A geometric-analytic study of linear differential equations of order two

Víctor León; Bruno Scárdua; Víctor León; Bruno Scárdua

doi:10.3934/era.2020107

Electronic Research Archive

2021, Volume 29, Issue 2: 2101-2127. doi: 10.3934/era.2020107

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A geometric-analytic study of linear differential equations of order two

Víctor León ^{1
,},
Bruno Scárdua ^{2
,
,}

1.
Instituto Latino-Americano de Ciências da Vida e da Natureza, Centro Interdisciplinar de Ciências da Natureza, Universidade Federal da Integração Latino-Americana, Parque tecnológico de Itaipu, Foz do Iguaçu-PR, 85867-970, Brazil
2.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro-RJ, 21945-970, Brazil

Received: 01 February 2020 Revised: 01 August 2020 Published: 19 October 2020
Primary:34A05, 34A25;Secondary:34A30, 34A26

We study second order linear differential equations with analytic coefficients. One important case is when the equation admits a so called regular singular point. In this case we address some untouched and some new aspects of Frobenius methods. For instance, we address the problem of finding formal solutions and studying their convergence. A characterization of regular singularities is given in terms of the space of solutions. An analytic-geometric classification of such linear polynomial homogeneous ODEs is obtained by the use of techniques from geometric theory of foliations means. This is done by associating to such an ODE a rational 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. We also address the problem of deciding which such polynomial equations admit a Liouvillian solution. A normal form for such a solution is then obtained. Our results are concrete and (computationally) constructive and are aimed to shed a new light in this important subject.
- Frobenius method,
- regular singularity,
- Riccati equation,
- holonomy of a second order equation,
- liouvillian solutions
Citation: Víctor León, Bruno Scárdua. A geometric-analytic study of linear differential equations of order two[J]. Electronic Research Archive, 2021, 29(2): 2101-2127. doi: 10.3934/era.2020107

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Abstract

We study second order linear differential equations with analytic coefficients. One important case is when the equation admits a so called regular singular point. In this case we address some untouched and some new aspects of Frobenius methods. For instance, we address the problem of finding formal solutions and studying their convergence. A characterization of regular singularities is given in terms of the space of solutions. An analytic-geometric classification of such linear polynomial homogeneous ODEs is obtained by the use of techniques from geometric theory of foliations means. This is done by associating to such an ODE a rational 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. We also address the problem of deciding which such polynomial equations admit a Liouvillian solution. A normal form for such a solution is then obtained. Our results are concrete and (computationally) constructive and are aimed to shed a new light in this important subject.

References

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