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Question #40528

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a differentiable function on the interval $\left(-\frac{1}{2}, \infty\right)$ that satisfies the differential equation $(x^2 - 1) \frac{dy}{dx} = y^2 - 1$ and the initial condition $f(0) = 2$ . The value of $f(2)$ is:

Options

A.
$\frac{4}{5}$
B.
$\frac{5}{4}$
C.
$-\frac{4}{3}$
D.
$0$

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

Denominator Restriction|Variable Separable Constraints|Integrating Factor (IF) Sign Sensitivity|Improper Integrals

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle variable separable differential equations with domain restrictions and absolute value sensitivities.

The integral of $1/(x^2-1)$ introduces logarithmic terms with absolute values.

A common trap is to drop the absolute values or assume the argument remains positive, leading to the incorrect relation $(y-1)/(y+1) = (x-1)/(3(x+1))$ .

Because the solution must cross the singularity at $x=1$ , the sign of $(x-1)/(x+1)$ flips.

Differentiability forces the unique analytic continuation $y = (x+2)/(2x+1)$ , correctly yielding $4/5$ .

Ignoring the sign flip yields the trap answer $5/4$ , while using the incorrect integration formula (arctan) yields $-4/3$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Denominator Restriction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free