Is it possible to separate two variables? Here is the following problem:
$$ (\frac{du}{dv})^2=u(v-u)$$
Is it possible to separate these two variables?
 A: $\left(\dfrac{du}{dv}\right)^2=u(v-u)$
$\dfrac{du}{dv}=\pm\sqrt u\sqrt{v-u}$
$\pm\sqrt{v-u}\dfrac{dv}{du}=\dfrac{1}{\sqrt u}$
Let $w=\pm\sqrt{v-u}$ ,
Then $v=w^2+u$
$\dfrac{dv}{du}=2w\dfrac{dw}{du}+1$
$\therefore w\left(2w\dfrac{dw}{du}+1\right)=\dfrac{1}{\sqrt u}$
$2w^2\dfrac{dw}{du}+w=\dfrac{1}{\sqrt u}$
Let $x=\sqrt u$ ,
Then $\dfrac{dw}{du}=\dfrac{dw}{dx}\dfrac{dx}{du}=\dfrac{1}{2\sqrt u}\dfrac{dw}{dx}=\dfrac{1}{2x}\dfrac{dw}{dx}$
$\therefore\dfrac{w^2}{x}\dfrac{dw}{dx}+w=\dfrac{1}{x}$
$\dfrac{w^2}{x}\dfrac{dw}{dx}=\dfrac{1}{x}-w$
$\dfrac{w^2}{x}\dfrac{dw}{dx}=\dfrac{1-wx}{x}$
$\left(\dfrac{1}{w}-x\right)\dfrac{dx}{dw}=w$
This belongs to an Abel equation of the second kind.
Let $z=\dfrac{1}{w}-x$ ,
Then $x=\dfrac{1}{w}-z$
$\dfrac{dx}{dw}=-\dfrac{1}{w^2}-\dfrac{dz}{dw}$
$\therefore z\left(-\dfrac{1}{w^2}-\dfrac{dz}{dw}\right)=w$
$-\dfrac{z}{w^2}-z\dfrac{dz}{dw}=w$
$z\dfrac{dz}{dw}=-\dfrac{z}{w^2}-w$
Let $t=\dfrac{1}{w}$ ,
Then $\dfrac{dz}{dw}=\dfrac{dz}{dt}\dfrac{dt}{dw}=-\dfrac{1}{w^2}\dfrac{dz}{dt}$
$\therefore-\dfrac{z}{w^2}\dfrac{dz}{dt}=-\dfrac{z}{w^2}-w$
$z\dfrac{dz}{dt}=z+w^3$
$z\dfrac{dz}{dt}-z=\dfrac{1}{t^3}$
This belongs to an Abel equation of the second kind in the canonical form.
Please follow the method in https://arxiv.org/ftp/arxiv/papers/1503/1503.05929.pdf or in http://www.iaeng.org/IJAM/issues_v43/issue_3/IJAM_43_3_01.pdf
