A ridiculously awesome non linear differential equation

Non linear DEs often have fascinating solution developments and this one here is no exception. The icing on the cake is the function solving the equation being far from boring.

[archiebrew]

You can get a cool solution for y in terms of x using the Lambert W function if you want an extra challenge

[videolome]

That sum of arctangents is not true in general

[shivanshnigam]

I think "A" is now the most powerful constant ever to exist

[kshitishp]

I think the following method would be simpler and it gives a different answer .
We have, y''(1+y²)+(y')³+y'=0
Taking the y' out,

y''(1+y²)= - y'(1+(y')²)
Rearranging -
y''/(1+(y')²) = - y'/(1+y²)
integrating both sides w.r.t dy' and dy respectively, we get -
y' = -y from which we can say
y = e^-x .
You can even check it's validity by substituting in the equation .

[SuperSilver]

This is probably a dumb question, but when using the arc tangent sum formula in this problem, do we have to consider all of the possible cases for where we need to add or subtract a constant term pi? For example the formula

arctan(u)+arctan(y) = arctan((u+y)/(1-u*y)) when u*y<1

But this formula changes changes when u*y>1, you just add or subtract a pi based on the sign of u I think? Would that pi just become absorbed into A and you can proceed with problem as normal?

In general, would you have to consider those cases if say the constant A wasn’t there to begin with?

[erfanmohagheghian]

You should mention that the arctan identity you're using is correct up to a constant and it's not the same for all combinations of u and y

[giuseppemalaguti]

Le equazioni differenziali non è il mio forte..le ho studiate 30anni fa...

[Dheeraj]

That seems to be terrible at first glance but as the video progresses it becomes easy to solve.
😂🎉❤ Thanks 🙏 bro for that videos