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The function of Jost Fl is the theoretical concept that allows to study in an unified way the bound, virtual, scattered and resonant states that can originate in the interactions between two quantum systems. In theory of collisions the function of Jost Fl plays a very important role, since it is related in a direct way with the scattering matrix S. In most of the existent methods in theory of collisions for the calculation of the function Fl first is necessary to know the regular solution of the treated system, which is obtained of the solution of the radial equation of Schrödinger, to be able to find the function Fl later. With the methodology proposed in this work an ordinary lineal differential equation of second order it is obtained whose solution in the asymptotic boundary coincides with the function Fl. The advantage of the present work is that solving the differential equation mentioned before one can obtain in a direct way the function Fl without having to find the regular solution of the problem. Another advantage is that not caring the initial conditions (real) that are chosen for the solution of the differential equation, the same elements of the matrix S, are always obtained. As an example and test of the methodology, it is solved this differential equation numerically for the elastic scattering of electrons by hydrogen atoms in the ground state to low energy (e- + H(1s)), obtaining for this system the function Fl, the elements of the matrix S and the phase shifts. The data obtained for the phase shifts are compared with the calculated by Klaus Bartschat [1]. [ABSTRACT FROM AUTHOR]