Problem 4: [10p] Consider the motion of a semi-infinite string with a constant external force f(t) = 1 acting on it. One end is fixed while the other end moves freely in the vertical direction. If the string is initially undeformed and at rest, the deformation of the string is governed by: Ju c at + 1, 0 < x < o, t> 0 x2 (1) u(x, 0) = 0, (x, 0) = 0 at (2) u(0,t) = 0, (x, t) 0, x (3) where u(x,t) is the displacement of the string. What law of physics is modeled by equation (1)? 4.2. Use the method of the Laplace transform to solve problem (1)-(3).