Consider an elastic string of length l. the end x = 0 is held fixed, while the end x = l is free; thus the boundary conditions are u(0,t) = 0 and u_x(l, t) = 0. the string is set in motion with no initial velocity from the initial position u(x, 0) = f{x), where f(x) = {1, 0, l/2 - 1 < x l/2 + 1 (l > 2), otherwise. find the displacement u(x, t).