Theoretical or experimental studies of the interactions between quantum particles are very important in physical sciences because from them it is possible to know the properties of the systems, therefore, the Jost functions are very important in Quantum Scattering Theory because they allow to carry out unified studies on bound states, scattered states, virtual states and resonance states. In this work a new method to calculate the Jost functions is presented, this method is based on a system of first order differential equations and has the advantage that the solutions of the problem can be calculated by solving the system of differential equations from the origin or from infinity, therefore, it facilitates calculations when systems are modeled with singular potentials at the origin. We tested this method by modeling the collision of an electron with a hydrogen atom in its ground state at low energies (10 eV ) and found very good results.