On the solution of a second order functional differential equation with a state derivative dependent delay

Abstract

In this paper, the second order differential equation with a state derivative dependent delay of the form $a_{2}x"(z)+a_1{x}^{\prime }(z)+a_{0}x(z)=x(p(z)+b{x}^{\prime }(z))+h(z)$ has been studied. Considering a convergent power series $g(z)$ of an auxiliary equation $a_2{\gamma }^{2}g"(\gamma z)g^{\prime }(z)=[g({\gamma }^{2}z)-p(g(\gamma z))]\gamma g^{\prime }(\gamma z)(g^{\prime }(z){)}^2+b{h}^{\prime }(g(z))(g^{\prime }(z){)}^3+(a_{2}p"(g(z))+a_1{p}^{\prime }(g(z))+a_{0}p(g(z)))(g^{\prime }(z){)}^3-a_1\gamma g^{\prime }(\gamma z)(g^{\prime }(z){)}^2-a_{0}g(\gamma z)(g^{\prime }(z){)}^3+a_2\gamma g^{\prime }(\gamma z)g"(z)$ with the relation $p(z)+b{x}^{\prime }(z)=g(\gamma g^{-1}(z)),$ we obtain an analytic solution $x(z).$ Moreover, an analytic solution depends on a parameter $\gamma$ which satisfies one of the following conditions: $(H1)0<|\gamma |<1,$ $(H2)\gamma =e^{2\pi i\theta }$ where $\theta$ is a Brjuno number or $(H3)\gamma =e^{2\pi i\theta }$ where $\theta$ is a rational number.