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

TL;DR

This paper develops an analytic solution approach for a second order differential equation with a state derivative dependent delay, using a convergent power series and conditions on a parameter b3 key part of the solution.

Contribution

The paper introduces a method to find analytic solutions to a complex second order functional differential equation with a derivative-dependent delay, expanding the theoretical understanding of such equations.

Findings

Derived an explicit analytic solution using a convergent power series.

Identified conditions on the parameter b3 key to the solution's existence.

Established solution validity for b3 range of complex values including roots of unity.

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.