Let y=y(x) be the solution of the differential equation (d y)/(d x)+3( ^2 x) y+3 y= ^2 x y(0)=(1)/(3)+e^3 . Then y((π)/(4)) is equal to

Correct answer: B. (4)/(3) aligned & (d y)/(d x)+3( ^2 x) y= ^2 x, y(0)=(1)/(3)+e^3 & If =e^ 3 ∫ ^2 x d x =e^ 3 x aligned Solution is aligned & e^ 3 x y=∫ e^ 3 x ^2 x d x & e^ 3 x y= e^ 3 x 3 +c & y(0)=(1)/(3)+e^3 c=e^3 & y((π)/(4))=((e^3)/(3)+e^3)/(e^3)=(4)/(3) aligned

JEE Main 20253 AprilEvening Shift

Exam: JEE Main
Year: 2025
Date: 3 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Differential Equations

JEE Main 2025 · 3 April · Evening Shift · Mathematics · Differential Equations

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

Let

y=y(x)

be the solution of the differential equation

\frac{d y}{d x}+3\left(\tan ^2 x\right) y+3 y=\sec ^2 x

y(0)=\frac{1}{3}+e^3

. Then

y\left(\frac{\pi}{4}\right)

is equal to

Options

A $\frac{2}{3}$
B $\frac{4}{3}$
C $\frac{4}{3}+\mathrm{e}^3$
D $\frac{2}{3}+\mathrm{e}^3$

Correct answer

B. $\frac{4}{3}$

Step-by-step solution

\begin{aligned}& \frac{d y}{d x}+3\left(\sec ^2 x\right) y=\sec ^2 x, y(0)=\frac{1}{3}+e^3 \\ & \text{ If }=e^{3 \int \sec ^2 x d x}=e^{3 \tan x}\end{aligned}

\therefore

Solution is

\begin{aligned}& e^{3 \tan x} y=\int e^{3 \tan x} \sec ^2 x d x \\ & e^{3 \tan x} y=\frac{e^{3 \tan x}}{3}+c \\ & \because y(0)=\frac{1}{3}+e^3 \Rightarrow c=e^3 \\ & \therefore y\left(\frac{\pi}{4}\right)=\frac{\frac{e^3}{3}+e^3}{e^3}=\frac{4}{3}\end{aligned}

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