power series whose general term involves xk. Look at problems 6.1.11-12
7. Determine the radius and interval of convergence for the power series. Look at problems 6.1.1-4.
8. Evaluate a gamma function. See appendix 1.
9. Find the general solution to a Bessel Equation. Look at problems 6.3.1-10.
10. Find two power series solutions to the given differential equation about the ordinary point x=0. Look at problems 6.1.17-28.
11. Determine the singular points of the differential equation and classify each point as regular or irregular. Two parts. Look at problems 6.2.1-10.
12. x=0 is a regular singular point of a differential equation. Identify p(x) = xP(x) and q(x) = x2Q(x). Write the indicial equation and find the indicial roots. Use the shortcut r(r-1)+a0r+b0=0 for the indicial equation. Look at problems 6.2.11-14.
13. x=0 is a regular singular point of a differential equation. The substitution necessary to solve by the Method of Frobenius is made and the resulting simplified equation is given. Take the simplified equation and find the indicial roots, find a recurrence equation for the remaining constants, find at least the first three non-zero constants for each indicial root, and write the general solution to the differential equation. Look at problems 6.2.15-24.

## Take Home Test

Because of the length of many of these problems, there is a take home exam. Work the following problems from your textbook. Each problem is worth 5 points and the take home test is due the day of the in-class test.

1. 5.1.10
2. 5.2.4
3. 6.1.20
4. 6.1.28
5. 6.2.30

## Notes

You may have a note card with certain formulas on it. These cards may not have worked out examples. Here are the formulas you may have:

• You may not have the formula for a spring/mass system on a note card, I want you to know this.
• From section 6.2, you may have formula 14 along with the following definition of a0 and b0: "where a0 and b0 are the constants in the power series expansion of p(x) and q(x)."
• From section 6.3, you may have formulas 1, 9, 11, 12, and 13.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 Total 6 3 9 6 6 4 4 3 4 9 6 6 9 75