# Asymptotic bounds of a simple "Delay Differential Equation" [duplicate]

Define the function $$x(t)$$ for $$t\ge0$$: $$x(0)=1\\ x'(t)=-x(t/2)$$ I could do a power series from $$t=0$$ like this (thanks to @JeanMarie for pointing this old question out), but I ultimately want asymptotic bounds on the excursions as $$t\to\infty$$, so I don't think the power series helps. Is there any other research on this function? How quickly do these excursions grow with $$t$$?

• See as well the excellent answer by Robert Israel here Commented May 22, 2021 at 17:23
• It is not a delay differential equation : it should have a term $f(t-a)$ for that... Commented May 22, 2021 at 17:25
• @JeanMarie Only this question is relevant, but the power series answer (modified for the important -1 in my equation) does not help me find the asymptotic bounds I want. Commented May 22, 2021 at 18:09
• Another track: math.stackexchange.com/q/3784061 Commented May 22, 2021 at 18:14
• If you take the Laplace Transform with ${\frak L}(f)=F$, $F$ must verify the functional equation $sF(s)+1=-2F(2s)$. Is there a solution ? Commented May 22, 2021 at 20:55