Consider c(x)(x) u t = x K0(x) u x + q(x)u + f(x, t) u(x, 0) = g(x), u(0, t) = (t) u(L

Chapter 8, Problem 8.4.4

(choose chapter or problem)

Consider c(x)(x) u t = x K0(x) u x + q(x)u + f(x, t) u(x, 0) = g(x), u(0, t) = (t) u(L, t) = (t). Assume that the eigenfunctions n(x) of the related homogeneous problem are known. (a) Solve without reducing to a problem with homogeneous boundary conditions. (b) Solve by first reducing to a problem with homogeneous boundary conditions.Assume that the solution u(x, t) has the appropriate smoothness, so that it may be represented by a Fourier cosine series, u(x, t) = n=0 cn(t) cos nx L . Solve for dcn/dt. Show that cn satisfies a first-order nonhomogeneous ordinary differential equation, but part of the nonhomogeneous term is not known. Make a brief philosophical conclusion.