Analytic Solutions of a First Order Iterative Functional Differential Equation

Abstract.

This paper is concerned with a first order iterative functional differential equation of the form x′ (z) = f(x(p(z) + bx(z))). By constructing a convergent power series solution of an auxiliary equation of the form $\beta y^\prime(\beta z)-p^\prime(y(z))y^\prime(z) = by^\prime(z)f\left[\frac{1}{b}(y(\beta^{2}z)-p(y(\beta z)))\right]$, analytic solutions of the form $x(z) = \frac{1}{b}[y(\beta y^{-1}(z))-p(z)]$ for the original equation are obtained.