Solving second order CC non-homogenuous pde

[iqjump123]

Homework Statement

Uxx+Uyy-c^2*u=0
for -inf<x<inf
y>0, subject to boundary conditions
Uy(x,0)=f(x), u(x,y) bounded as x-> +/- inf or y -> inf

Homework Equations

Fourier transform
greens function?

The Attempt at a Solution

I would think that I would have to go through two Fourier transforms to get this to be solved, or use greens function. However, I remember going through in class where since the PDE has constant coefficients, I can go ahead and solve assuming U=e^(rx+sy).
Is this the right path to go? I tried using the CC method using U=e^(rx+sy), but it isn't giving me a straightforward answer.

 

[Adrmay]

It's giving me U=e^(rx+sy)*(Acos(cy)+Bsin(cy)) which doesn't make sense to me. Also, for the boundary conditions, I know that Uy(x,0)=f(x), but what would happen with u(x,y) as x-->+/-inf or y->inf? I don't think I can just set it equal to 0, because that would mean that u(x,y) would have to be 0 everywhere, which isn't true.