Polynomial matrices and differential equations

Abstract

Differential equations (E.D) and systems of differential equations (S.E.D) arise naturally when studying problems in physics, mechanics, economics, biology, etc.Its resolution is not an easy undertaking to carry out, requiring numerous calculations. It is due to these drawbacks that this proposal arises, which consists of five sections, which we will briefly explain. Section one is divided into two subsections, subsection 1.1 recalls concepts seen in Linear Algebra, such as polynomial matrix, companion matrix and linearization, in subsection 1.2, a theorem is stated that establishes the solution of a SED that makes use of the concepts recalled in subsection 1.1. In section two interesting situations are mentioned in which S.E.D of different orders appear, paying attention to the systems of second order equations that arise in mechanics, as for example, the study of coupled masses joined by springs. In section three there are solved examples together with an application problem, which are solved by means of higher order E.D systems applying the theorem. Section four is intended for final conclusions, it will be noted fundamentally the practicality of using the theorem in the resolution of nonhomogeneous second order SEDs, which usually require numerous calculations and tedious work for resolution, finally in section five Bibliographic references are listed.