Solution of the differential equation $y\, e^{x/y}dx= (x\, e^{x/y}+y^2 sin y )dy, $ is

$e^{x/y}=-cos\, y + C$

The correct answer is option (4) : $e^{x/y}=-cos\, y + C$

$y\, e^{x/y}dx= (x\, e^{x/y}+y^2 sin y )dy$

$⇒(y\, dx-x\, dy)e^{x/y}e^{x/y}=y^2sin\, y \, dy$

$⇒e^{x/y}\begin{Bmatrix}\frac{y-dx-xdy}{y^2}\end{Bmatrix}= sin y\, dy $

$⇒e^{x/y}d\left(\frac{x}{y}\right) = - d(cos y )$

On integrating, we obtain

$e^{x/y}= -cos y + C$, as the required solution.