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Question #41496

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be the solution curve of the differential equation $\left(xy \cos\left(\frac{y}{x}\right) - x^2 \sin\left(\frac{y}{x}\right)\right) \frac{dy}{dx} = y^2 \cos\left(\frac{y}{x}\right) - xy \sin\left(\frac{y}{x}\right) + y^2$ for $x > 0$ , satisfying $y(1) = \frac{\pi}{2}$ . If $y(\alpha) = \frac{\pi}{6} \alpha$ , then the value of $\alpha$ is

Options

A.
$e^{1/\pi}$
B.
$e^{2/\pi}$
C.
$e^{3/\pi}$
D.
$e^{1/(2\pi)}$

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

Integration by Parts (LIATE Rule)|Substitution Logic in Homogeneous DE|Homogeneous Function Homogeneity

Why This Question Is Tricky

Common Mistakes

This question constructs a lethal trap within a standard homogeneous differential equation.

By substituting $y=vx$ , students arrive at the integral $\int \frac{v \cos v - \sin v}{v^2} dv$ .

The trap triggers when students split this into $\int \frac{\cos v}{v} dv - \int \frac{\sin v}{v^2} dv$ and blindly apply the LIATE rule.

The first term is the non-elementary Cosine Integral $\text{Ci}(v)$ , which creates an illusion of unsolvability if evaluated in isolation.

The core JEE Advanced logic tested here is recognizing that the homogeneous structure often yields an exact differential bypass—specifically, the telescoping nature of integration by parts where $d(\frac{\sin v}{v})$ perfectly resolves the integrand without circular loops.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Integration by Parts (LIATE Rule)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free