Chapter 10.2, Problem 30E

In Problems 27-30, a partial differential equation (PDE) is given along with the form of solution having separated variables. Show that such a solution must satisfy the indicated set of ordinary differential equation.

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

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

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

Where $\lambda$, $\mu$ are constants.