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Question #40717

Difficulty: 8/10 | 🟢 Free

The Question

Let $H(x, y)$ be a homogeneous function of degree 2 such that $H(x, 1) = \frac{1}{x}$ for all $x > 0$ . For any point $(x, y)$ on a curve $y = y(x)$ (where $x > 0, y > 0$ ), $n$ arithmetic means $A_1, A_2, \dots, A_n$ are inserted between $x \frac{\partial H}{\partial x}$ and $y \frac{\partial H}{\partial y}$ . Let $S_n = \sum_{i=1}^n A_i$ . If the curve $y = y(x)$ satisfies the differential equation $\frac{x^2}{n} S_n \frac{dy}{dx} + y^4 = x^2 y^3 \ln x$ and passes through the point $(1, 2)$ , then the value of $y(e)$ is:

Options

A.
$\frac{2e^2}{9} + \frac{19}{9e}$
B.
$\frac{e^2}{3} + \frac{17}{9e}$
C.
$\frac{2e^2}{3} + \frac{19}{9e}$
D.
$\frac{e^2}{9} + \frac{19}{3e}$

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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

Insertion of Means|Linear Differential Equation Structure|Homogeneous Function Homogeneity

Why This Question Is Tricky

Common Mistakes

This question fuses the insertion of arithmetic means with Euler's Theorem for homogeneous functions to form the coefficients of a Linear Differential Equation.

The primary trap lies in the sequence aspect: students often attempt to manually calculate the common difference $d = \frac{y H_y - x H_x}{n+1}$ and sum the AP using standard formulas, which frequently leads to index-shifting errors (e.g., dividing by $n$ instead of $n+1$ ) and an unmanageable differential equation.

The optimal path forces the recognition that the sum of inserted means is simply $\frac{n}{2}$ times the sum of the extremes.

By applying Euler's Theorem, the extremes sum to $k \cdot H(x,y)$ , instantly collapsing the complex partial derivatives into a clean algebraic term that transforms the non-linear looking DE into a standard exact/linear DE.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Insertion of Means
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free