Question

Find the Integrating factor of differential equation given that $\tan x\frac{dy}{dx}+y=2x\tan x+x^2$

Answer 1

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

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

Here, $P=\frac{1}{\tan x}=\cot xQ=v=2x+x^2\cot x$

Therefore,

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