solving second order CC non-homogenuous pde

iqjump123

1. The problem statement, all variables and given/known data
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


2. Relevant equations
Fourier transform
greens function?


3. 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.