Consider the boundary value problem given by azu at2 axz . U(t, 0) = U(t, L) = 0,t > 0, x E [0, L], where U(t,x) is the vertical displacement of a string at position x and time t. (i) Assume the partial differential equation has a solution of the form U(t, x) = T(t)X(x). Use separation of variables to determine a pair of differential equations that correspond to the partial differential equation. (ii) By considering the boundary conditions given, show that An = X(x) = Bn sin (x). Br ER, NEN, where 1, and X(x) are eigenvalues and eigenfunctions of a Sturm-Liouville problem in part (i) . (iii) Solve the second Sturm-Liouville problem in part (i) . Hence, show that U(t, x) = [ {[Cm. cos (Smut) + Dn sin (Sent ) ] Br. sin ("In ) n=1 where ByC ER