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Question #40707

Difficulty: 8/10 | 🟢 Free

The Question

Let $S(x) = \sum_{n=1}^{\infty} \frac{1}{(n - \log_2 x)(n + 1 - \log_2 x)}$ . Let $y=y(x)$ be the solution to the initial value problem $x \ln 2 \frac{dy}{dx} - S(x) y = -S(x) y^2, \quad y(1) = \frac{1}{2}$ defined on its maximal interval of existence. Find the number of solutions to the equation $y(x) = x$ .

Options

A.
0
B.
1
C.
2
D.
3

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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

V-N Method (Partial Fractions)|Bernoulli Equation Transformation|Implicit Differentiation Implicit Domain

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of the Method of Differences (V-N method), Bernoulli differential equations, and implicit domain constraints.

The trap is set by the series $S(x)$ , which is undefined at $x=2$ because its first term involves division by zero.

Students who blindly apply the telescoping sum will find the algebraic solution $y(x) = \frac{1}{2 - \log_2 x}$ and solve $y(x) = x$ to find two roots ( $x=1/4$ and $x \in (2,4)$ ).

However, the fundamental theorem of ODEs dictates that the solution to an initial value problem is only valid on the maximal connected interval containing the initial condition where the differential equation is defined.

Since $x=2$ is a singularity, the maximal interval of existence is $(0, 2)$ .

Thus, the second root is extraneous, and ignoring this 'Implicit Domain' leads to an incorrect count of solutions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	V-N Method (Partial Fractions)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free