In this paper, we look further into one of the relativistic wave equations we have introduced recently. Our relativistic wave equation is a PDE that is rooted in the relativistic energy Compton momentum relation, rather than the standard energy momentum relation. They are two sides of the same coin, but the standard momentum is just a derivative of the Compton momentum, so this simplifies things considerably. Here the main focus is to rewrite our relativistic PDE wave equation in spherical polar coordinates, then by some separation of variables we end up with three ordinary differential equations (ODEs) for which we present possible solutions, and we also provide a table where we compare our ODEs with the ODEs one gets from the Schro ̈dinger equation [1]. This approach is used to describe hydrogen-like atoms. We encourage other researchers to check our calculations and the predictions from our solutions and see how they fit compared to observations from hydrogen-like atoms.