The equation is Uxx + Uyy = 0 And domain of solution is 0 < x < a, 0 < y < b Boundary conditions: Ux(0,y) = Ux(a,y) = 0 U(x,0) = 1 U(x,b) = 2 What I've done is that I did separation of variables: U(x,y)=X(x)Y(y) Plugging into the equation gives: X''Y + XY'' = 0 Rearranging: X''/X = -Y''/Y = k For case k > 0, I saw that it gives no non-trivial solutions. For case k = 0, I solved it and found U(x,y) = y/b + 1 For case k < 0, I'm slightly lost. X'' + kX = 0 Y'' - kX = 0 Using the X boundary conditions: Using the Y boundary condition: Using Fourier Series to find the coefficient: But the integral just gives Dn = 0, and this doesn't satisfy u(x,0) = 1. Can someone explain where I went wrong? Thanks!