I am currently working on solving $$ \begin{cases} \Delta u = 0,\ \ 1 < r < 2\\ u(1,\theta) = 0, \ \ u(2, \theta) = \sin(\theta) \end{cases} $$ After making the ansatz $u(r,\theta) = R(r)\Theta(\theta)$ I have separated the equation into $$ \Theta(\theta) = \mathcal{A}e^{im\theta},\ \ \forall\mathcal{A}\ \in\ \mathbb{C} $$ and solved the Cauchy-Euler equation to find $$ R(r) = Br^m + Cr^{-m} $$ On a circular disk I am used to discarding the constant $C$ as the solution must be bounded at $r = 0$, but in this case however, I am unsure as to how to use the boundary conditions to find my constants $\mathcal{A}, B$ and $C$. To find $\mathcal{A}$ I suppose the second boundary condition should suffice, but the radial equation has got me confused.
Any insights are much appreciated.