Let y=y(x) be the solution of the differential equation 2 x ~d y ~d x = 2 x-4 y x, x ∈(0, (π)/(2)) . If y((π)/(3))=0 , then y^ ((π)/(4))+y((π)/(4)) is equal to ________.

Correct answer: 0 aligned & (d y)/(d x)+2 y x= x & I.F. =e^ 2 ∫ x d x = ^2 x & y ^2 x=∫ ( x)/( ^2 x) d x & =∫ x x d x & = x+C & C=-2 aligned aligned & y= x-2 ^2 x & y((π)/(4))= 1 2 -1 & y^ =- x+4 x x & y^ ((π)/(4))=- 1 2 +2 & y^ ((π)/(4))+y((π)/(4))=1 aligned

JEE Main 202524 JanuaryEvening Shift

Exam: JEE Main
Year: 2025
Date: 24 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Differential Equations

JEE Main 2025 · 24 January · Evening Shift · Mathematics · Differential Equations

Let y=y(x) be the solution of the differential equation 2 x ~d y ~d...

Let

y=y(x)

be the solution of the differential equation

2 \cos x \frac{\mathrm{~d} y}{\mathrm{~d} x}=\sin 2 x-4 y \sin x, x \in\left(0, \frac{\pi}{2}\right)

. If

y\left(\frac{\pi}{3}\right)=0

, then

y^{\prime}\left(\frac{\pi}{4}\right)+y\left(\frac{\pi}{4}\right)

is equal to

\qquad

________.

Correct answer

Step-by-step solution

\begin{aligned} & \frac{d y}{d x}+2 y \tan x=\sin x \\ & \text{ I.F. }=e^{2 \int \tan x d x}=\sec ^2 x \\ & y \sec ^2 x=\int \frac{\sin x}{\cos ^2 x} d x \\ & =\int \tan x \sec x d x \\ & =\sec x+C \\ & C=-2\end{aligned}

\begin{aligned} & y=\cos x-2 \cos ^2 x \\ & y\left(\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}-1 \\ & y^{\prime}=-\sin x+4 \cos x \sin x \\ & y^{\prime}\left(\frac{\pi}{4}\right)=-\frac{1}{\sqrt{2}}+2 \\ & y^{\prime}\left(\frac{\pi}{4}\right)+y\left(\frac{\pi}{4}\right)=1\end{aligned}

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