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Question #41507

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be the solution to the differential equation $x(y+x)\frac{dy}{dx} = y^2 + xy + x^2$ for $x > e^{-2}$ , satisfying the initial condition $y(1) = -3$ . Find the value of $y(e^6)$ .

Options

A.
$-5e^6$
B.
$3e^6$
C.
$-3e^6$
D.
$5e^6$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Completing the Square in Quadratics|Substitution Logic in Homogeneous DE|Implicit to Explicit Conversion

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to navigate the implicit to explicit conversion trap after solving a homogeneous differential equation.

By using the substitution y=vx, the integral naturally leads to a quadratic in v, requiring completing the square.

The critical filter occurs when taking the square root of (v+1)^2: students frequently default to the principal (positive) root.

However, the initial condition y(1) = -3 strictly demands the negative branch.

Failing to verify the branch selection leads directly to the trap answer 3e^6.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Completing the Square in Quadratics
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free