Séries de potências como método de solução de EDO: fundamentos e aplicações

Abstract

The study of Ordinary Differential Equations is extremely important for various fields, such as Physics and Engineering. However, there are some Ordinary Differential Equations (ODEs) for which the most well-known methods are not effective, making it necessary to explore knowledge tools for an alternative approach. One of the tools that can be used is solving through Power Series, which allows representing the solution of the ODE in the form of Series around a specific point. In this context, we sought to investigate how the power series method can be applied in solving second-order ODEs with variable coefficients. To prove its functionality, the method was applied to two very important ODEs in the field of mathematics, the Airy equation and the Legendre equation. As this study is related to ODEs, it falls within the field of pure mathematics, specifically in the subfield of analysis.