A Semi-Algorithm to Find Elementary First Order Invariants of Rational Second Order Ordinary Differential Equations

Here we present a method to find elementary first integrals of rational second order ordinary differential equations (SOODEs) based on a Darboux type procedure [3, 4, 5]. Apart from practical computacional considerations, the method will be capable of telling us (up to a certain polynomial degree) if the SOODE has an elementary first integral and, in positive case, finds it via a single quadrature.