1. (a) (3 points) For which values of k > 0 is the spring-mass system y\"+y'+ky=0, (i) underdamped; (ii) critically damped; (iii) overdamped? For which values of k > 0 can a solution y % 0 of y\" + y' + kg = 0 vanish at rst at t = 1 and next at t = 3, but at no time in between? (b) (7 points) 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) (4 points) Determine a recurrence relation for the coefcients in the power series about x0 = 0 for the general solution of y"+a;y'+2y=0. (b) (3 points) Solve the initial value problem for this differential equation with y(0) = 0, y'(0) = 1- (c) (3 points) Use the ratio test to show that the radius of convergence of the solution from part (b) is innite. Then show that the particular series solution from part (b) converges . . 2 to (1.e., can be written as) y = ire\"7" /2. 3. Consider the differential equation 0 = (1 1132):)\" 2323/ + 203/ with solution of the form 00 y = Z anac\". n=0 2 (a) (6 points) Show that the recurrence relation is an+2 = wan, n 2 0. (b) (4 points) Find the solution obeying y(0) = 1, y'(0) = 0. [Hintz The solution is a polyno- mial.]