Question

The solution of differential equation $\frac{d y}{d x}=\cos x(2-y \operatorname{cosec} x)$ is

• A. $y=\tan \frac{x}{2}+\cot \frac{x}{2}+C$
• B. $y=\frac{1}{\sqrt{2}} \sec \frac{x}{2}+\sqrt{2} \cos \frac{x}{2}+C$
• C. $y=\sin x+C \operatorname{cosec} x$
• D. $y=\sin x-\cos x+C$

Answer 1

Answer: (C)

$\frac{d y}{d x}=2 \cos x-y \cot x$
$ \frac{d y}{d x}+y \cot x=2 \cos x$
IF $=e^{\int \cot x d x}=e^{\ell n \sin x}=\sin x$
complete solution
$y \cdot \sin x=\int \sin x \cdot 2 \cos x d x$
$y \sin x=\sin ^2 x+C$
$y=\sin x+C \operatorname{cosec} x$