In this second of a three-part article, we use Maple to separate variables in a partial differential equation, then show how to obtain a solution of the resulting Sturm-Liouville eigenvalue problem. The PDE is Laplace's equation in a cylinder, and separation of variables leads to an eigenvalue problem involving Bessel's equation. We solve this singular Sturm-Liouville eigenvalue problem for the two cases of homogeneous Dirichlet and Neumann boundary conditions, and for the two cases of axial symmetry and asymmetry.