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Question #41318

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be a continuously differentiable function defined on the entire interval $(0, \infty)$ that satisfies the differential equation $x^2 \frac{dy}{dx} = y^2 - 5xy + 9x^2$ Then the value of $f(e^2)$ is:

Options

A.
$e^2$
B.
$\frac{5e^2}{2}$
C.
$3e^2$
D.
$2e^2$

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

Logarithmic Differentiation Domain|Homogeneous Function Homogeneity|Even Multiplicity Roots|Cross Multiplication Trap

Why This Question Is Tricky

Common Mistakes

This question targets the 'Even Multiplicity Root' and 'Cross Multiplication' traps in homogeneous differential equations.

Students routinely divide by $(v-3)^2$ and cross-multiply without considering the root $v=3$ , thereby losing the singular solution.

When they obtain the general solution $y = 3x - x/(\ln x + C)$ , they fail to analyze the domain of the logarithmic integration constant.

Because $\ln x$ spans all real numbers, the denominator must hit zero for some $x > 0$ , making the general solution discontinuous.

The question forces the realization that the 'lost' singular solution is the only one globally defined on $(0, \infty)$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Logarithmic Differentiation Domain
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free