**Math 3226Test 3Fall 1996**

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

**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 and damping coefficient , satisfying the initial conditions

**(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:

**(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:

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

**Problem 5 (15 points) ** **(A) ** For which value of the spring constant is the harmonic oscillator
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).

Fri Dec 6 11:26:40 MST 1996