I am an experimental physicist and I am trying to solve the following differential equation:
$$ z'' + w_0^2 z + c_1 z^3 - c_2 y = 0$$ $$ y'' + c_3y' + w_0^2 y + c_4\delta(t) + c_5 z' = 0$$
These equations represent a system of two coupled oscillators, where one is non-linear ($x$) and the other is damped and noise-driven ($y$).
All coefficients $w_0$ and $c_i$ are real and constant, $\delta(t)$ is the Dirac-Delta function. As final result I am interested in the time-average of the Fourier Transformation of $y'(t)$.
I am a bit overwhelmed by the combination of coupled, non-linear and stochastic. Where do I start here? Is this even analytically attainable?