Consider the following heat equation on a rod (which includes heat loss through the lateral side and fixed heat flow on the boundary):   ∂u ∂t    =    ∂2u ∂x2   − 9u, 0  <  x  <  1, t  >  0, subject to the boundary conditions ux(0, t)  =  3, ux(1, t)  =  18, t  >  0, and the initial condition u(x, 0)  =  0, 0  <  x  <  1. (a) Find the equilibrium solution uE(x) for this problem by solving a certain boundary value problem on (0, 1). (b) Use integration by parts and the given boundary conditions that uE(x) satisfies to compute A0  =  1 ∫ 0 u′′E (x) dx Enter A0. (Do not use the explicit formula for uE(x) obtained in part (a).) (c) Use integration by parts and the given boundary conditions that uE(x) satisfies to show that 1 ∫ 0 u′′E (x) cos(nπx) dx  =  An + Bn2 1 ∫ 0 uE(x) cos(nπx) dx, n  ≥  1, (1) for some constants B and An. (Do not use the explicit formula for uE(x) obtained in part (a).) Enter the values of B and An (in that order) into the answer box below, separated with a comma.