Analytic solutions of a second-order iterative functional differential equation

This paper is concerned with a second-order iterative functional differential equation x″(z) = (xm(z))2. Its analytic solutions are discussed by locally reducing the equation to another functional differential equation with proportional delays μ2y″(μz)y′(z) = μy′(μz)y″(z) + [y′(z)]3[y(μmz)]2 and by constructing a convergent power series solutions for the latter equation.