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Question #41595

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be the solution to the differential equation $x \frac{dy}{dx} = y + x + 3|x - y|$ for $x > 0$ , satisfying the initial condition $y(1) = 0$ . Find the value of $y(2)$ .

Options

A.
$\frac{8}{3}$
B.
$\frac{7}{3}$
C.
$\frac{10}{3}$
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

Homogeneous Function Homogeneity|Parametric Form Manipulation|Modulus Inequality Logic

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to rigorously handle domain restrictions imposed by modulus functions within homogeneous differential equations.

The trap lies in the absolute value $|x-y|$ , which translates to $|1-v|$ in the parametric form.

Students commonly drop the modulus or assume the initial branch $v < 1$ holds globally, leading to the extraneous solution $v = \frac{4}{3}(1 - x^{-3})$ for all $x$ .

By evaluating at $x=2$ , which lies beyond the critical transition point $x = 4^{1/3}$ , the problem forces the student to recognize that $v$ exceeds $1$ , requiring a piecewise integration and recalculation of the integration constant to maintain continuity of the solution curve.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Homogeneous Function Homogeneity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free