arXiv · 1508.06405
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 $a_2 γ^{2} g"(γz) g'(z) = [g (γ^2 z) - p(g(γz))] γg'(γz)(g' (z))^{2} + bh'(g(z))(g' (z))^{3} + \Big( a_2p"(g(z))+ a_1p'(g(z)) +a_0p(g(z))\Big) (g'(z))^{3} - a_1γg'(γz) (g' (z))^{2} - a_0g(γz)(g'(z))^{3} + a_2γg'(γz)g"( z)$ with the relation $p(z) + bx'(z) = g(γg^{-1}(z)),$ we obtain an analytic solution $x(z).$ Moreover, an analytic solution depends on a parameter $γ$ which satisfies one of the following conditions: $(H1) \ 0<|γ|<1,$ $(H2) \ γ= e^{2πi θ}$ where $θ$ is a Brjuno number or $(H3) \ γ= e^{2πi θ}$ where $θ$ is a rational number.
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Jiraphorn Somsuwan, Keaitsuda Maneeruk Nakprasit. 2015-08-26. On the solution of a second order functional differential equation with a state derivative dependent delay. https://arxiv.org/abs/1508.06405
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