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Question #71561

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be the continuous solution to the differential equation $\frac{dy}{dx} = y \frac{ \left[ \frac{x}{a} \right] \sqrt{x^2 - 4ax + 4a^2} }{ (x - 2a)(x^2 - 4ax + 3a^2) }$ for $x \in (a, 3a) \setminus \{2a\}$ , where $a > 0$ is a constant and $[ \cdot ]$ denotes the greatest integer function. If $y\left(\frac{3a}{2}\right) = c$ (where $c > 0$ ), then $y\left(\frac{5a}{2}\right)$ is equal to:

Options

A.
$c \cdot 3^{-\frac{1}{2a}}$
B.
$c \cdot 3^{-\frac{3}{2a}}$
C.
$c$
D.
$c \cdot 3^{-\frac{1}{a}}$

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

General vs Particular Solutions|Partial Fractions Decomposition|Greatest Integer Function (GIF)|The Zero-Factor Cancellation Trap

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

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

Difficulty Boost : +4

Common Mistakes

This problem tests the rigorous evaluation of piecewise differential equations, specifically targeting the 'Zero-Factor Cancellation' trap and GIF discontinuities.

Students often incorrectly simplify $\sqrt{(x-2a)^2}/(x-2a)$ to $1$ , ignoring the absolute value which introduces a negative sign for $x < 2a$ .

Furthermore, the presence of the Greatest Integer Function $[x/a]$ forces the domain to be split at $x=2a$ .

Failing to evaluate the piecewise constants of integration by ensuring continuity at the boundary $x=2a$ leads to incorrect general solution constants.

The combination of these two traps ensures only students who meticulously check domain constraints and boundary conditions arrive at the correct particular solution.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	General vs Particular Solutions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free