Using the idea of separating the variables, you should be able to see from the PDE that
A(r,θ) = Cn(r) cos(nθ) + Sn(r) sin(nθ)
is a solution for the right hand side terms (1/r)(Xn cos(nθ) + Yn sin(nθ))
That will give you ordinary differential equations to solve for Cn(r) and Sn(r).
