# An easy way to solve this Poisson equation?

Problem : $$-\Delta u=1-\sqrt{x^2-y^2}$$ in $$B_1(0)$$; $$u=0$$ on $$\partial B_1(0)$$

I could use the Poisson integral formula : $$u(x)=\int\limits_{B_1(0)}G(x,y)\Delta u(y)dy$$ where $$G$$ is the Green's function for the unit ball. But the integral looks daunting.

On the other hand I could write $$u(x,y)=X(x)Y(y)$$ and then solve the eigenvalue problem $$u_{xx}+u_{yy}=-\lambda u$$ which leads me to $$\frac{X''}{X}+\frac{Y''}{Y}=-\lambda$$ or $$X''+\mu X=0, Y''+(\lambda-\mu)Y=0$$ If it were a rectangular region, I would have had nice boundary conditions. I also thought about using polar coordinates but didn't go through with it.