2. First show that for the completely insulated bar ut = c2uxx; 0 < x < L; t > 0: ux(0; t) = 0; t > 0 ux(L; t) = 0; t > 0 u(x; 0) = f(x); 0 < x < L the separation of variables gives the solution u(x; t) = A0 + 1 Ancos n x e ( cn )2t L n=1 L X where 2 Z0 L n x An = f(x)cos dx; n = 1; 2; 3:::; L L then nd the temperature in the bar with L = ; c = 1; and f(x) = x