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

e^{x / y} = - sin y + C

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

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e^{y / x} = cos y + C

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e^{y / x} = - sin y + C

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Solution

The correct option is B $e^{x / y} = - cos y + C$
$y e^{x / y} d x = (x e^{x / y} + y^{2} sin y) d y$
$\Rightarrow (y d x - x d y) e^{x / y} = y^{2} sin y d y$
$\Rightarrow (\frac{y d x - x d y}{y^{2}}) e^{x / y} = sin y d y$
$\Rightarrow e^{x / y} d (\frac{x}{y}) = sin y d y$
Integrating both side we get, $e^{x / y} = - cos y + c$

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