Question

The integrating factor of differential equation $\cos xdy=y(\sin x-y)dx$ is

• A. $\tan x$
• B. $\sec x$
• C. $e^{\tan x}$
• D. $e^{\sec x}$

Answer 1

Answer: (B)

$\sec x$

$\frac{dy}{dx}=y\tan x-y^2\sec x$

$\Rightarrow$ $\frac{dy}{dx}-y\tan x=-y^2\sec x$

$\Rightarrow$ $\frac{1}{y^2}\frac{dy}{dx}-\frac{1}{y}\tan x=-\sec x$ (*)

Substitute $\frac{1}{y}=t$

$\Rightarrow$ $-\frac{1}{y^2}\frac{dy}{dx}=\frac{dt}{dx}$ $\therefore$ By (*) we have

$\frac{dt}{dx}+t\tan x=\sec x$, which is linear in t

$\therefore$ I.F. $=e^{\int \tan xdx}=e^{\log \left(\sec x\right)}=\sec x$