Hi, I need help with parts b and c.

*Handwritten solutions only.

4. i) The temperature in a bar of length L is described by the heat equation du 20zu = at for 0 0. (1) The ends of the bar are insulated, which are boundary conditions with no heat flux, thus au =0 at x = 0 and at x = L for t > 0. (2) a) Using the method of separation of variables, let u(x, t) = X(x) T(t) and derive the following ODEs X" = KX, T' = ka'T. b) Consider all possibilities of the constant of separation