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Question #41607

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be the solution to the differential equation $x \frac{dy}{dx} + (1 - 2x|x-1|) y = x^2(1-x) y^2$ for $x > 0$ , satisfying the condition $\lim_{x \to 0} x y(x) = \frac{2}{2e - 1}$ . Find the value of $y(2)$ .

Options

A.
$1$
B.
$\frac{1}{2e - 1}$
C.
$\frac{2}{2e - 1}$
D.
$e - \frac{1}{2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Bernoulli Equation Transformation|Integrating f(x) + xf'(x) forms|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize the $d(xy)$ exact differential form hidden within a Bernoulli equation, followed by a critical modulus trap in the resulting linear differential equation.

The integrating factor depends on $|x-1|$ , forcing a domain bifurcation at $x=1$ .

Students who blindly integrate without splitting the domain will incorrectly assume a single constant of integration, leading to the trap answer $1/(2e-1)$ .

The problem rigorously evaluates piecewise continuity of the transformed variable $u(x)$ to find the correct global solution.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Bernoulli Equation Transformation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free