A Comparative Study of the Frobenius Method and a Lie Symmetry Analysis Method for Solving Second-Order Ordinary Differential Equations with Regular Singularities

A. This article compares two approaches for solving second-order ordinary differential equations (ODEs) with regular singularities: the Frobenius method and a method of Lie symmetry analysis of successive reduction of order.We provide a theoretical overview of each method, emphasizing their mathematical foundations and computational frameworks.The comparison is illustrated through four example ODEs with regular singularities, allowing a direct evaluation of their performance.The Frobenius method provides series expansions that can often be expressed compactly in terms of special functions, such as Bessel or hypergeometric families.In contrast, Lie symmetry analysis offers a unified and algorithmic framework that is applicable across all cases, provided the ODE admits the necessary symmetries.This difference highlights the Frobenius method's case-specific flexibility and the generality of the Lie symmetry approach.