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 $\lim_{n\to\infty}a_n=1$.


Question: BJT switch connections
Answer: If the load is connected between the emitter and ground then the configuration is what is known as common collector and in that configuration the emitter voltage is approximately 0.6v lower than the base voltage.



It is also called an emitter follower because the emitter voltage always "follows" the voltage of the base minus the base-emitter diode voltage drop (about 0.6v).

This mode is not convenient when you try to switch a load that uses a voltage higher than the voltage you connect to the base, so if you try to drive a 12v relay connected to the emitter with a microcontroller that feeds the base of the transistor with 3.3v or 5v then you will see that the relay actually gets 2.7v or 4.4v (for 3.3v and 5v input) which is much lower than needed.

On the other hand it can be very useful if you want to make a regulator, you can use a zener to drive the base with a constant voltage and the emitter output will keep a constant voltage approximately 0.6v lower than the base.