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Question #41302

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be the solution curve of the differential equation $x \frac{dy}{dx} = y + |x| \sqrt{1 - \left(\frac{y}{x}\right)^2}$ satisfying the initial condition $y(-1) = 0$ . Then the value of $y(-e^{\pi/4})$ is:

Options

A.
$\frac{1}{\sqrt{2}} e^{\pi/4}$
B.
$-\frac{1}{\sqrt{2}} e^{\pi/4}$
C.
$e^{\pi/4}$
D.
$-e^{\pi/4}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

The Zero-Factor Cancellation Trap|Homogeneous Function Homogeneity|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the 'Zero-Factor Cancellation Trap' and modulus breakdown in homogeneous differential equations.

Students often mechanically substitute y=vx and divide by x without considering the domain.

By placing the initial condition in the negative x-domain (x=-1), the term |x| must be resolved as -x.

Failing to account for this sign change leads to integrating dx/x instead of -dx/x, resulting in an inverted sign for the inverse sine argument and ultimately the wrong sign for the final coordinate (Option B).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	The Zero-Factor Cancellation Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free