d^2 f/dx^2 - d^f/dy^2 = x

This is a non homogeneous PDE. I first tried to find the solution to the homogeneous PDE associated with it.

d^2 f/dx^2 - d^f/dy^2 = 0

I equated d^2 f/dx^2 = d^f/dy^2 = k (a constant)

Now integrated both the sides to get f = k [(x^2)/2 + (y^2)/2]

Is this correct?