Particular solution of a non-homogeneous second-order ordinary Cauchy Eulerian differential equation using the ECA method

Abstract

Differential equations have been a fundamental tool in multiple areas of science and technology, as they allow us to describe dynamic phenomena and establish connections between variables in complex systems. This article addresses the specific solution of a second-order nonhomogeneous ordinary differential equation of this type. x2y′′+xy′=f(x)x^2 y'' + x y' = f(x)x2y′′+xy′=f(x), known as the Cauchy-Euler equation. To solve it, the ECA method is used, which is based on a change of variable with the aim of transforming it into a linear differential equation with constant coefficients. Subsequently, the solution is expressed again in terms of the original variable, thus obtaining a particular solution equivalent to that which would be achieved by the method of variation of parameters or the use of computational tools. The approach adopted is analytical and considers different scenarios depending on the roots of the auxiliary equation, whether they are distinct real roots or repeated real roots. The results were compared with those obtained by other methods such as variation of parameters, transformation to constant coefficients, and mathematical programs such as Symbolab and Wolfram. It is concluded that the ECA method offers an effective alternative for Solve Cauchy-Euler equations, optimizing calculations without sacrificing accuracy.