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Question #40549

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be the solution of the differential equation $x \ln x \frac{dy}{dx} + (1 - \ln x) y = x^2 (\ln x)^2$ on its maximal interval of existence containing $x = e^{-1}$ , satisfying $y(e^{-1}) = 5e^{-2}$ . Find the number of roots of the equation $y(x) = 0$ .

Options

A.
0
B.
1
C.
2
D.
3

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Closed vs Open Intervals|Integrating Factor (IF) Sign Sensitivity|Homogeneous Function Homogeneity

Why This Question Is Tricky

Common Mistakes

This question tests the 'zero-point' domain trap in linear differential equations masked by a homogeneous-style substitution.

By substituting $v = y/x$ , the DE transforms into a linear equation with an integrating factor of $|\ln x|$ .

Students often find the algebraically correct function $y(x)$ and solve $y(x) = 0$ to find exactly one root.

However, they fail to recognize that the IF becomes zero at $x=1$ , creating a non-removable singularity where the solution blows up.

Consequently, the maximal interval of validity is strictly $(0, 1)$ .

Since the algebraic root lies at $x > 1$ , it falls outside the domain of the valid solution, making the true number of solutions zero.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Closed vs Open Intervals
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free