1. The problem statement, all variables and given/known data Use separation of variables to find the solution to Laplaces equation satisfying the boundary conditions u(x,0)=0 (0<x<2) u(x,1)=0 (0<x<2) u(0,y)=0 (0<y<1) u(2,y)= asin2πy(0<y<1) 3. The attempt at a solution I am able to perform the separation of variables technique on the wave equation. The heat equation is a little harder, I struggle a bit, but eventually I get there. Laplace's equation is pretty much impossible. From my understanding, the method is very similar in all three cases, but I think there are some differences which I don't see, which is why I can't do this question. So I managed to separate the variables, deriving two ODE's, one in terms of x and one in terms of y, with the separation constant λ. F''(x) - λF(x) = 0 G''(x) - λG(x) = 0 For the case where λ=0, there are no solutions because nothing can satisfy the last boundary condition listed. For the case where λ<0, I think there are no solutions, I could sort of tell by having a look at the final answer given. I don't understand how to show this? For the case where λ>0 I get F(x) = A*cosh(sigma*x) + B*sinh(sigma*y) G(y) = (Ccos(sigma*y) + Dsin(sigma*y)) where sigma is the roots of the ODE's. so now u(x,t) = F(x)*G(y) I have this function with 5 unknowns, A,B,C,D, and sigma When I apply all the boundary conditions, I don't really get anywhere. No helpful information appears. What am I doing wrong? Or, what am I not doing?