Copyright © 2009 Tongbo Liu and Hong Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the existence of analytic solutions of a class of second-order differential equations involving iterates of the unknown function x′′(z)+cx′(z)=x(az+bx(z)) in the complex field ℂ. By reducing the equation with the Schröder transformation to the another functional differential equation without iteration of the unknown function λ2g′′(λz)g′(z)−λg′(λz)g′′(z) + c(g′(z))2(λg′(λz)−ag′(z)) = (g′(z))3(g(λ2z)−ag(λz)), we get its local invertible analytic solutions.