Analytic solutions of an iterative functional differential equation

摘要

This paper is concerned with a functional differential equation x′(z)=1/x(az+bx(z)), where a, b are two complex numbers. By constructing a convergent power series solution y(z) of a auxiliary equation of the form b2y′(z)=(y(μ2z)−ay(μz))(μy′(μz)−ay′(z)), analytic solutions of the form 1b(y(μy−1(z))−az) for the original differential equation are obtained.