_ ___ __ \\ Elonsider the follotivingheat equation u; = um, :13 6 [0,1], t E [0, +00). with boundary conditions u$(t, 0) = 0, 1.530, 1) = 0. 1. (7 points) Assume u(t,:z:) = x) - F(t), nd the boundary value problem that (:z:) should satisfy. 2. (7 points) Find the eigenvalues and their corresponding nontrivial solutions (:r), F02). 3. (6 points) Find the solution u(t, as) satisfying the initial condition u(0, x) = 1+cos(71ra:)