Question

Fine the integrating factor for the differential equation $cotx\frac{dy}{dx}+y=2x+x^2$

Answer 1

$\cot x\frac{dy}{dx}+y=2x+x^2$

$\frac{dy}{dx}+\frac{y}{\cot x}=2x+x^2$

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

Here, $P=\tan x,Q=2x+x^2$

Therefore,

Integrating factor $=e^{\int Pdx}=e^{\int \tan xdx}=e^{\ln \left(\cos x\right)}=\cos x$

Hence the integral factor of the given function is $\cos x$.