There was a lengthy series of posts [link], in late 2008, concerning the paper of Demetrios Panyotounakos that claimed to have solved a large class of Abel equations. As I read the posts, it would seem that the final conclusion had not yet occurred, as to the validity of his claim.
In particular there were questions to Edgardo Cheb-Terrab as to why this method had not yet been implemented in the Maple dsolve package. His responses seemed to be that someone else needed to show an actual example where this method would work. I and my student have been working on an Abel equation that does not seem to be solvable by Maple, but would fit into the class of equations that Dr. Panyotounakos considered.
Therefore, we wondered if any other conclusions had been reached, since 2008.
We would especially wonder if Dr. Cheb-Terrab, who was working for Maple then, at least, could answer this question for us.
Thanks very much, Daniel Finley,
Dept. of Physics & Astronomy, University of New Mexico