I am wondering if there is a general solution for this ODE


$\ddot X +2\gamma \alpha \dot X + (\alpha+S(t)) X = \beta $


the dot represents time derivative, and $\gamma>1$, so it is in the over-damped regime.


It is a form of parametric oscillator, but I am wondering if there may exist a general solution for any function $S(t)$, as there exist for first order systems.


Thanks in advance.



Solution of General Parametric Oscillator
No, I'm fairly confident there is no known general solution. Even in some very simple cases, such as $S(t) = t + t^3$, $\beta = 0$, $\alpha = 0$, I don't think the solutions can be written in closed form (Maple 18 doesn't find such a form). 


