Question

The integrating factor of $x\cos x\left(\frac{dy}{dx}\right)+(x\sin x+\cos x)y=1$ is

• A. $x\cos x$
• B. $x\sin x$
• C. $x\sec x$
• D. $x\csc x$

Answer 1

Answer: (C)

$x\sec x$

$x\cos x\frac{dy}{dx}+\left(x\sin x+\cos x\right)y=1$

$\Rightarrow \frac{dy}{dx}+\frac{\left(x\sin x+\cos x\right)y}{x\cos x}=\frac{1}{x\cos x}$

It is first degree linear D.E on comparing with

$\frac{dy}{dx}+Py=Q$ ,we get

$P=\frac{\left(x\sin x+\cos x\right)}{x\cos x}$

$=\tan x+\frac{1}{x}$

$I.F=e^{\int Pdx}$

$=e^{\int \left(\tan x+\frac{1}{x}\right)dx}$

$=e^{\left(\log \left(\sec x\right)+\log \left(x\right)\right)}$

$=e^{\log \left(x\sec x\right)}$

$=x\sec x$