A generalization of the S-function method applied to a Duffing–Van der Pol forced oscillator

In Duarte et al. (2016) and Avellar et al. (2019), we have developed a method (we call it S-function method) that is successful in treating certain classes of rational second order ordinary differential equations (rational 2ODEs) that are particularly ‘resistant’ to canonical Lie methods and to Darbouxian approaches (extensions of the Prelle–Singer method). In this present paper, we generalize the S-function method making it capable of dealing with a class of elementary 2ODEs. We also present a Maple implementation of the method in a computational package – S++ – that is designed to provide a set of tools to allow the user to analyze the intermediary steps of the generalized S-function method. Finally, we apply this method to a Duffing–Van der Pol forced oscillator, obtaining an entirely new class of first integrals.