Let y=y(x) be a solution of the differential equation (x cos x) d y+(x y sin x+y cos x-1) d x=0,0 < x < (π)/2. If (π)/3 y((π)/3)=√(3), then |(π)/6 y^()((π)/6)+2 y^()((π)/6)| is equal to _ _ _ _ _ _.

Differential Equations: JEE Main 2023 Mathematics question

JEE Main 2023 · 6 Apr, Morning shift · Q34

JEE Main 20236 Apr, Morning shiftMathematicsMediumDifferential Equations163× since 2002

Let

y=y(x)

be a solution of the differential equation

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

. If

\frac{\pi}{3} y\left(\frac{\pi}{3}\right)=\sqrt{3}

, then

\left|\frac{\pi}{6} y^{\prime \prime}\left(\frac{\pi}{6}\right)+2 y^{\prime}\left(\frac{\pi}{6}\right)\right|

is equal to ____________.

Your answerType your answer. Integers and decimals are accepted.