Consider Laplace equation $\Delta u=0$ in the cylinder$\{r\le a,\ 0<z<b,\ 0\le \theta \le 2\pi\}$. Separate variables $u=R(r)Z(z)\Theta(\theta)$.
1. Write down ODE which should satisfy $\Theta$ and solve it (using periodicity).
2. Write down ODE which should satisfy $Z$ and solve it using $Z(0)=Z(b)=0$.
3. Write down ODE which should satisfy $R$.
Ans:
$$\eqalign{
& \Delta u = {u_{rr}} + {1 \over r}{u_r} + {1 \over {{r^2}}}{u_{\theta \theta }} + {u_{zz}} = 0 \cr
& \Rightarrow {{{r^2}R'' + rR'} \over R} + {{\Theta ''} \over \Theta } + {{{r^2}Z''} \over Z} = 0 \cr} $$
(1) Let:
$$\left. \matrix{
{{\Theta ''} \over \Theta } = - {m^2} < 0 \hfill \cr
\Theta (0) = \Theta (2\pi ),\Theta '(0) = \Theta '(2\pi ) \hfill \cr} \right\}\matrix{
{{\Theta _{1,m}} = \cos (m\theta )} \cr
{{\Theta _{2,m}} = \sin (m\theta )} \cr
} $$
(2) Let:
$$\left. \matrix{
{{{r^2}Z''} \over Z} = - l \Rightarrow Z'' + {l \over {{r^2}}}Z = 0 \hfill \cr
Z(0) = Z(b) = 0 \hfill \cr} \right\}\matrix{
{{l \over {{r^2}}} = {{{\pi ^2}{n^2}} \over {{b^2}}}} & { \Rightarrow l = {{{r^2}{\pi ^2}{n^2}} \over {{b^2}}}} \cr
{z = \sin ({{\pi nz} \over b})} & {n = 1,2,...} \cr
} $$
(3) from (1) and (2) we have:
$$\matrix{
{{r^2}R'' + rR' - \left( {{m^2} + {{{r^2}{\pi ^2}{n^2}} \over {{b^2}}}} \right)R = 0,} & {n = 1,2,...} \cr
} $$
