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

Abstract EN

We investigate an analytic solution of the second-order differential equation with a state derivative dependent delay of the form x″(z)=x(p(z)+bx′(z)).

Considering a convergent power series g(z) of an auxiliary equation γ2g″(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 relation p(z)+bx′(z)=g(γg-1(z)), we obtain an analytic solution x(z).

Furthermore, we characterize a polynomial solution when p(z) is a polynomial.