In your answers below, for the variable i type the word lambda, for y type the word gamma; otherwise treat these as you would any other variable. We will solve the heat equation u, = 4uxx: 0

Find Eigenfunctions for X(x). The problem splits into cases based on the sign of i. (Notation: For the cases below, use constants a and b) • Case 1: 1 = 0 X(x) = Plugging the boundary values into this formula gives 0 = X(0) = 0 = X(2) = So X(x) = which means u(x,1) = We can ignore this case. Case 2: 1 = -y2 (In your answers below use gamma instead of lambda) X(x) = Plugging the boundary values into this formula gives 0 = X(0) = 0 = X(2) = So X(x) = which means u(x, 1) = We can ingore this case. Case 3: 1 = y2 (In your answers below use gamma instead of lambda) X(x) = Plugging in the boundary values into this formula gives 0 = X(0) = 0 = X(2) = Which leads us to the eigenvalues yn = and eigenfunctions X,(x) = (Notation: Eigenfunctions should not include any constants a or b.) Solve for T()

Solve for T(t). Plug the eigenvalues in = rî from Case 3 into the differential equation for T(t) and solve: Ty(t) = (Notation: use c for the unknown constant.) Combining all of the X, and Tn we get that u(x, t) = Ž B, where Bn are unknown constants. n=1 Fourier Coefficients. We compute B, by plugging t = 0 into the formula for u(x, t) and setting equal to the initial heat distribution given in the problem. So, 0 < x < 1 u(x, 0) = 2 B, 3, 1