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Question #73850

Difficulty: 8/10 | 🟢 Free

The Question

Let $y(x)$ be a differentiable function on the interval $(0, 2a)$ satisfying the differential equation $(x-a) dy - y dx = (x-a)^2 \left[ \frac{x}{a} \right] dx$ where $a > 0$ is a parameter and $[\cdot]$ denotes the greatest integer function. If $y\left(\frac{a}{2}\right) = \alpha$ and $y\left(\frac{3a}{2}\right) = \beta$ , which of the following relations holds true?

Options

A.
$4(\alpha + \beta) = a^2$
B.
$4(\alpha + \beta) = 3a^2$
C.
$2(\alpha + \beta) = a^2$
D.
$4(\alpha - \beta) = a^2$

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

Exact Differential Recognition|Integration Constant Trap|Indeterminate Forms|Step Function Discontinuities

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the recognition of exact differentials combined with piecewise integration across a step function discontinuity.

The primary trap lies in assuming a single global integration constant 'C' for the entire domain, which leads to the incorrect relation $4(\alpha + \beta) = 3a^2$ .

Because the differential equation has a singularity at $x=a$ , the function naturally evaluates to an indeterminate $0/0$ form when finding the derivative.

Students must assign independent constants for each interval and enforce differentiability at the junction $x=a$ by evaluating the left-hand and right-hand limits of the derivative.

This resolves the indeterminate form and correctly links the piecewise constants.

Quick Facts About This Question

Question Information

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