Mathematics - Differential Equation Question with Solution | TestHub

$\frac{dy}{dx}+\frac{y}{2}\sec x=\frac{\tan x}{2y}$
$2y\frac{dy}{dx}+y^2\sec x=\tan x$
Put$y^2=t \Rightarrow 2y\frac{dy}{dx}=\frac{dt}{dx}$
$\frac{dt}{dx}+t\sec x=\tan x$
$I.F=e^{\int \sec xdx}=e^{\ln (\sec x+tanx)}=\sec x+\tan x$
$\Rightarrow t\left(\sec x+\tan x\right)=\int \left(\sec x+\tan x\right)\tan xdx$
$\Rightarrow t\left(\sec x+\tan x\right)=\int \left(\sec x\tan x\right)dx+\int {\tan }^{2}xdx$
$\Rightarrow y^2\left(\sec x+\tan x\right)=\sec x+\tan x-x+c$
$y\left(0\right)=1 \Rightarrow c=0$
$\Rightarrow y^2=1-\frac{x}{\sec x+\tan x}$