Elementos básicos de la teoría analítica y cualitativa de las ecuaciones diferenciales ordinarias

Abstract

In general, it is well known that solving systems of differential equations can be a very complex task, and sometimes it is impossible to obtain explicit solutions. Even when this is possible, studying such solutions can be very complicated. Because of this, it is very important to have different tools for study from both quantitative and qualitative perspectives. This work conducts a study of the fundamental concepts of the theory of linear ordinary differential equation systems (LODES) using two approaches: quantitative and qualitative. It aims to serve as an integral and essential guide to understanding and applying these concepts in different fields of study. We begin the study with theorems that allow us to guarantee the existence, uniqueness, and method for calculating the solutions of linear equation systems, which are fundamental in the theory of ODEs, as they establish conditions under which it is possible to find a unique and valid solution for a given system. We also address Jordan forms, which will allow us to find the solution of a system of linear differential equations with constant coefficients, using the eigenvalues and eigenvectors of the system's coefficient matrix. Finally, we present a complete classification of phase diagrams of linear differential equation systems in the plane.