I am trying to solve the following Boundary Value Problem - a particle is moving on the surface of a sphere:
$$ \triangle y(\theta)=\frac{d²y(\theta)}{d \theta²} + \cot{\theta } \frac{dy(\theta)}{d\theta} = -1 $$
where $\triangle$ is the laplacian in spherical coordinates, with the BC $y(\theta_0) = 0$ and $y'(\pi) = 0$.
I found a solution to the homogeneous equation:
$$ y(\theta)_H = A_1 \ln \Big(\frac{\cos {\theta/2}}{\sin{\theta/2}} \Big) + A_2. $$
However, $\theta = \pi$ is a singular point; $\theta = 0$ is not within the domain so I don't have to worry about it.
I don't know how I should apply the boundary conditions. Is the homogenous solution I found valid? How should I deal with the $\theta = \pi$ point? Should I expect $y(\theta)_H$ to be trivially zero and try find a particular solution instead? I would really appreciate any suggestion on what to try next!
I don't know if it helps, but:
$y''+y'\cot\theta=-1\\ y''\sin\theta+y'\cos\theta=-\sin\theta\\ (y'\sin\theta)'=-\sin\theta\\ y'\sin\theta=\cos\theta+C\\ y'=\dfrac{C+\cos\theta}{\sin\theta}\\ $
For $C=1$ we get $\lim\limits_{\theta\to\pi}\dfrac{1+\cos\theta}{\sin\theta}=0=y'(\pi)$