I have the differential equation

$$y''y+n(y')^2=0\tag 1$$

I haven't found any references to equations of this type by searching.  Does anyone have a suggestion as to how to proceed?

It comes about from the following progression:

$$y=(y')^n-\frac 1n\tag 2\\
y'=ny''(y')^{n-1}\implies1=ny''(y')^{n-2}\\
0=n(n-2)(y'')^2(y')^{n-3}+ny'''(y')^{n-2}\\
0=(n-2)(y'')^2+y'''y'$$

Taking $y$ in place of $y'$ and $n$ in place of $n-2$ I get the form shown in $(1)$.  Are there any good ways to solve any of the other intermediate forms, or even the initial form shown in $(2)$?

Further constraints:

$y(0)\approx 1.27$ is an otherwise-unknown constant that I am attempting to build an expression for.

The initial problem is the recurrence relation with $a_1\approx 1.27$ the unknown constant and

$$a_{n+1}=(a_n)^{n+1}-\frac 1{n+1}$$

with the additional constraint $\lim_{n\to\infty}a_n=1$.


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