Usando polinômios de Darboux na obtenção de integrais primeiras liouvillianas de equações diferenciais ordinárias racionais de segunda ordem

Abstract

In the late XIX century Darboux laid the theoretical foundations that connected the search for first integrals of systems of differential equations with special polynomials (later they became known as Darboux polynomials). It was the first generalist (nonclassificatory) method, in the history of determining solutions/invariants of systems of differential equations. This procedure has been extended over time and the ’heart’ of all these ’Darbouxian’ approaches (for the search for first solutions/integrals of differential equations) was the determination of Darboux polynomials associated with the differential equation (or system) in question. Ironically, this is the more complex step of this kind of approach. Thus, in recent decades has been a significant increase in the number of studies that sought strategies to compute these polynomials. In the specific case of ordinary second order differential equations, Darboux polynomials are in three variables, which makes the process more difficult (it is already expensive when it comes to Darboux polynomials in two variables). In this thesis, me and my reseracrh group present alternative approaches to direct calculation (the so-called method of undetermined coefficients – MUC) for computation of Darboux’s polynomials. The techniques we build fit the context of order reducing order (search for first integrals) of rational ordinary differential equations of second order presenting at least one first Liouvillian integral.