On Szebehely's Equation and its Connection with Dainelly's / Whittaker's Equations

In this paper we prove that Szebehely's equation implies the condition necessary and sufficient in order that the Dainelli-Whittaker formulas be obtained from a potential function. Thus, we prove that Szebehely's equation is not only a necessary condition to be satisfied by the potential U(ěc x) but is also a sufficient condition. This was shown by Broucke and Lass (1977) using a procedure different from ours. We obtain also a first order partial differential equation for the function g² (ěc x) appearing in the paper of Broucke and Lass. This equation, being of a quite simple structure, is quite adequate for its integration, and this is shown by examples.