Mathematics · Differential Equations

JEE Main 2026 — 2 April, Morning Shift — Question 34

If the curve y = f(x) passes through the point (1, e) and satisfies the differential equation dy = y (2 + log_e x) dx, x > 0, then f(e) is equal to :

Option A:
$e^e$
Option B:
$e^{e^2}$
Option C:
$e^{2e}$
Correct
Option D:
$e^{e/2}$

Answer: C

Step-by-step solution

$\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}(2+\ell \mathrm{nx})$ $\int \frac{\mathrm{dy}}{\mathrm{y}}=\int(2+\ell \mathrm{nx}) \mathrm{dx}$ $\ell \mathrm{ny}=2 \mathrm{x}+\mathrm{x} \ell \mathrm{nx}-\mathrm{x}+\mathrm{C}$ $\ell \mathrm{ny}=\mathrm{x}+\mathrm{x} \ell \mathrm{nx}+\mathrm{C}$ Since it passes through $(1, \mathrm{e})$ $1=1+0+C \Rightarrow C=0$ $\ell \mathrm{ny}=\mathrm{x}+\mathrm{x} \ell \mathrm{nx}$ $f(x)=y=e^{x+x \ln x}$ $\Rightarrow \mathrm{f}(\mathrm{e})=\mathrm{e}^{\mathrm{e} \mathrm{e}}=\mathrm{e}^{2 \mathrm{e}}$

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Exam: JEE Main 2026
Paper: 2 April, Morning Shift
Subject: Mathematics
Chapter: Differential Equations
Topic: Methods of solving a First Order,First Degree Differential

Let y = y(x) be the solution curve of the differential equation (1 + sin x) dy/dx + (y + 1) cos x = 0, y(0) = 0. If the curve y = y(x)…

Answer: C

Step-by-step solution

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