5. Consider a metal rod with a temperature distribution (x, t) as a function of distance x along the rod and time t, which obeys the heat equation ae Ət = α a²A მე2 ს where a > 0 is a thermal diffusivity constant. The rod has a length l (so that x = [0, l]). Suppose that at the end x = 0 it is insulated so that no heat escapes, and at the end xl it is held at the temperature 0. = (a) Express the boundary conditions as equations. (b) Using the method of separation of variables, show that solutions of the differential equation, subject to the boundary conditions from part (a), can be written as (2k + 1)²²α 4l2 [ }] (x,t) = Σ bk cos (2k+1)πx COS exp 2l k=0 (c) Suppose that the metal rod has the initial condition (x, 0) = 0, independent of x. Find the coefficients {b} in this case. l Hint: COS [cos (7(2n + 1) z) cos COS π(2m+1) 2l l X бпт, dx = 28nm, n, mЄ N. 2