Consider the heat equation %= 16 :27: 00 and we solve the equation by assuming seperation of variables of the form u(m,t) = F(m) -G(t), forO S a: S L, t 2 0. It is easy to see that u(a2,t) = 0 for all m, t is a solution to this partial differential equation. We are interested in non-trivial solution, so we assume that G(t) is not constantly zero. If the boundary conditions u(0,t) = Oand u(L,t) = 0 for all t 2 0 are enforced, select all the restrictions that these place on following solutions. Case 1: If the general solution of F(a:) is given by F(:1:) = Aep'\" + Brapm, assuming 11 a 0, then boundary conditions imply: . A is - B is . p IS In this case, the solution of the heat equation is Case 2: If the general solution of F(a:) is given by F (at) = A2: + B, then boundary conditions imply: . A is - B is In this case, the solution of the heat equation is Case 3: If the general solution of F(a:) is given by F(a:) = Acos(pm) + Bsin(pm), then boundary conditions imply: . A is - B is . I) Is In this case, the solution of the heat equation is