The general solution of the differential equation y $(x^{2} y + e^{x}) d x - e^{x} d y = 0$ is ___ {c is arbitrary constant}

x^{3} y - 3 e^{x} = c y

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x^{3} y + 3 e^{x} = 3 c y

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y^{3} x - 3 e^{y} = c x

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x y^{3} + 3 e^{x} = 3 c y

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Solution

The correct option is B $x^{3} y + 3 e^{x} = 3 c y$
$(x^{2} y^{2} + e^{x} y) d x - e^{x} d y = 0$
$x^{2} d x + \frac{y e^{x} d x - e^{x} d y}{y^{2}} = 0$
$x^{2} d x + d (\frac{e^{x}}{y}) = 0$
$\frac{x^{3}}{3} + \frac{e^{x}}{y} = c \to y x^{3} + 3 e^{x} = 3 c y$

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