Question

For each the differential equations given, find the general solution :
$\frac{dy}{dx}+\left(\sec x\right)y=\tan x(0\le x\le \frac{\pi }{2})$

Answer 1

$\frac{dy}{dx}+\left(\sec x\right)y=\tan x$

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

$y(\sec x+\tan x)=\int (\sec x\tan x+{\tan }^{1}x)$

$=\sec x+\int {\sec }^{2}x-1$

$=\sec x+\tan -x$