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My system is described by the following ODE:

$$ \frac{\mathrm{d}y(t)}{\mathrm{d}t} = a-y(t)x(t) $$

Where $a$ is a constant and x(t) is a poisson process so that:

$$ E[x(t)x(s)] = qI\delta(t-s) $$

The solution is of the form:

$$ y\left( t \right) = e^{-\int^t{x(z)dz}}{\int^t{a\,e^{\int^z{x(w)dw}}}dz+ c} $$

I would like to find a close form for the output total power $E[y^2(t)]$ in function of time.

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  • $\begingroup$ Where is the $t$ in the final equation? After $\int x(t) dt$, the $t$ is integrated away so it is just a number at that point, call it $z$. Then according to your equation you'd have $y(t)=e^{-z}\int ae^zdt + c$, and this isn't too bad to solve. I feel like there is a typo or I am misreading something $\endgroup$
    – Engineer
    Commented Apr 21, 2020 at 17:58
  • $\begingroup$ Maybe it should be rewritten as $y(t) = e^{-\int^t{x(z)dz}}{\int^t{a\,e^{\int^z{x(w)dw}}}dz+ c}$ $\endgroup$
    – Knyq
    Commented Apr 21, 2020 at 18:09

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