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Question #41606

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be the solution to the initial value problem $\frac{dy}{dx} + \frac{1}{x \ln x} y = \frac{1}{x^2 |\ln x|}$ with $f(e) = 1 - \frac{1}{e}$ , defined on its maximal interval of existence. Let $S$ be the set of all possible values of $f(1/e)$ . Find $S$ .

Options

A.
{ e - 1 }
B.
{ 1 - e }
C.
{ 0 }
D.
\emptyset

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

Integrating Factor (IF) Sign Sensitivity|Variable Separable Constraints|Extraneous Solution Rejection

Why This Question Is Tricky

Common Mistakes

This question tests Integrating Factor (IF) sign sensitivity and Extraneous Solution Rejection.

$Students often blindly compute the IF as \ln x without the absolute value, ignoring the domain restriction x > 1.$

By forcing a variable separable transformation via the IF method, the DE reveals a singularity at x=1.

$While the solution for x > 1 has a removable singularity at x=1 (limit is 1), the valid solution for x < 1 (where IF = -\ln x) has a limit of -1.$

This jump discontinuity means the initial value problem cannot be extended across x=1.

Students who ignore the IF sign will incorrectly match the branches and obtain an extraneous value for f(1/e).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Integrating Factor (IF) Sign Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free