# MA2051--C97: TEST II

**1.** (30 points)
Work with the following differential equation:

- (a)
- Find two linearly independent solutions for the
corresponding homogeneous equation.
Prove that your solutions really are linearly independent.
- (b)
- Find a
*particular solution* for the differential equation.
- (c)
- Find the solution that satisfies the initial conditions
*y*(0) = 1/2, .
- (d)
- Describe the behavior of the general solution for large values of
*t*.
What part of the solution is transient? Is there a limiting periodic
solution?

2. (30 points)
A 6 kg mass is suspended from a spring that extended 2 cm
when the weight was attached.
Assume that friction damping can be ignored.
You pull the mass *down* 3 cm (more) and release it from rest.
- (a)
- Write down the mathematical model for this
system. Use the given data to
specify the numerical values for the parameters in the model.
(Be careful with units.)
- (b)
- Solve the initial value problem derived in part (a).
- (c)
- What is the amplitude of the motion?
How long does it take for the mass to travel from
its highest position to its lowest position?
- (d)
- What is the velocity of the mass when it passes
through the equilibrium position (
*x*=0)?

**3.** (30 points)
Consider the following initial value problem

- (a)
- Give the characteristic equation and find its roots. Use the roots to
find two linearly independent solution pairs and use them to give the
general solution for the system.
- (b)
- Find the solution satisfying the given initial conditions.
- (c)
- Is it possible to find initial conditions (different than the
given initial conditions) so that the solution
curve converges to the origin (
*y*,*z*) = (0,0) as *t* goes to infinity?

**4.** (10 points)
Choose *p* so that the following spring-mass model is critically
damped:
Show that the solution can cross through the equilibrium position
(*x* = 0 ) *at most* once. Determine conditions
on the initial position and velocity that allow the mass to pass
through equilibrium.

Arthur Heinricher <heinrich@wpi.edu>
Last modified: Wed Oct 8 08:23:52 EDT 1997