Analytic solutions of a second-order iterative functional differential equation

This paper is concerned with a second-order iterative functional differential equation x″(x[r](z))=c0z+c1x(z)+⋯+cmx[m](z), where r and m are nonnegative integers, x[0](z)=z, x[1](z)=x(z), x[2](z)=x(x(z)), etc., are the iterates of the function x(z). By constructing a convergent power series solution y(z) of a companion equation of form α2y″(αr+1z)y′(αrz)=αy′(αr+1z)y″(αrz)+[y′(αrz)]3∑i=0m ciy(αiz), analytic solutions of the form y(αy−1(z)) for the original differential equation are obtained.