On singular Frobenius for linear differential equations of second and third order, part 1: ordinary differential equations

TL;DR

This paper advances the understanding of Frobenius methods for second and third order linear differential equations, focusing on convergence, classification, and solutions, with applications to classical equations like Bessel and Legendre.

Contribution

It provides new results on convergence of formal solutions, a classification of regular singularities, and a Frobenius-like theorem for third order equations, with a computational approach.

Findings

Characterization of regular singularities via solution spaces

Analytic classification using Riccati equations and holonomy groups

Existence and convergence results for non-homogeneous equations

Abstract

We study second order and third order linear differential equations with analytic coefficients under the viewpoint of finding formal solutions and studying their convergence. We address some untouched aspects of Frobenius methods for second order as the convergence of formal solutions and the existence of Liouvillian solutions. A characterization of regular singularities is given in terms of the space of solutions. An analytic classification of such linear homogeneous ODEs is obtained. This is done by associating to such an ODE a Riccati differential equation and therefore a global holonomy group. This group is a computable group of Moebius maps. These techniques apply to classical equations as Bessel and Legendre equations. In the second part of this work we study third order equations. We prove a theorem similar to classical Frobenius theorem, which describes all the possible cases and solutions to this type of ODE. Once armed with this we pass to investigate the existence of solutions in the non-homogeneous case and also the existence of a convergence theorem in the same line as done for second order above. Our results are concrete and (computationally) constructive and are aimed to shed a new light in this important, useful and attractive field of science.