The integrating factor of the differential equation siny $\left(\frac{\mathrm{d} y}{\mathrm{~d}…
The integrating factor of the differential equation siny $\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)=\operatorname{cosy}(1-x \cos y)$ is
$\mathrm{e}^{-x}$
$\mathrm{e}^{-\cos y}$
$\mathrm{e}^{-y}$
$\mathrm{e}^{\text {siny }}$
Solution
$\sin y \frac{d y}{d x}=\cos y(1-x \cos y) \Rightarrow \sin y \frac{d y}{d x}=\cos y-x \cos ^{2} y$
Dividing both sides by $\cos ^{2} y$, we get
$\sec y \tan y \frac{d y}{d x}=\sec y-x$
$\sec y \cdot \tan y \frac{d y}{d x}-\sec y=-x$
Put $\sec y=v \Rightarrow \sec y \tan y d y=d x$
$\therefore \frac{\mathrm{dv}}{\mathrm{d} \mathrm{x}}-\mathrm{v}=-\mathrm{x}$...(1)
$\therefore$ I.F. $\quad=\mathrm{e}^{-\int 1 \mathrm{dx}}=\mathrm{e}^{-\mathrm{x}}$