I am trying to solve the equation

$$ \frac{dy}{dt}=\sqrt{(\gamma-1+\frac{2\alpha\beta}{2\alpha-1})e^{-2\alpha y}-\frac{2\alpha\beta}{2\alpha-1}e^{-y}+1}\tag{1} $$

$y(0) = 0$; $t_{0}=0$; $\alpha$, $\beta$ and $\gamma$ are positive constants ($\beta$ is also less than 1). There is also a special form for $\alpha=\frac{1}{2}$ but I will not dwell on this right now.

Provided I am not mistaken, letting $x = e^{-y}$ and rearranging leads to

$$ \int_{e^{-y(t)}}^{1}{\frac{dx}{x\sqrt{(\gamma-1+\frac{2 \alpha\beta}{2\alpha-1})x^{2\alpha}-\frac{2 \alpha\beta}{2\alpha-1}x+1}}}=t\tag{2} $$

This integral is solvable for particular values of $\alpha$ (ones that allow a partial fraction decomposition of the integrand(?))

My questions are:

1) Is there a relatively compact way that may determine if the integral $(2)$ can be expressed in closed form?

2) Can $x(t)$ be expressed in terms of relatively well-studied special functions?

3) Even if a general method for solving this equation (which I believe does not exist), is there a way of obtaining an approximate solution for $x(t)$ in closed form?

I have considered taking the Taylor series of the radical, but unfortunately this method does not work very well because the integrand does contain a region of rapid change within the integration region.

Any ideas on how to treat this problem will be greatly appreciated!


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