Edit: Note, I accidentally wrote $\phi$ instead of $\theta$ throughout this problem. I do not wish to go back and change them all. Please consider $\phi$ as if I had written $\theta$!
As usual, let's separate variables. We let:
\begin{equation}
u(x,t) = \Phi(\phi)R(r)
\end{equation}
Plugging this into the given equation yields:
\begin{equation}
\Phi R'' + \frac{1}{r} \Phi R' + \frac{1}{r^2} \Phi'' R = 0 \longrightarrow \left( \frac{r^2R''}{R} + \frac{rR'}{R} \right) + \frac{\Phi''}{\Phi} = 0
\end{equation}
Now as usual, we have a term that depends on $R$ and a term that depends on $\Phi$. So we set the $\Phi$ term equal to $\lambda$ and solve the resulting ODE.
\begin{equation}
\Phi'' - \lambda\Phi = 0
\end{equation}
Let's first consider $\lambda = - \omega^2 < 0$. This yields a solution:
\begin{equation}
\Phi(\phi) = A\cos\omega\phi + B\sin\omega\phi
\end{equation}
Using the BCs, we show that:
\begin{equation}
\Phi(0) = A = 0
\end{equation}
\begin{equation}
\Phi(\pi) = B\sin\omega\pi = 0 \longrightarrow \omega\pi = n\pi \longrightarrow \omega = n
\end{equation}
So for negative eigenvalues we conclude that $\Phi_n(\phi) = B_n\sin(n\phi)$ with $\lambda = -n^2$.
Let's check the positive case ($\lambda = \omega^2 > 0$). Wellllll, this will yield:
\begin{equation}
\Phi(\phi) = C\cosh\omega\phi + D\sinh\omega\phi
\end{equation}
Using the boundary conditions here gives us:
\begin{equation}
\Phi(0) = C = 0
\end{equation}
\begin{equation}
\Phi(\pi) = D\sinh\omega\pi = 0 \longrightarrow \omega\pi = 0 \longrightarrow \omega = 0
\end{equation}
But we are checking the case where $\omega$ is a positive (i.e. greater than zero) eigenvalue, so this doesn't make sense. We conclude that there are no positive eigenvalues.
For $\lambda = 0$, the solution will be $\Phi(\phi) = E\phi + F$. The first BC gives $\Phi(0) = F \longrightarrow F = 0$. The second gives $\Phi(\pi) = E\pi = 0 \longrightarrow E = 0$. We conclude that there is no zero eigenvalue.
Now we must consider the $R$ equation. Recall that we have:
\begin{equation}
\left( \frac{r^2R''}{R} + \frac{rR'}{R} \right) = - \lambda \longrightarrow r^2R'' + rR' + \lambda R = 0
\end{equation}
This is a typical Euler equation, so we assume a solution $R = r^m$. Plugging this in gives:
\begin{equation}
r^2m(m-1)r^{m-2} + rmr^{m-1} + \lambda r^m = 0 \longrightarrow m(m-1) + m + \lambda = 0 \longrightarrow m^2 - n^2 = 0 \longrightarrow m = \pm n
\end{equation}
The solutions are then:
\begin{equation}
R(r) = Gr^n + Hr^{-n}
\end{equation}
Well where are we? In the half-disk! So we discard the $H$ term, which will blow up at the origin. This leaves us with $R_n(r) = G_nr^n$.
Now let's combine the solutions, as we usually do. We can write our general solution as:
\begin{equation}
u(r, \phi) = \sum_1^{\infty}a_nr^n\sin(n\phi)
\end{equation}
Now we shall make use of this Robin boundary condition. We have that:
\begin{equation}
u(r, \phi) |_{r=1} = \sum_1^{\infty}a_n\sin(n\phi)
\end{equation}
\begin{equation}
u_r(r, \phi) |_{r=1} = \sum_1^{\infty}a_nn\sin(n\phi)
\end{equation}
Altogether, this is:
\begin{equation}
(u_r + u) |_{r=1} = \sum_1^{\infty} \left(a_nn\sin(n\phi) + a_n\sin(n\phi) \right) = 1 \longrightarrow \sum_1^{\infty} \left(a_n\sin(n\phi)(n+1) \right) = 1
\end{equation}
Now we'd better find out what $a_n$ is. So:
\begin{equation}
a_n = \frac{2}{\pi(n+1)} \int_0^{\pi}1\sin(n\phi)d\phi = \frac{2(1 - \cos(n\pi))}{\pi(n+1)n} = \frac{2(1 - (-1)^n)}{\pi(n+1)n}
\end{equation}
So we finally get that:
\begin{equation}
u(r, \phi) = \sum_1^{\infty} \frac{2(1 - (-1)^n)}{\pi(n+1)n} r^n\sin(n\phi)
\end{equation}
If anyone notices any errors, please let me know!
