I am trying to find the solution of $ \nabla^2U = U_{rr}+\frac{1}{r}U_r+U_{zz}=0$, where $0\leq r <\infty$, and $z>0$. The boundary conditions are given by $U(r,0)=A$ (A is constant), for $0\leq r <a$, and $U(r,0)=0$ , for $r >a$.
Let $H(U(r,z)) = u(\alpha,z)$, $H(f(r)) = f(\alpha)$, then we have $$ f(\alpha) = -\alpha^2u+u_{zz}=0 $$ Am I on the right track? I'm not exactly sure what to do next. Thanks for the help!