PROBLEM SET #2. AVAILABLE 1/30; DUE 2/13 at 4:30 PM in Cu I Room 100.

Be sure to show your work.

1. All of the functions y(x) = exp(x), exp(-x), sinh(x) and cosh(x) are solutions of y'' - y = 0. There are 6 ways to pick a pair of
distinct solutions from this set of 4 functions. For each possible pair, determine whether it is a fundamental system of solutions.

2. y1(t) = t and y2(t) = t^2 -1 are both solutions of (t^2 + 1)y'' -2ty' + 2y = 0. Use Abel's theorem (3.3.2 on page 155) to find the wronskian W(y1, y2).
Then compute it directly.

3. Solve the IVP 100y'' - y = 0, y(0) = 0, y'(0) = 1. Express the answer in terms of sinh or cosh functions.

4. Solve the IVP y'' - 10y' + 25y = 0, y(0) = 1, y'(0) = 0. How does y(t) behaves as t goes to infinity?

5. Now answer problem 4. if the initial conditions are y(0) = y'(0) = 0.

6. y1(t) = t^2 is one solution of t^2y'' - 3ty' + 4y = 0, t > 0. Use the method of reduction of order to find a second solution y2(t).
Verify that the two solutions are linearly independent.

7. Use the method of undetermined coefficients to find the general solution of y'' - 2y' + y = 3exp(x).

8. Now do problem 7. using variation of parameters.

9. Find a particular solution of y'' - (2/t)y' + (2/t^2)y = 1. Note that y1(t) = t and y2(t) = t^2 are solutions of the corresponding homogeneous equation.

10. Consider the equation *) y'' + y = exp(-x^2)sec(x). Solutions of the corresponding homogeneous equation are of course cos(x) and sin(x).
We want a particular solution yp of *).
a) Try to use undetermined coefficients to find yp.
b) Try to do it using variation of parameters.
c) Both methods fail. Explain why.