Solution of a system of differential equations with constant coefficients using in-verse moments problem techniques

It is known that given a system of simultaneous linear differential equations with constant coefficients you can apply the Laplace method to solve it. The Laplace transforms are found and the problem is reduced to the resolution of an algebraic system of equations of the determining functions, and applying the inverse transformation the generating functions are determined, solutions of the given system. This implies the need to know the analytical form of the inverse transform of the function. In this case the initial conditions consist in knowing the value that the generating function and its derivatives takes at zero. A generalization of this method is proposed in this work, which is to define a more general integral operator than the Laplace transform, the initial conditions consist of Cauchy conditions in the contour. And finally, we find a numerical approximation of the inverse transformation of the generating functions, using the techniques of inverse moment problems, without being necessary to know the analytical form of the inverse transform of the function.