solve the Following Differential equation: $y''=2yy'$ Solve the following equation: $y''=2yy'$

My attempt:

$y''=2yy'$

integrate on both sides:

$y'=\int(2yy')$

Let y'=s

We have $\int 2y \dfrac{ds}{dy}dy = 2 \int yds = y^2$

Is that a way to integrate $2yy'$?

I fthat's the case don't we get $y=\dfrac{1}{3}y^5$ which is false
Indeed,

$$\frac{dy'}{dx}=2y\frac{dy}{dx}$$ 
can be integrated by

$$\int dy'=\int2y\,dy,$$ 

$$y'=y^2+C.$$

Then

$$\frac{dy}{dx}=y^2+c^2,$$

$$\int\frac{dy}{y^2+c^2}=dx,$$

$$\frac1c\arctan\left(\frac yc\right)=x+c',$$

$$y=c\tan(cx+c').$$

Note that the constant was denoted as $c^2$ to ensure positiveness. A similar solution holds for a negative constant, involving the hyperbolic tangent.
Yeah. More generally if you have a 2nd degree nonlinear autonomous equation (no independent variable) do the following substitutions

y = y

y' = s

y'' = s*ds/dy

This reduces the order.