Let $r,\theta$ be the usual polar coordinates in $\mathbb{R^2}$, let $\Omega$ be the unit disc $r<1$ and recall that the Laplacian is given by $$\nabla^2u=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2}.$$ Use separation of variables to solve, \begin{align} -\nabla^2u&=\lambda u \ \ \ \ \text{in}\ \ \Omega \\ u&=0\ \ \ \ \ \ \ \text{in}\ \ \partial\Omega. \end{align}
We seek $u=X(x)Y(y)$ for \begin{align} -\nabla^2u-\lambda u&=0 \\ -\left(\partial_{xx}u+\partial_{yy}u\right)-\lambda u&=0 \\ -\frac{X''}{X}-\lambda&=\frac{Y''}{Y}=\mu \ \ \ \ \text{($\mu$ is a separation constent)} \end{align} We then consider three cases ($\mu=0, \ \mu<0, \ \mu>0$) for each equation. But what are the boundary conditions for $X$ and $Y$ if $u=0$?