Annals of Mathematical Sciences and Applications

Volume 6 (2021)

Number 1

Analysis of third order linear analytic differential equations with a regular singularity: Bessel and other classical equations

Pages: 51 – 83



Víctor León (Instituto Latino-americano de Ciências da Vida e da Natureza, Universidade Federal da Integração Latino-Americana, Foz do Iguaçu, Paraná, Brazil)

Alexis Rodriguez (Departamento de Matemáticas, Universidad Nacional de Trujillo, Peru; and Departamento de Ciencias, Universidad privada del Norte, Trujillo, Peru)

Bruno Scárdua (Instituto de Matemática, Universidade Federal do Rio de Janeiro, RJ, Brazil)


We study third order linear differential equations with analytic coefficients under the viewpoint of explicitly constructing solutions and studying their convergence. We consider both homogeneous and non-homogeneous cases. In the homogeneous case with a regular singular point we address some untouched aspects of the classical theory like the complete description of the space of solutions. For this we identify the influence of resonances. We also study the convergence of formal solutions and a concrete way of constructing a third solution from two given solutions. These techniques apply, via classical order reduction and variation of parameters, to a complete description of the solutions of an analytic non-homogeneous equation with a regular singularity in terms of the given coefficients. We also propose models of order three for the classical differential equations like Airy, Chebychev, Laguerre, Lagrange and Hermite.We obtain explicit solutions to these models enforcing the properties they share with their order two counterparts. The final part contains some computer generated graphs of the solutions of these models. We believe our results are constructive and concrete making them useful in applied sciences and engineering.


Frobenius method, regular singularity, analytic equation, Bessel equation

2010 Mathematics Subject Classification

Primary 34A05, 34A25. Secondary 34A26, 34A30.

Received 27 January 2021

Accepted 30 March 2021

Published 6 October 2021