Question

The integrating factor of the differential equation $x\frac{dy}{dx}+y-x+xy\cot x=0,x\ne 0$ can be

• A. $x\sin x,x\in (0,\frac{\pi }{2})$
• B. $x\text{cosec}x,x\in (0,\frac{\pi }{2})$
• C. $x\sin x,x\in (\pi ,\frac{3\pi }{2})$
• D. $-x\sin x,x\in (0,\frac{\pi }{2})$

Answer 1

The correct option is A $x\sin x,x\in (0,\frac{\pi }{2})$

$x\frac{dy}{dx}+y-x+xy\cot x=0,x\ne 0$
$\Rightarrow \frac{dy}{dx}+y(\frac{1}{x}+\cot x)=1$

$I.F.=e^{\int (\frac{1}{x}+\cot x)dx}$
$\Rightarrow I.F.=e^{\ln |x|+\ln |\sin x|}=|x\sin x|$