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**1. Homework Statement**

Solve the Laplace equation inside a sphere, with the boundary condition:

\begin{equation}

u(3,\theta,\phi) = \sin(\theta) \cos(\theta)^2 \sin(\phi)

\end{equation}

**2. Homework Equations**

\begin{equation}

\sum^{\infty}_{l=0} \sum^{m}_{m=0} (A_lr^l + B_lr^{-l -1})P_l^m(\cos \theta) [S_m\sin(m\phi) + C_m\cos(m\phi)]

\end{equation}

**3. The Attempt at a Solution**

I have derived the general solution for the Laplace equation in spherical problems and that went okej. The problem is when I try to match it with the boundary conditions. If I use the method of identification it is imminently obvious that:

\begin{equation}

C_m=0, m=1 \text{ and that for} \quad l\neq1 \rightarrow A=0

\end{equation}

, m has to equal one so the arguments of sin is correct and A=0 so the solution does not blow up, when r approaches 0. Which means I have two different solutions: one where l=1 and another solution for all the rest of l´s. When l equals one the Legendre polynomials dose not match the boundary conditions so that solution can be discarded.

Which means that I have an equation that looks like:

\begin{equation}

\sum^{\infty}_{l=0} ( B_l3^{-l -1})P_l^1(\cos \theta) S_1\sin(\phi) = \sin(\theta) \cos(\theta)^2 \sin(\phi)

\end{equation}

When I look at table's for Legendre polynomials I do not find any that looks like it can solve my equation. I would really appreciate some guidance in the right direction.