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Question #41601

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(x, 0)$ (where $x > 0$ ) be a point and $L$ be its polar with respect to the circle $x^2 + y^2 = 1$ . Let $B$ be the intersection of $L$ with the $x$ -axis. A function $P(x)$ is defined as $P(x) = \left| 1 - \frac{OB}{OA} \right|$ , where $O$ is the origin. Let $y = y(x)$ be the solution to the differential equation $\frac{dy}{dx} + P(x) y = 1 - \frac{1}{x^2}$ for $x > 0$ , satisfying $y\left(\frac{1}{2}\right) = e^{5/2} - 1$ . Find the value of $y(2)$ .

Options

A.
1 + e^{3/2} - 2e^{-1/2}
B.
1 - e^{3/2} + 2e^{-1/2}
C.
-1 + e^{3/2} - 2e^{-1/2}
D.
1 + e^{5/2} - 2e^{1/2}

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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

Linear Differential Equation Structure|Polar and Pole Duality|Geometrical Interpretation of Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of coordinate geometry and calculus by embedding a geometric locus property into the coefficient of a Linear Differential Equation.

The core trap is the modulus function inside the integrating factor P(x), which arises from the relative position of the pole A to the conic section.

Students commonly make the mistake of ignoring the modulus and applying a single integrating factor globally.

To solve it correctly, one must split the domain at x=1 (where the pole crosses the conic), solve two distinct LDEs, and use the continuity of the differentiable function y(x) to bridge the constants of integration.

Quick Facts About This Question

Question Information

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