Problem 3.17 - Separation of Variables, Spherical Coordinates: Introduction to Electrodynamics

Here we go! The solution to the Laplace's equation in spherical leads us down a complicated path. When using separation of variables, the radial equation has a fairly simply solution, but the angular equation is not so simple. The solutions are Legendre polynomials in the variable cos θ. These polynomials are most conveniently defined by the Rodrigues formula. Here we see how to find the polynomials! This is from a power series solution, in which these polynomials are found from the recursion relation. You will see these for other differential equations as well!
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