If y=y(x) is the solution of the differential equation d y ~d x +2 y= (2 x), y(0)=(3)/(4) , then y ((π)/(8)) is equal to:

Correct answer: C. e^ -π / 4 aligned & (d y)/(d x)+2 y= 2 x, y(0)=(3)/(4) & I.F =e^ ∫ 2 d x =e^ 2 x & y · e^ 2 x =∫ e^ 2 x 2 x d x & y · e^ 2 x = e^ 2 x (2 2 x-2 2 x) 4+4 +C & x=0, y=(3)/(4) (3)/(4) · 1=(1(0-2))/(8)+C & (3)/(4)=-(1)/(4)+C & 1=C & y=(2 2 x-2 2 x)/(8)+1 · e^ -2 x & x=(π)/(8), y=(1)/(8)(2 (π)/(4)-2 (π)/(4))+e^ -2((π)/(8)) & y=0+e^ -(π)/(4) aligned

JEE Main 20245 AprilMorning Shift

Exam: JEE Main
Year: 2024
Date: 5 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Differential Equations

JEE Main 2024 · 5 April · Morning Shift · Mathematics · Differential Equations

If y=y(x) is the solution of the differential equation d y ~d x +2...

y=y(x) is the solution of the differential equation

\frac{\mathrm{d} y}{\mathrm{~d} x}+2 y=\sin (2 x), y(0)=\frac{3}{4}

, then

\left(\frac{\pi}{8}\right)

is equal to:

Options

A $\mathrm{e}^{\pi / 8}$
B $\mathrm{e}^{\pi / 4}$
C $\mathrm{e}^{-\pi / 4}$
D $\mathrm{e}^{-\pi / 8}$

Correct answer

C. $\mathrm{e}^{-\pi / 4}$

Step-by-step solution

\begin{aligned} & \frac{d y}{d x}+2 y=\sin 2 x, y(0)=\frac{3}{4} \\ & \text{ I.F }=e^{\int 2 d x}=e^{2 x} \\ & y \cdot e^{2 x}=\int e^{2 x} \sin 2 x d x \\ & y \cdot e^{2 x}=\frac{e^{2 x}(2 \sin 2 x-2 \cos 2 x)}{4+4}+C \\ & x=0, y=\frac{3}{4} \Rightarrow \frac{3}{4} \cdot 1=\frac{1(0-2)}{8}+C \\ & \frac{3}{4}=-\frac{1}{4}+C \\ & 1=C \\ & y=\frac{2 \sin 2 x-2 \cos 2 x}{8}+1 \cdot e^{-2 x} \\ & x=\frac{\pi}{8}, y=\frac{1}{8}\left(2 \sin \frac{\pi}{4}-2 \cos \frac{\pi}{4}\right)+e^{-2\left(\frac{\pi}{8}\right)} \\ & y=0+e^{-\frac{\pi}{4}}\end{aligned}

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