The solution of the differential equation $y\, dx + (x+ x^2 y ) dy = 0,$ is

$-\frac{1}{xy}+log\, y = C$

The correct answer is option (2) : $-\frac{1}{xy}+log\, y = C$

We have,

$ydx+(x+x^2 y)dy=0$

$⇒y\, dx + x\, dy + x^2 y\, dy = 0 $

$⇒\frac{d(xy)}{(xy)^2}+\frac{1}{y}dy = 0 $  [Dividing throughout by $(xy)^2 $]

On integrating, we get $-\frac{1}{xy}+\log \, y = C$