JEE 2023 — Mathematics PYQ

(x\cos x)dy+(xy\sin x+y\cos x-1)dx=0, 0<x<\frac{\pi}{2}

$\Rightarrow \frac{dy}{dx}+\left(\frac{x\sin x+\cos x}{x\cos x}\right)y=\frac{1}{x\cos x}$

I.F.$\int$ $\frac{x\sin x+\cos x}{x\cos x}$ $dx$
II.F.$\int$ Q[IF] $dy=\int y\sec xdx=\int \frac{y{\sec }^{2}x}{x\cos x}dx=\tan x+C$
Since $y\left(\frac{\pi }{3}\right)=\frac{3\sqrt{3}}{\pi }$ Hence $C=\sqrt{3}$
Hence $\frac{\pi }{6}y\left(\frac{\pi }{6}\right)+y\left(\frac{\pi }{6}\right)=-|2|=2$