IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 14, Issue 4 Ver. I (Jul - Aug 2018), PP 31-34 www.iosrjournals.org DOI: 10.9790/5728-1404013134 www.iosrjournals.org 31 | Page The Cauchy - Euler Differential Equation and Its Associated Characteristic Equation Edel Alexander Silva Pontes 1 1 (Department of Mathematics, Federal Institute of Alagoas, Brazil) Corresponding Author: Edel Alexander Silva Pontes Abstract: Many natural phenomena, from physics to biology, through the fields of medicine and engineering, can be described by means of differential equations. This article aims to present the solutions of a homogeneous Cauchy-Euler differential equation from the roots of the characteristic equation associated with this differential equation, in order to facilitate the life of the university student. The great algebraic difficulty that a graduate student encounters in solving the homogeneous Cauchy-Euler differential equations is that his solution depends on a polynomial equation of degree n called the characteristic equation. It is hoped that this work can contribute to minimize the lags in teaching and learning of this important Ordinary Differential Equation. Keywords: Ordinary Differential Equation. Equation of Cauchy-Euler homogeneous. Characteristic equation. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 28-06-2018 Date of acceptance: 16-07-2018 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction [1] Many of the principles, or laws, that govern the behavior of the physical world are propositions, or relationships, involving the rate at which things happen. Expressed in mathematical language, relations are equations and rates are derived. Equations containing derivatives are Differential Equations. The ordinary differential equations , ,   , ⋯ ,       =0 encompass a very broad area of mathematics and of fundamental importance to explain various models of everyday life. [1] [2] [3] Geometrically, the general solution of a differential equation represents a family of curves that are called the integral curves. This solution is called primitive or integral of the differential equation. Solving a differential equation means finding an adequate family of curves. Many natural phenomena, from physics to biology, through the fields of medicine and engineering, can be described by means of these differential equations. The solutions of these equations are used, for example, to design automobiles, construct buildings and bridges, identify population growth, explain electric circuits, among many other applications. [4] [5] A single differential equation can serve as a mathematical model for several different phenomena. The purpose of this paper is to present the solution of an Ordinary Differential Equation, called the Cauchy-Euler Equation from the roots of the Characteristic Equation associated with this differential equation. We first define the homogeneous Cauchy-Euler equation of order n. Then we will use the particular case, n = 2, to present its solutions, according to the roots of its characteristic equation. In the following section, several Equations Characteristics associated with their Cauchy-Euler Equations will be presented, and finally a conclusion will be reported. II. The Equation of Cauchy - Euler homogeneous [6] Leonhard EULER (1708-1783), physicist, mathematician, German-speaking Swiss geometer had as his advisor the mathematician Johann Bernoulli. He developed several works in the area of Calculus and Graph Theory and spent much of his life in St. Petersburg and Berlin. He wrote several works, among them, Complete Treaty of Mechanics, Institutions of Integral Calculus and Introduction to Analysis of Infinitesimals. [6] Augustin Louis CAUCHY (1789-1857), French mathematician, was the creator of Theory of Analytical Functions and the Method to determine the number of real roots. He created the modern notion of continuity for the functions of real or complex variable. The Equation of Cauchy - Euler homogeneous of nth order is any Ordinary Differential Equation of the form: 0 0 1 2 2 2 2 1 1 1 1           y A dx dy x A dx y d x A dx y d x A dx y d x A n n n n n n n n  , (1) Where 1 2 1 , , , , A A A A n n   and 0 A are constant.