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Question #70562

Difficulty: 8/10 | 🟢 Free

The Question

Let $y(x)$ be a positive differentiable function satisfying the differential equation $x(x^2-a^2) \frac{dy}{dx} + (a^2-3x^2) y \ln y = p x^3 y$ for $x > a > 0$ , where $p, q$ are positive real constants. If $\lim_{x \to a^+} \frac{\ln y(x) + \frac{pa}{2}}{x - a} = q$ , then $y(2a)$ is equal to:

Options

A.
$e^{a\left(\frac{p}{2} + 3q\right)}$
B.
$e^{-pa} + \frac{3a(2q+p)}{2} e^{-\frac{pa}{2}}$
C.
$e^{2a(p + 3q)}$
D.
$e^{\frac{a}{4}(6q - p)}$

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

General vs Particular Solutions|Logarithmic Differentiation|Indeterminate Forms|Integration Constant 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 question tests the execution of logarithmic differentiation on a Bernoulli-type differential equation and the evaluation of boundary conditions involving indeterminate forms.

The primary trap is the 'Integration Constant Trap' in logarithmic contexts.

Students often incorrectly assume the integration constant applies additively outside the exponential, writing $y = e^{f(x)} + C$ instead of $y = e^{f(x)+C}$ .

If they make this structural error, applying L'Hôpital's rule to the $0/0$ boundary limit yields a completely different integration constant, leading them directly to the meticulously crafted distractor option.

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