Let u be the solution to the initial boundary value problem for the Heat Equation, with Dirichlet boundary conditions u(t,0) = 0 and u(t, 1) = 0, and with initial condition d,u(t, x) = 3 du(t, x), The solution u of the problem above, with the conventions given in class, has the form Cn = 2/(npi)^2sin(npi/2)-1/(npi) cos(npi/2) u(0, x) = f(x) = . u(t, x) = n=1 t (0,00), X. 0, XE = [0,1), with the normalization conditions v (0) = 1 and w() = 1. Find the functions Un, W, and the constants C Un(t) = e^(-4x^2pi^2t) w(x) = sin(npix) x (0, 1); * [1]. XE Cn Un(t) wn(x),