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Question #41393

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be the continuous solution to the differential equation $\frac{dy}{dx} = \frac{y}{x} + 3 - \left[ \frac{y}{x} \right], \quad x \ge 1$ with the initial condition $f(1) = 0$ , where $[ \cdot ]$ denotes the greatest integer function. Let $S$ be the set of all points $x > 1$ where $f(x)$ is not differentiable. Which of the following represents the product of all elements in $S$ ?

Options

A.
e^{5/2}
B.
e^3
C.
e^{7/2}
D.
e^4

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

Concavity Analysis|Homogeneous Function Homogeneity|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle piecewise continuous differential equations driven by the Greatest Integer Function.

The primary trap lies in assuming the solution is globally smooth or failing to properly match the constants of integration at the boundaries of each interval to maintain the continuity of the function.

Furthermore, the 'finite-time stabilization' (where the derivative of the substitution variable $v$ becomes zero, trapping $v$ at 3) tests if students blindly continue the pattern infinitely, which would lead to an incorrect infinite set of non-differentiable points.

Quick Facts About This Question

Question Information

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