Solving Laplace's equation within an infinite domain

Question

I try to solve Laplace's equation $\Delta u=0$ in the half plane: $${d^2u\over dx^2}+{d^2u\over dy^2}\:=\:0 \quad\text{where } -\infty<x<\infty \text{ and }y>0 $$ and the boundary condition $$\left.u\right|_{y=0}\:=\:\frac{x^2-1}{(x^2+1)^2} = \frac 1{x^2+1} - \frac{2}{(x^2+1)^2}$$ We can separate variables $u(x,y)=X(x)Y(y)$ and get ${X''\over X} = -{Y''\over Y} = k$, but what to do next?
These conditions will be enough or not, maybe I have to use polar coordinates and the Poisson integral?