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**General** $\displaystyle

y^2(1-x^2)dx+x(x^2y+2x+y)dy=0

$

$\displaystyle

y^2dx-x^2y^2dx+x^3ydy+2x^2dy+xydy=0

$

Re-arrange:

$\displaystyle

y^2dx+xydy+2x^2dy+x^3ydy-x^2y^2dx=0

$

Taking common factors:

$\displaystyle

y(ydx+xdy)+2x^2dy+x^2y(xdy-ydx)=0

$

Or:

$\displaystyle

yd(xy)+2x^2dy+x^4y \, d\left(\dfrac{y}{x}\right)=0

$

Devide by $\displaystyle x^2y^3$ :

$\displaystyle

\dfrac{d(xy)}{(xy)^2}+2 \, \dfrac{dy}{y^3} + \dfrac{x^2}{y^2} \, d\left(\dfrac{y}{x}\right) = 0

$

Or:

$\displaystyle

\dfrac{d(xy)}{(xy)^2}+2 \, \dfrac{dy}{y^3} + \dfrac{1}{\left(\dfrac{y}{x}\right)^2} \, d\left(\dfrac{y}{x}\right) = 0

$

Integrate.