Solving Laplace's equation with Legendre polynomials in spherical coordinates

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Fjolvar
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Question on Solving Laplace Equation Potentials.

Hello, I'm learning how to solve Laplace's equation to find Potentials in Cartesian, Cylindrical, and Spherical Coordinates and let's just say it's not going as smoothly as I'd like. In particular, I'm having difficulty with the Spherical case which involves Legendre Polynomials, Method of Frobenius, Orthogonality, etc. Can anyone recommend any sources or even simply give me a hint/tips on how to approach these types of problems? Any advice would be greatly appreciated. Thank you.
 
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Perhaps my post is too vague, so let's try this: In the Spherical case, how do you determine Pl(X) in the Angular Equation of V(r,[tex]\vartheta[/tex]) where [tex]\Theta[/tex]([tex]\vartheta[/tex]) = Pl(cos([tex]\vartheta[/tex]))..

What determines l (lower case L) in the Legendre Polynomials when solving for Pl(X)..

I know that when l=0, Pl(X) = 1. When l=1, Pl(X) = X, etc. So what does l depend on and how does it relate to the order of the equation and the physics of a problem?
 
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Your textbook or teacher shold be able to answer that, but your question here is still too vague.