Determinação de integrais primeiras liouvillianas em equações diferenciais ordinárias de segunda ordem

• Autor(a) principal: Costa Filho, Jayr Avellar
• Data de Publicação: 2013
• Tipo de documento: Tese
• Idioma: por
• Título da fonte: Biblioteca Digital de Teses e Dissertações da UERJ
• Texto Completo: http://www.bdtd.uerj.br/handle/1/12834
• topic: Liouvillian ODE; Darboux; Semi-algorithm; EDO Liouvillianas; Darboux; Semi-algoritmo; CNPQ::CIENCIAS EXATAS E DA TERRA::FISICA::FISICA GERAL

In this thesis we have developed various `Darboux type' approaches to look for first integrals (elementary and Liouvillian) of rational second order ordinary differential equations (rational 2ODEs). The algorithms (semi-algorithms) we developed follow the line of Prelle and Singer's work. Basically, the methods to search for elementary first integrals are an extension of the technique developed by Prelle and Singer to find elementary solutions of rational first order ordinary differential equations (rational 1ODEs). The procedure that handles rational 2ODEs presenting Liouvillian first integrals is based on a extension of our method for finding Liouvillian solutions of rational 1ODEs. The fundamental idea behind our work is that the integrating factors for polynomial 1-forms generated by differentiation of elementary and Liouvillian functions are formed by certain polynomials called Darboux polynomials. We will show how to combine these Darboux polynomials to construct integrating factors and using them to determine first integrals via quadratures. We will also discuss some computational implementations of the semialgorithms developed.