(e) Show that v(x, t) = u(x, t) — uĘ(x, t) is solution of a homogeneous problem (with a different initial condition) and solve it using the method of separation of variables and the superposition principle. The solution has the form 00 v(x,t) = Go(t) + Σ Gn(t) cos(nлx) n=1 Enter the functions Go (t) and Gn(t) (in that order) into the answer box below, separated with a comma. (f) The solution u(x, t) of the original problem, written as a cosine Fourier series with coefficients that are functions of t, has the form 80 u(x,t) = H₁(t) + Σ H₂(t) cos(nлx) n=1 Enter the functions Ho(t) and Hn(t) (in that order) into the answer box below, separated with a comma. : Consider the following heat equation on a rod (which includes heat loss through the lateral side and fixed heat flow on the boundary): ди at 22 и = 2х2 - 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 uĘ(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 uĘ(x) satisfies to compute A₁ = √√√u (x ) d x Ao Enter Ao. (Do not use the explicit formula for uĚ (x) obtained in part (a).) (c) Use integration by parts and the given boundary conditions that uĘ (x) satisfies to show that Lug (x) cos(nлx) dx ) cos(nлx) dx = A + Bn² uд(x) cos(nлx) 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. (d) Use the differential equation that uд(x) satisfies together with the results obtained in (b) and (c) to deduce the Fourier cosine series representation for uд(x) on (0, 1) which is of the form 2 + 80 Σan cos(ηπx). n =1 Enter the values of ao and an (in that order) into the answer box below, separated with a comma. - (e) Show that v(x,t) = u(x,t) – u£ (x, t) is solution of a homogeneous problem (with a different initial condition) and solve it using the method of separation of variables and the superposition principle. The solution has the form