I'm struggling to find the solution to the follow Poisson problem in spherical coordinates:
$$ \Delta\, f\left(r,\theta\right) = \sum_{l=1}^{4}k_{l} \left(r\right) P_{l}^{1}\left( \theta \right)$$
In the above equation $P_{l}^{1}\left( \theta \right)$ are the Associated Legendre polynomials of the first kind and the functions $k_{l}\left(r\right)$ are polynomials decreasing as $\frac{1}{r^{n}}$ with $n>7$.
Note that the problem is axysimmetric since the function $f$ and the right hand side do not depend the angle $\phi$, nevertheless the problem is $3$D.
The boundary conditions to the equations are:
$$ \begin{cases}f\left(1,\theta \right)&= -\sin\left( \theta \right) \\ f\left(\infty,\theta \right) & = 0 \end{cases} $$
I'm quite clueless on how to proceed, do you have any suggestion? maybe a particular form of $\ f\left(r,\theta\right)\;$ is helpful to find the solution?
(EDIT: this problem arise from the solution of the stokes equation with a body force where a sphere is free to move under the effect of the above mentioned body force field; if you feel that more details are required to the question I will gladly add them.)