Um novo método para buscar soluções liouvillianas de equações diferenciais ordinárias de primeira ordem que apresentam funções elementares

Abstract

In this thesis we present a new approach to deal with first order ordinary differential equations (1ODEs) presenting elementary functions. The method we developed is na alternative to the procedure that was presented in (1, 2). In (3), a theoretical basis was established to deal with second order rational ordinary differential equations (rational 2ODEs) - the S-function method - that presents (at least) a Liouvillian first integral. Here, we combine the technique used in (3) with an idea analogous to that used in (1, 2) in order to produce a method that proved to be very efficient in a large number of cases in which the most powerful traditional generalist methods (the Lie and Darboux methods) present difficulties. The procedure we present here for solving 1ODEs is placed on a solid mathematical basis and two of the main algorithms were implemented in a computational package (LeapS1ode) in Maple. The LeapS1ode package includes commands that allow obtaining all the intermediate steps of the 1ODEs resolution process and has a design that makes it very useful for research (both in physics and mathematics) and especially effective in finding regions of integrability for 1ODEs that presenting parameters.