This seems like a long shot, but is there any closed form solution to $$y''x^{2r} - {\frac{r}{2}}y'x^r - cyx^{\frac{5r}{2}} = 0 ?$$
Here, we can take $x>0,r>0$ if it helps. This sort of looks like some kind of generalized Bessel equation... but not quite?
Assume $c,r\neq0$ for the key cases.
This is an very complicated ODE.
Some special cases:
$r=1$ :
$x^2y''-\dfrac{xy'}{2}-cx^\frac{5}{2}y=0$
ODE of the form http://eqworld.ipmnet.ru/en/solutions/ode/ode0209.pdf and can transform to Bessel ODE.
$r=2$ :
$x^4y''-x^2y'-cx^5y=0$
$x^2y''-y'-cx^3y=0$