Problem 2: Consider the elastic string of unit length, which is fixed at x = 0 and free at x = 1. The vertical displacement of the string satisfies the wave equation Utt = CUxx, 00 where c > 0 is the wave speed. Suppose the string is deformed initially as u(x, 0) = x and is left on its own. Find the vertical displacement u as the series of eigenfunctions for t > 0 and obtain the equation of motion of the free end at x = 1