[13] Consider Ut - Uzz = 0, 0 0, u,(0, t) = 0, u(1, t) = 0, t> 0, u(x,0) = f(x), 0≤x≤ 1. (a) Find all product solutions u(x, t) = T(t)X(r) satisyfing the PDE and boundary conditions. (b) Find the general solution formula for u(x, t). (c) Solve the problem when f(x) = {1 0 ≤ x ≤ 1/2, 1/2 < x < 1.

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