Math 3226
Test 3
Fall 1996


No books, notes etc. are permitted.
Show all your work! Box in your answers!
The test has 6 problems.

Read the problems very carefully.

Problem 1 (15 points) (A) What can you say about the eigenvalues associated with the linear system depicted below?
(B) Find a set of eigenvectors for the system!

Problem 2 (20 points) (A) Find the motion y(t) of a spring-mass system with mass m=1, spring constant tex2html_wrap_inline80 and damping coefficient tex2html_wrap_inline82 , satisfying the initial conditions

displaymath74

(B) Is the spring underdamped, critically damped, or overdamped?

Problem 3 (15 points) (A) Find the general solution of the following system of linear differential equations:

eqnarray40

(B) What can you say about the equilibrium point (0,0)?

Problem 4 (15 points) (A) Find the general solution of the following system of linear differential equations:

eqnarray46

(B) What can you say about the equilibrium point (0,0)?

Problem 5 (15 points) (A) For which value of the spring constant tex2html_wrap_inline88 is the harmonic oscillator tex2html_wrap_inline90 critically damped?

(B) Describe the long-term behavior of the solutions to the critically damped harmonic oscillator!
(C) How does the behavior of the solutions to the harmonic oscillator change as the oscillator changes from being underdamped to being overdamped?

Problem 6 (20 points) Find all equilibrium points of the following system, and then--using the technique of linearization--classify each of them according to its type (if possible).

eqnarray59


Helmut Knaust
Fri Dec 6 11:26:40 MST 1996