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Question #70536

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be the particular solution to the differential equation $\frac{dy}{dx} = y^2 (x^2 - 1)$ satisfying the initial condition $f(-1) = -1$ . Let $I$ be the maximal interval of existence of $f(x)$ containing $x = -1$ . Which of the following represents the range of $f(x)$ on the interval $I$ ?

Options

A.
$(-\infty, -1]$
B.
$(-\infty, -1] \cup [3, \infty)$
C.
$\mathbb{R} \setminus \{0\}$
D.
$[-1, 3]$

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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

General vs Particular Solutions|Variable Separable Constraints|Monotonicity and Boundedness|First Derivative Test Trap

Why This Question Is Tricky

Examiner Levers

Logical Assertion

Tests the causal relationship between two true statements.

Difficulty Boost : +4

Boundary Condition

Tests behavior at limits like 0, 1, infinity, or empty sets.

Difficulty Boost : +4

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the intersection of Differential Equations and Function Analysis, specifically targeting the 'First Derivative Test Trap' and domain boundedness.

Students often blindly differentiate the general solution $y = \frac{-3}{x^3 - 3x + 1}$ to find critical points at $x = -1$ and $x = 1$ , yielding local extrema values of $-1$ and $3$ .

Ignoring the vertical asymptotes (roots of $x^3 - 3x + 1 = 0$ ) that restrict the maximal interval of existence $I$ around the initial condition, they incorrectly assume the function is continuous on $\mathbb{R}$ .

This leads them to either choose the global range $\mathbb{R} \setminus \{0\}$ or falsely assume the range is determined solely by the local extrema $(-\infty, -1] \cup [3, \infty)$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	General vs Particular Solutions
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free