I have a second order differential equation given by:

$x''(t) = \displaystyle\frac{\exp(-\frac{x(t)^2}{4t})}{A \sqrt{t}}$

I would like to be able to obtain an analytic solution to this equation, which may not be easy, but it's worth asking if anyone has any ideas! Frustratingly if instead of the $\frac{1}{\sqrt{t}}$ we have $\frac{1}{t^{3/2}}$ I think some progress can be made by defining $x(t) = \phi(s) \sqrt{t}$, where $s=\log(t)$. In this case applying the chain rule gives that:

$x''(t) = \displaystyle\frac{(\phi''(s) - \frac{\phi(s)}{4})}{t^{3/2}}$

and you get:

$\phi''=\displaystyle\frac{\phi}{4} -\exp(-\frac{\phi^2}{4})$

from which you can use:

$\int \phi'' d\phi = \frac{1}{2} \phi'^2 + C$.

This still does not give an exact solution but it's a lot nicer. I've not been able to so anything similar for the $\frac{1}{\sqrt{t}}$ case stated though, and in fact can't really make any progress at all. Any bright ideas?