How do I solve this Dirichlet problem?
$$\left\{ \begin{array}{l l} u_{xx} + u_{yy} = 1 & \quad \mbox{ on $ x^2 + y^2< a^2\ $,} \\ \quad u(x,y) = 0 & \quad \mbox{ on $ x^2 + y^2 = a^2\ $}. \end{array} \right. $$
would the solution be $u(r) = \frac{r^2-1}{4}$?
HINT:
Symmetry suggests that $u(\rho,\phi)=f(\rho)$ is independent of $\phi$. Hence, we have
$$\nabla^2 u(\rho,\phi)=\frac{1}{\rho}\frac{\partial }{\partial \rho}\left(\rho \frac{\partial f(\rho)}{\partial \rho}\right)=1$$
with $f(a)=0$.
Can you finish now?