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Question #40356

Difficulty: 8/10 | 🟢 Free

The Question

Let $y(x)$ be a continuous function on $\mathbb{R}$ satisfying the differential equation $(x^2+1) \frac{dy}{dx} + 2x \text{sgn}(x) y = 2x$ for all $x \neq 0$ , with the boundary condition $y(-1) = -1$ . If $A = \lim_{x \to \infty} y(x)$ , then the value of $\frac{1}{\pi} \int_0^\infty (A - y(x)) dx$ is equal to

Options

A.
0
B.
$\frac{1}{2}$
C.
1
D.
2

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Rational Function Asymptotes|Integrating Factor (IF) Sign Sensitivity|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the integration of first-order linear ODEs with piecewise continuous coefficients (specifically involving the signum function).

The primary trap lies in the jump discontinuity of the integrating factor's form across $x=0$ .

Students frequently integrate both sides independently and either ignore the constant of integration for $x>0$ or incorrectly assume the same constant applies globally.

By forcing the evaluation of the piecewise constant through the continuity of $y(x)$ at $x=0$ , the problem shifts the horizontal asymptote profile.

Failing to match $y(0^-) = y(0^+)$ leads to an incorrect numerator in the rational function, directly altering the area computed in the final improper integral.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Rational Function Asymptotes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free