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Question #41593

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be a continuously differentiable function on $(0, 2\pi)$ satisfying the differential equation $\sin x \frac{dy}{dx} - y \cos x = \sin x \sqrt{1 - \cos^2 x}$ with the condition $y\left(\frac{\pi}{2}\right) = \frac{\pi}{2}$ . Then the value of $y\left(\frac{3\pi}{2}\right)$ is:

Options

A.
$-\frac{\pi}{2}$
B.
$-\frac{3\pi}{2}$
C.
$\frac{\pi}{2}$
D.
$\frac{3\pi}{2}$

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

Variable Separable Constraints|Logarithmic Integration Modulus|General vs Particular Solutions|Symmetry and Periodicity

Why This Question Is Tricky

Common Mistakes

This question targets the classic trap of ignoring the modulus during logarithmic integration and trigonometric simplification.

Students often incorrectly simplify $\sqrt{1 - \cos^2 x}$ as $\sin x$ globally, ignoring that it equals $|\sin x|$ .

This leads to a uniform general solution $y = x \sin x$ and the trap answer $-3\pi/2$ .

The correct approach requires recognizing the piecewise nature of the differential equation due to the periodic sign-flip of $\sin x$ .

Furthermore, because $y(\pi) = 0$ for any integration constant, continuity alone is insufficient to find the particular solution for the second branch.

Students must apply the 'continuously differentiable' condition to match the left and right derivatives at the periodic root $x = \pi$ , forcing the re-evaluation of the integration constant.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Variable Separable Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free