Algorithmic solution of Abel’s equation

Abstract

The solution scheme for Abel’s equation proposed in this article avoids to a large extent thead hoc methods that have been discovered in the last two centuries since Abel introduced the equation named after him. On the one hand, it describes an algorithmic method for obtaining almost all closed form solutions known in the literature. It is based on Lie’s symmetry analysis. Secondly, for equations without a symmetry, a new method is proposed that allows to generate solutions of all equations within an equivalence class if a single representative has been solved before. It is based on functional decomposition of the absolute invariant of the equation at hand for which computer algebra algorithms have become available recently.