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

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

We have,

$y e^{x / y} d x=\left(x e^{x / y}+y^2 \sin y\right) d y$

$\Rightarrow (y d x-x d y) e^{x / y}=y^2 \sin y d y$

$\Rightarrow e^{x / y}\left\{\frac{y d x-x d y}{y^2}\right\}=\sin y d y$

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

On integrating, we obtain

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