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The solution of $y e^{- x / y} d x - (x e^{- x / y} + y^{3}) d y = 0$ is:

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$\frac{x^{2}}{2} + e^{x / y} = k$

$\frac{y^{2}}{2} + e^{x / y} = k$

$\frac{y^{2}}{2} + e^{- x / y} = k$

$\frac{x^{2}}{2} + e^{- x / y} = k$

answer is A.

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Detailed Solution

$y e^{- x / y} d x - (x e^{- x / y} + y^{3}) d y = 0 e^{- x / y} (y d x - x d y) = y^{3} d y$

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

$e^{- x / y} d (\frac{x}{y}) = y d y .$ Integrating both sides, we get

$\begin{array}{r} k - e^{- x / y} = \frac{y^{2}}{2} \\ \Rightarrow \frac{y^{2}}{2} + e^{- x / y} = k \end{array}$

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