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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.

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    $\begingroup$ Already the equation $X''+S(t)X=0$ has no general solution method. $\endgroup$
    – Christian Remling
    Commented Aug 6, 2014 at 20:36
  • $\begingroup$ "over dumped" $\rightarrow$ over-damped? $\endgroup$
    – Noam D. Elkies
    Commented Aug 7, 2014 at 7:50

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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).

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