PROBLEM SET #3. AVAILABLE 2/13; DUE 2/27 at 4:30 PM in Cu I Room 100.

Be sure to show your work.

1. A mass weighing 128 pounds stretches a certain spring 2 feet. Suppose the mass is suspended from this spring at equilibrium. It is then
pushed upward 6 inches and released from rest. There is no friction. Find the function u(t) = distance to equilibirum position at time t.
Also, find the period of the motion. (Use the fps system, and let the downward orientation be positive.)

2. Now suppose everything is the same as in problem 1, except that starting at time t = 0, an external force of 8sin(4t) pounds is applied.
Now find u(t). What kind of motion is this?

3. Now let the motion of problem 1. (NOT 2.) take place in a viscous medium which imparts a frictional drag of 4|v| pounds on the mass.
Find u(t). Find the pseudo - period, and compare to the period in problem 1.

4. Finally, suppose that the motion of problem 2. occurs in the viscous medium of problem 3. The steady state motion is simple harmonic.
Find its amplitude and period.

5. The function y(x) = cosh(x) is a solution of the IVP y'' - y = 0, y(0) = 1, y'(0) = 0. Use this fact to find a power series expansion of cosh(x).

6. *) (x^3 - 1)y'' + xy' + y = 0 has an ordinary point at x = 0. If we find a series solution of *), what is a guaranteed minimum for its radius of convergence.
(For those inclined to wit : "0" is not a satisfactory answer.)

7. Find the first three nonzero terms of the series solution of the IVP *) y'' - xy' + 2y = 0, y(0) = 1, y'(0) = 0.

8. a) Solve the IVP (1 - x^2)y'' + xy' = 0, y(0) = 1, y'(0) = 0 by power series methods.
b) Now solve it by reduction of order, letting u = y' and solving the resulting first order linear equation.

9. Give the general solution of x^2y'' - 6xy' = 0, x > 0.

10. Find the solution of 4x^2y'' + 4xy' - y = 0, x > 0 , y(1) = 0, y'(1) = 1.