1. (a) For which values of k > 0 is the spring-mass system y" ty' t ky =0, (i) underdamped; (ii) critically damped; (iii) overdamped? For which values of k > 0 can a solution y * 0 of y" + y' + ky =0 vanish at first at t = 1 and next at t = 3, but at no time in between? (b) The current I = I(t) in a certain LRC circuit obeys 41" + 121' + 251 = 10 cos(5t/2) + 10 sin(5t/2), I(0) = I'(0) = 0. Determine I and identify its transient and steady state parts. 2. (a) Determine a recurrence relation for the coefficients in the power series about x0 = 0 for the general solution of y" try + 2y =0. (b) Solve the initial value problem for this differential equation with y(0) = 0, y'(0) = 1.(c) Use the ratio test to show that the radius of convergence of the solution from part (b) is infinite. Then show that the particular series solution from part (b) converges to (i.e., can be written as) y = xe-22/2 3. Consider the differential equation 0 = (1-x2) y" - 2xy' + 20y with solution of the form y = _ anan n=0 (a) Show that the recurrence relation is an+2 = (+2) (nthan, n 2 0. Find the solution obeying y(0) = 1, y'(0) = 0. [Hint: The solution is a polyno- mial.]