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Question #40307

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a continuously differentiable function defined on $(0, \infty)$ which satisfies the differential equation $\frac{dy}{dx} = \frac{y}{x} - \sqrt{\left(\frac{y}{x}\right)^2 - 1}$ with the initial condition $f\left(\frac{1}{2}\right) = \frac{5}{8}$ . Find the value of $f(2)$ .

Options

A.
2
B.
$\frac{5}{2}$
C.
$\frac{17}{8}$
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

Set Complement Boundaries|Homogeneous Function Homogeneity|Substitution Logic in Homogeneous DE

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of substitution in homogeneous differential equations and the identification of set complement boundaries.

The standard algorithmic approach of substituting $y=vx$ , integrating, and squaring to find the general solution introduces a hidden domain restriction ( $v \le 1/x$ ).

Students commonly ignore this restriction and blindly evaluate the resulting algebraic curve at $x=2$ , falling into the trap of answering $5/2$ .

A complete understanding requires recognizing that the general solution branch becomes invalid at $x=1$ , forcing the function to continuously transition to the singular solution $y=x$ to maintain the non-negativity of the square root and the monotonicity of $v$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Set Complement Boundaries
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free