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Question #41403

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a differentiable function on $(0, \infty)$ satisfying the differential equation $x \frac{dy}{dx} - y \ln|y| = y (x \ln x - 1)$ with the initial condition $f(1) = -e$ . Find the value of $f(e)$ .

Options

A.
$-e^{\frac{e+2}{2}}$
B.
$e^{\frac{e+2}{2}}$
C.
$-e^{e+1}$
D.
$-e^{\frac{e}{2}}$

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

Implicit Differentiation Rate|Linear Differential Equation Structure|Logarithmic Integration Modulus|Integration Constant Trap

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle implicit derivatives embedded within a Linear Differential Equation and rigorously apply the modulus operator during logarithmic integration.

The primary trap occurs when substituting $u = \ln|y|$ .

Students often carelessly write $u = \ln y$ .

When evaluating the initial condition $f(1) = -e$ , they encounter $\ln(-e)$ , drop the negative sign to 'fix' the undefined domain, and proceed to find the positive branch $y = e^{\frac{e+2}{2}}$ .

The correct method requires maintaining the modulus $|y|$ to identify that the function remains on the negative branch $y < 0$ for all $x > 0$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Implicit Differentiation Rate
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free