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Question #41314

Difficulty: 8/10 | 🟢 Free

The Question

Consider the differential equation $\frac{dy}{dx} = \frac{2x - x^3 - x y^3}{3y^2}$ Let $y(x)$ be the real-valued solution satisfying $y(0) = \sqrt[3]{4}$ . If $(a, b)$ is the maximal open interval on which $y(x)$ is a valid differentiable solution, find the value of $b - a$ .

Options

A.
4
B.
$\infty$
C.
2
D.
8

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

Chain Rule Inner Function Differentiability|Bernoulli Equation Transformation|Implicit to Explicit Conversion

Why This Question Is Tricky

Common Mistakes

This question tests the implicit-to-explicit conversion trap in solving differential equations via substitution (such as Bernoulli equations).

Students often correctly find the explicit function $y(x) = (4 - x^2)^{1/3}$ and incorrectly assume the maximal interval of existence is $\mathbb{R}$ because the cube root is defined for all real numbers.

However, they fail to verify the differentiability of the inner function and the domain restrictions imposed by the original ODE (where $y=0$ causes a singularity).

The chain rule application $u' = 3y^2 y'$ breaks down at $y=0$ , restricting the valid differentiable solution to the interval $(-2, 2)$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Chain Rule Inner Function Differentiability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free