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Question #40943

Difficulty: 8/10 | 🟢 Free

The Question

Let $y: [1, e] \to \left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$ be a differentiable function satisfying the differential equation $\lim_{h \to 0} \frac{\arcsin(\sin(y(x+h))) - \arcsin(\sin(y(x)))}{h} + \frac{y(x)}{x} = \frac{y^2(x)}{x^2}$ for all $x \in [1, e]$ , with the initial condition $y(1) = \pi$ . Find the value of $y(e)$ .

Options

A.
$\frac{\pi e}{\pi + 1}$
B.
$\frac{2\pi e}{\pi + (2-\pi)e^2}$
C.
$\frac{2\pi e}{(\pi+2)e^2 - \pi}$
D.
$\frac{\pi + 1}{\pi e}$

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

First Principle of Derivative|Bernoulli Equation Transformation|Variable Transformation Substitution

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle variable transformations within inverse trigonometric domains alongside the First Principle of derivatives and Bernoulli differential equations.

The primary trap lies in the limit expression: students who blindly evaluate $\arcsin(\sin y)$ as $y$ will obtain $y'$ instead of $-y'$ , leading to a fundamentally different Bernoulli equation.

Recognizing the branch cut $\arcsin(\sin y) = \pi - y$ for $y \in (\pi/2, 3\pi/2)$ is crucial before applying the standard $v = y^{1-n}$ substitution.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	First Principle of Derivative
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free