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Question #41497

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be the continuous solution to the differential equation $2x \frac{dy}{dx} = [x] (y^2 - 1)$ for $x \ge 1$ , where $[x]$ denotes the greatest integer less than or equal to $x$ . If $y(1) = 0$ , then the value of $y(\sqrt{6})$ is:

Options

A.
$-\frac{1}{2}$
B.
$-\frac{1}{3}$
C.
$-\frac{5}{7}$
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

Partial Fractions Decomposition|Implicit to Explicit Conversion|General vs Particular Solutions|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle piecewise differential equations driven by the Greatest Integer Function (GIF).

The primary trap lies in the implicit-to-explicit conversion and the assumption of a global constant of integration.

Students often incorrectly integrate the equation by treating $[x]$ as a constant or failing to recalculate the integration constant $C$ across the integer boundary at $x=2$ .

By forcing the evaluation at $x=\sqrt{6}$ , the student must first find the particular solution for $x \in [1, 2)$ , use continuity to find the boundary condition $y(2)$ , and then solve the new ODE for $x \in [2, 3)$ .

A secondary trap involves improperly dropping the absolute value from the partial fraction integration $\ln|y-1| - \ln|y+1|$ , which leads to a sign error (Option D).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Partial Fractions Decomposition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free