Solve the differential equation:
$(x - sin y) d y + tan y d x = 0$

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Solution

$(x - s i n y) d y + t a n y d x = 0$
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$\Rightarrow \frac{d x}{d y} + \frac{x - s i n y}{t a n y} = 0$

$\Rightarrow \frac{d x}{d y} + x c o t y = \frac{1}{s e c y} = c o s y$

$\Rightarrow \frac{d x}{d y} + x c o t y = c o s y$

Integrating factor $= e^{\int c o t y d y} = e^{l o g_{e} s i n y} = s i n y$

$∴ x s i n y = \int c o s y . s i n y d y = \frac{1}{2} \int 2 s i n y c o s y d y$
$= \frac{1}{2} \int s i n 2 y d y$
$= \frac{1}{2} \frac{(- c o s 2 y)}{2} + c$

$\Rightarrow x s i n y = \frac{- 1}{4} c o s 2 y + c$

$\Rightarrow x s i n y + \frac{1}{4} c o s 2 y = c$

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