Chapter 10.2, Problem 31E

For the PDE in Problem 27, assume that the following boundary conditions are imposed: $u\left(r,0\right)=u\left(r,\pi \right)=0$

$u\left(r,\theta \right)$ remains bounded as $r\to 0^{+}$.

Show that a nontrivial solution of the form $u\left(r,\theta \right)=R\left(r\right)\theta \left(\theta \right)$ must satisfy the boundary conditions

$\theta \left(0\right)=\theta \left(\pi \right)=0,$

$R\left(r\right)$ remains bounded as $r\to 0^{+}$.

$\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\theta }^2}=0$

With $u\left(r,\theta \right)=R\left(r\right)\theta \left(\theta \right)$ yields

$\begin{array}{l}{r}^2{R}^{″}\left(r\right)+r{R}^{\prime }\left(r\right)-\lambda R\left(r\right)=0,\\ {\theta }^{″}\left(\theta \right)+\lambda \theta \left(\theta \right)=0,\end{array}$

Where $\lambda$ is a constant.