Before You Begin

Question #68590

Difficulty: 8/10 | 🟢 Free

The Question

Let $\alpha = \int_{-1}^{1} x^2 \sin(x^3) \,dx$ . Consider the differential equation for $x > 0, y > 0$ : $x \frac{dy}{dx} - y = x \left( \left[ \frac{y}{x} \right] - 1 + \alpha \right) \left( \frac{y}{x} - 2 \right)$ where $[ \cdot ]$ denotes the greatest integer function. Let $y_c(x)$ denote the solution curve satisfying the initial condition $y(1) = c$ . Which of the following expressions for $y_c(x)$ (valid in a neighborhood of $x=1$ ) is INCORRECT?

Options

A.
$y_{5/2}(x) = 2x + \frac{1}{2}x^2$
B.
$y_{1/2}(x) = 2x - \frac{3}{2}$
C.
$y_{3/2}(x) = \frac{x^2+2x}{x+1}$
D.
y_{2}(x) = 2x

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

Geometric Interpretation|Homogeneous Function Homogeneity|The Zero-Factor Cancellation Trap|Greatest Integer Function (GIF)

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Common Mistakes

This question tests the geometric interpretation of homogeneous differential equations combined with the Greatest Integer Function (GIF), specifically targeting the 'Zero-Factor Cancellation Trap'.

Students often algorithmically separate variables by dividing by $([y/x] - 1)(y/x - 2)$ without checking for intervals where the GIF causes the term to vanish.

For the initial condition $y(1) = 1.5$ , the ratio $y/x \in [1, 2)$ , making $[y/x] - 1 = 0$ .

This implies the derivative of the homogeneous variable $v$ is zero, yielding a straight-line singular flow $y = 1.5x$ .

The incorrect option represents the solution obtained if a student completely ignores the GIF and integrates $x v' = (v-1)(v-2)$ , a classic distractor that punishes algorithmic calculus over geometric analysis.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Geometric Interpretation
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free