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$The general solution of the differential equation y (x^{2} y + e^{x}) dx - e^{x} dy = 0 is$

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$y^{3} {x-3e}^{y} =cx$

$x^{3} {y+3e}^{x} =3cy$

$x^{3} {y-3e}^{x} =cy$

$y^{3} {x+3e}^{y} =cx$

answer is B.

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Detailed Solution

$\begin{array}{l} x^{2} dx + \frac{{ye}^{x} dx - e^{x} dy}{y^{2}} = 0 \to \int x^{2} dx + \int d (e^{x} / y) = \int 0 \\ \to \frac{x^{3}}{3} + \frac{e^{x}}{y} = c \to x^{3} y + 3 e^{x} = 3 cy \end{array}$

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