If y = y(x) is the solution of the differential equation (dy)/(dx) + ( x)y = x, 0 ≤ x ≤ (π)/(3), with y(0) = 0, then y((π)/(4)) is equal to

Correct answer: B. ( 1 2 2 ) _ e 2 (dy)/(dx) + ( x)y = x; 0 ≤ x ≤ (π)/(3) I . F . = e ^ ∫ tanxdx = e ^ x = x ysecx = ∫ tanxdx + C ysecx = | x| + C x = 0, y = 0 ⇒ C = 0 ysecx = | x| y = x · | x| y _ x = (π)/(4) = ( 1 2 ) · ln 2 y _ x = (π)/(4) = 1 2 2 _ e 2

JEE Main 202116 MarchEvening Shift

Exam: JEE Main
Year: 2021
Date: 16 March
Shift: Evening Shift
Subject: Mathematics
Chapter: Differential Equations

JEE Main 2021 · 16 March · Evening Shift · Mathematics · Differential Equations

If y = y(x) is the solution of the differential equation (dy)/(dx)...

If $y = y\left(x\right) is the solution of the differential equation$ \dfrac{dy}{dx} + \left(\tan x\right)y = \sin x, 0 \le x \le \dfrac{\pi}{3}, $with$ y\left(0\right) = 0, then $y\left(\dfrac{\pi}{4}\right)$ is equal to

Options

A $\dfrac{1}{4}\log_{e}2$
B
$\left(\dfrac{1}{2\sqrt{2}}\right)\log_{e}2$
C $\log_{e}2$
D $\dfrac{1}{2}\log_{e}2$

Correct answer

$\left(\dfrac{1}{2\sqrt{2}}\right)\log_{e}2$

Step-by-step solution

$\dfrac{dy}{dx} + \left(\tan x\right)y = \sin x; 0 \le x \le \dfrac{\pi}{3}$

$I . F . = {e}^{∫ tanxdx} = {e}^{\ln \sec x} = \sec x$

$ysecx = ∫ tanxdx + C$

$ysecx = \ln \left|\sec x\right| + C$

$x = 0, y = 0 ⇒ C = 0$

$ysecx = \ln \left|\sec x\right|$

$y = \cos x \cdot \ln \left|\sec x\right|$

${y}_{x = \dfrac{\pi}{4}} = \left(\dfrac{1}{\sqrt{2}}\right) \cdot ln\sqrt{2}$

${y}_{x = \dfrac{\pi}{4}} = \dfrac{1}{2\sqrt{2}}\log_{e}2$

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