Problem 3 (15 points) a) (5 points) Find all the standing wave solutions to the wave equation -Z²uxx Utt = 0, x = R, t > 0, that are periodic in

x (for fixed t) with the period 1 > 0. b) (10 points) Find all the standing wave solutions for a finite string that is free at both ends: Utt − c² Uxx = 0, x ≤ (0, π), t > 0, ux (0, t) = 0, t≥ 0, ux (π, t) = 0, t≥ 0.