The solution of differential equation $(e^{x} + 1) y d y = (y + 1) e^{x} d x$ is :

(e^{x} + 1) (y + 1) = c e^{y}

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(e^{x} + 1) (y + 1) = c e^{- y}

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(e^{x} + 1) (y + 1) = c e^{2 y}

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none of the above

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Solution

The correct option is A $(e^{x} + 1) (y + 1) = c e^{y}$
The given differential equation is $(e^{x} + 1) y d y = (y + 1) e^{x} d x$
$\Rightarrow \frac{y d y}{(y + 1)} = \frac{e^{x}}{(e^{x} + 1)} d x;$ Integrating both sides
$\Rightarrow y - l o g | y + 1 | = l o g (e^{x} + 1) + l o g k$
$\Rightarrow y = l o g | (y + 1) (e^{x} + 1) | + l o g k$
$\Rightarrow (y + 1) (e^{x} + 1) = c e^{y}$

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