Analytic Solutions of a Second-Order Functional Differential Equation with a State Derivative Dependent Delay

We investigate an analytic solution of the second-order differential equation with a state derivative dependent delay of the formx″(z)=x(p(z)+bx′(z)). Considering a convergent power seriesg(z)of an auxiliary equationγ2g″(γz)g′(z)=[g(γ2z)-p(g(γz))]γg′(γz)(g′(z))2+p′′(g(z))(g′(z))3+γg′(γz)g″(z)with the relationp(z)+bx′(z)=g(γg-1(z)), we obtain an analytic solutionx(z). Furthermore, we characterize a polynomial solution whenp(z)is a polynomial.