Study on non-homogeneous linear ordinary differential equations of constant coefficients by the method of successive integrals

Abstract

Introduction:  When solving non-homogeneous linear differential equations with constant coefficients, it is necessary to find a solution that corresponds to the homogeneous part, denoted as , and a part that corresponds to the non-homogeneous part, denoted as . In the case where the non-homogeneous part has the form , where "a" is one of the roots corresponding to the characteristic equation factors of the differential equation, a peculiarity arises in the solution. Depending on the order in which the factors are found, when setting up the successive integral, it may appear that there are two solutions for . However, upon verifying them in the differential equation, they both satisfy it. This may lead to the belief that it does not comply with the Cauchy's theorem, which states that the differential equation has a unique solution. However, when applying the initial conditions to the general solution: , it does comply with the theorem, confirming that the non-homogeneous linear differential equation with constant coefficients has a unique solution when solved using the method of successive integrals. Thus, the study confirms and verifies that Cauchy's theorem guarantees the existence and uniqueness of a solution that satisfies the initial conditions of an ordinary differential equation. Objective:  Objective: Check that the solution solved by successive integration of a linear ordinary differential equation with constant coefficients, leads to a particular solution of the unique differential equation. Methodology: In the present study, for the resolution of non-homogeneous linear ordinary deferential equations with constant coefficients, the method of successive integrals is used, in addition to using the criterion of verification of the solution of a differential equation, to determine by means of Cauchy's theorem, that the solution is unique in a differential equation when it has specific initial conditions. Results: Starting from a specific case when the non-homogeneous part has the form  where a is a root of the characteristic equation, apparently two solutions are found for the solution, it is verified that the solution is unique by obtaining the solution of the differential equation, with a given initial condition. Conclusion: It is verified that the solution solved by successive integration of a linear ordinary differential equation of constant coefficients, leads to a particular solution of the unique differential equation. General area of ​​study: mathematical analysis. Specific area of ​​study: non-homogeneous linear Ordinary Differential Equations with constant coefficients.