Let x=x(y) be the solution of the differential equation y=(x-y ~d x ~d y ) ((x)/(y)), y 0 and x(1)=(π)/(2) . Then (x(2)) is equal to :

Correct answer: D. 2( _e 2)^2-1 aligned & y d y=(x d y-y d x) ((x)/(y)) & (d y)/(y)=((x d y-y d x)/(y^2)) ((x)/(y)) & (d y)/(y)= ((x)/(y)) d(-(x)/(y)) & arrow n y= (x)/(y)+C aligned aligned & x(1)=(π)/(2) 0= (π)/(2)+C C=0 & y = (x)/(y) & but y=2 (x)/(2)= n 2 & x=2 ^2 (x)/(2)-1 & =2( 2)^2-1 aligned

JEE Main 202523 JanuaryEvening Shift

Exam: JEE Main
Year: 2025
Date: 23 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Differential Equations

JEE Main 2025 · 23 January · Evening Shift · Mathematics · Differential Equations

Let x=x(y) be the solution of the differential equation y=(x-y ~d x...

Let

x=x(y)

be the solution of the differential equation

y=\left(x-y \frac{\mathrm{~d} x}{\mathrm{~d} y}\right) \sin \left(\frac{x}{y}\right), y\gt0

and

x(1)=\frac{\pi}{2}

. Then

\cos (x(2))

is equal to :

Options

A $1-2\left(\log _e 2\right)^2$
B $1-2\left(\log _{\mathrm{e}} 2\right)$
C $2\left(\log _e 2\right)-1$
D $2\left(\log _e 2\right)^2-1$

Correct answer

D. $2\left(\log _e 2\right)^2-1$

Step-by-step solution

\begin{aligned} & y d y=(x d y-y d x) \sin \left(\frac{x}{y}\right) \\ & \frac{d y}{y}=\left(\frac{x d y-y d x}{y^2}\right) \sin \left(\frac{x}{y}\right) \\ & \frac{d y}{y}=\sin \left(\frac{x}{y}\right) d\left(-\frac{x}{y}\right) \\ & \rightarrow \ell n y=\cos \frac{x}{y}+C\end{aligned}

\begin{aligned} & x(1)=\frac{\pi}{2} \Rightarrow 0=\cos \frac{\pi}{2}+C \Rightarrow C=0 \\ & \text{ \ln y }=\cos \frac{x}{y} \\ & \text{ but } y=2 \Rightarrow \cos \frac{x}{2}=\ell n 2 \\ & \cos x=2 \cos ^2 \frac{x}{2}-1 \\ & =2(\ln 2)^2-1\end{aligned}

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