1(a) If ut ? kuxx = f, vt ? kvxx = g, f ? g, and u ? v at x = 0,x = l and t = 0, prove that u ? v for 0 ? x ? l, 0 ? t < ?.
(b) If vt ? vxx ? cos x for ??/2 ? x ? ?/2, 0 < t < ?, andif v(??/2, t) ? 0, v(?/2, t) ? 0 and v(x, 0) ? cos x, use part (a)to show that v(x, t) ? (1 ? e ?t ) cos x.
