Let $0<a<b$ where a,b are constants.
I want to solve this bad boy:
$\nabla^2 u = 0 \enspace a<r<b, \enspace 0<\theta<\pi$
$u=g(\theta) \enspace on \enspace r=a$
$u=h(\theta) \enspace on \enspace r=b$
$u(r,0)=0 \enspace , \enspace u(r,\pi)=0$
How can I solve for $u(r, \theta) $
Ok so I got a solution of :
$u(r,\theta)=\frac{1}{2}(C_0 + D_0 \ln{r}) +\sum_{n=1}^\infty (C_nr^n+D_n r^{-n})\sin{n\theta}$
And some constants:
\begin{cases} \frac{1}{2}(C_0 + D_0 \ln{a})=\frac{2}{\pi}\int_0 ^\pi g(\theta) d\theta\\ \frac{1}{2}(C_0 + D_0 \ln{b})=\frac{2}{\pi}\int_0 ^\pi h(\theta) d\theta\\ \end{cases}
\begin{cases} C_na^n+D_n a^{-n}=\frac{2}{\pi}\int_0 ^\pi g(\theta)\sin{n\theta} d\theta\\ C_nb^n+D_n b^{-n}=\frac{2}{\pi}\int_0 ^\pi h(\theta) \sin{n\theta} d\theta\\ \end{cases}
Is this as far as I can go? Is this the final answer according to the world 'solve' ? THANKS!
Hint: Suppose $u(r, \theta)$ can be written in the form $$u(r, \theta) = \sum_{k = 1}^{\infty} v_k(r) \sin(k \theta),$$ substitute, and separate variables. (Thanks to Willie Wong for pointing out an oversight in the original solution.)