Find Uss (x). (b) The transient temperature utr(x, t) is defined to be Utr (x, 1) = u(x, 1) - Uss(x). Show that utr satisfies the boundary value problem dutr azutr = k at ax2 utr (0, t) = utr (L, t) = 0, Utr (x, 0) = g(x) = f(x) - Uss(x). (c) Conclude from the formulas in (30) and (31) that u(x, t) = Uss(x) + utr (x, 1) = Uss(x) + > on exp (_n2x2kt/12 ) sin nux L n=1 where nux Cn = [f(x) - Uss(x)] sin dx. L L17. (Steady-state and transient temperatures) Let a laterally insulated rod with initial temperature u(x, 0) = f(x) have fixed endpoint temperatures u (0, t) = A and u(L, t) = B. (a) It is observed empirically that as t - too, u(x, t) approaches a steady-state temperature uss (x) that corre- sponds to setting ut = 0 in the boundary value problem. Thus uss (x) is the solution of the endpoint value problem a2 Uss Uss (L) = B. ax2 = 0; Uss (0) = A