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Question #41598

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 $x > 0$ , satisfying the initial condition $y(1) = 0$ . Find the value of $y(e^{3/8})$ .

Options

A.
$\frac{1}{2} e^{3/8}$
B.
$\frac{3}{2} e^{3/8}$
C.
e^{3/8}
D.
$-\frac{1}{2} e^{3/8}$

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

Substitution Logic in Homogeneous DE|Implicit to Explicit Conversion|Even Multiplicity Roots

Why This Question Is Tricky

Common Mistakes

This question tests the critical transition from an implicit relation to an explicit function in homogeneous differential equations.

After separating variables, the integration yields an even power $(v-1)^2$ .

When taking the square root to find the explicit solution, students habitually default to the positive branch.

However, the initial condition $v(1)=0$ dictates that the expression $v-1$ is initially negative, mandating the selection of the negative root.

Failing to verify this sign constraint leads directly to the trap answer of 3/2 e^{3/8}.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Substitution Logic in Homogeneous DE
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free