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 $a_2x"(z) + a_1x'(z) + a_0x(z) = x(p(z) + bx'(z)) + h(z)$ has been studied. Considering a convergent power series $g(z)$ of an auxiliary equation $a_2 \gamma^{2} g"(\gamma z) g'(z) = [g (\gamma^2 z) - p(g(\gamma z))] \gamma g'(\gamma z)(g' (z))^{2} + bh'(g(z))(g' (z))^{3} + \Big( a_2p"(g(z))+ a_1p'(g(z)) +a_0p(g(z))\Big) (g'(z))^{3} - a_1\gamma g'(\gamma z) (g' (z))^{2} - a_0g(\gamma z)(g'(z))^{3} + a_2\gamma g'(\gamma z)g"( z)$ with the relation $p(z) + bx'(z) = g(\gamma g^{-1}(z)),$ 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.