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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?

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  • $\begingroup$ I don't know if this helps, but the equation for $\phi$ can be reduced, using the function $E(\phi) = \frac{1}{2} (\phi ')^2 - \frac{1}{8} \phi^2 $, to the integral $$ \int \frac{d\phi}{\sqrt{\frac{1}{4} \phi^2 - \sqrt{\pi} \mbox{erf} (\frac{1}{2} \phi) + C_1 }} = t + C_2$$ where $C_1$ and $C_2$ are constants. $\endgroup$ – David Hughes Sep 9 '18 at 17:50
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Maple does not find a closed-form solution (it does find one for the $1/t^{3/2}$ version). Moreover, it finds no symmetries for your differential equation, while the $1/t^{3/2}$ version has symmetries with infinitesimals $\xi = 2 t,\; \eta = x$. So I doubt that your differential equation has closed-form solutions.

By the way, your equation has a rather special series solution of the form

$$ \eqalign{x(t) &= \frac{4}{3}{\frac {{t}^{3/2}}{A}}-{\frac {16\,{t}^{7/2}}{315\,{A}^{3}}}+{ \frac {1504\,{t}^{11/2}}{280665\,{A}^{5}}}-{\frac {3996032\,{t}^{15/2} }{5746615875\,{A}^{7}}}+{\frac {552891776\,{t}^{19/2}}{5568470782875\, {A}^{9}}}\cr &-{\frac {419477321358848\,{t}^{23/2}}{27958094579597056875\,{ A}^{11}}}+{\frac {7840954485228544\; t^{27/2}}{3330302442569649421875\,{A}^{13}}}\cr &-{\frac {5273854278377204027392\; t^{31/2}}{ 13894884338733202927262765625\,{A}^{15}}}\cr &+{\frac { 38153364471684027322714112\; t^{35/2}}{611080980658633880276100359765625\,{A}^{17 }}} + \ldots} $$ but I don't know of a closed form for this.

EDIT: One thing you can do is remove the parameter $A$ by the change of variables $t = As$, $x(t) = \sqrt{A}\; y(s)$.

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