TITLE: Finding solutions of the differential equation $x\frac{d}{d x}\left(x\frac{d y}{d x}\right)=4(y-x^{2}y+y^{3})$
QUESTION [7 upvotes]: In my research I have come across the following non-linear differential equation:
$$x\frac{d}{d x}\left(x\frac{d y}{d x}\right)=4(y-x^{2}y+y^{3})$$
I want to find the general solution of this equation for $x\geq0$, but didn't manage to do it.
I have found some facts about the solutions by investigating the asymptotics:

For $x\ll1$, the solutions go like $y=Cx^2$ for some constant $C$, or diverge.
For $x\gg1$, the solutions go like $y=x$, or go to $0$ (probably after oscillating, found by numerics), or diverge.

In particular, I am interested in a one of the solutions. By looking upon the numerics, I believe that there is only one solution that goes like $y=x$ for $x\gg1$, and it does not diverge at $x=0$. I didn't manage to prove this by now.
I am interested in finding this solution analytically, or if impossible, just find appropriate $C$ analytically.
Thanks!

REPLY [3 votes]: Is this too naive? To obtain the large-$x$ behavior, I take a series expansion of $y(x)$ in powers of $1/x$,
$$y(x)=c x + a_0 + a_1/x + a_2/x^2+ a_3/x^3 +\cdots,$$
substitute into
$$R(x)=x\frac{d}{d x}\left(x\frac{d y}{d x}\right)-4(y-x^{2}y+y^{3}),$$
expand $R(x)$ in powers of $1/x$ and demand that the leading order terms vanish. It follows that $c=1$, $a_{n}=0$ for any even $n$, while for odd $n$ I find
$$a_1=-\frac{3}{8},\;\;a_3=-\frac{9}{128},a_5=-\frac{99}{1024},\;\;a_7=-\frac{11637}{32768},\;\;a_9=-\frac{627669}{262144},$$
so coefficients of the form $a_{2n+1}=-b_n/2^{2n+1}$ with integer $b_n$.