The solution of $(y (1 + x^{- 1}) + s i n y) d x + (x + l o g x + x c o s y) d y = 0$ is

(1 + y^{- 1} s i n y) + x^{- 1} l o g x = C

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(y + s i n y) + x y l o g x = C

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x y + y l o g x + x s i n y = C

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Solution

The correct option is C $x y + y l o g x + x s i n y = C$
The given equation can be written as $y (1 + x^{- 1}) d x + (x + l o g x) d y + s i n y d x + x c o s y d y = 0$
$\Rightarrow d (y (x + l o g x)) + d (x s i n y) = 0 \to y (x + l o g x) + x s i n y = C$

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(y (1 + x^{- 1}) + s i n y) d x + (x + l o g x + c o s y) d y = 0

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