Problem 8. Consider the following Laplace-like problem in a semi-infinite strip: Uxr + 2uyy + 47u = 0 a > 0, y E (0, 1) (P) u(x,0) = 0, u(r, 1) = 0, x >0 A condition at a o is also imposed: lim u(r, y) = 0 for each y. (L) X=0 - You are given the eigenfunctions/values (for -o" = Ao with appropriate BCs) Pn(y) sin nAy, An = n?n, n > 1. %3D along with the integral sin (naa) de = 1 for n >1 a) Find the "general" solution u(x, y) that satisfies (P) and the limit condition (L). Consider: for an ODE like c() AnCn(x) = 0, it may be useful to define kin = VAn. = 1 in the eigenfunction basis and write out the first b) Calculate the expansion for f(x) two non-zero terms explicitly. c) Now suppose that, in addition to (P) and (L), u(0, y) = 1+ A sin ry, y E (0, 1) where A is a constant. For what values of A, if any, does a solution exist? Explain.