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Question #41605

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be the continuous solution to the differential equation $\frac{dy}{dx} + x|x| y = x^2 e^{x^3/3}$ for $x \in \mathbb{R}$ , satisfying $y(\sqrt[3]{3}) = \frac{e^2 + 1}{2e}$ . Then the value of $y(-\sqrt[3]{6})$ is

Options

A.
$-e^{-2}$
B.
$\frac{e^4 + 1}{2e^2}$
C.
$-\frac{3}{2} e^{-2}$
D.
$-2e^{-2}$

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

Integrating Factor (IF) Sign Sensitivity|Substitution (Change of Variable)|Modulus Function Symmetry

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle linear differential equations with piecewise continuous coefficients due to a modulus function.

The primary trap lies in the 'Global Continuity' of the solution.

Students often ignore the sign change of $x|x|$ at $x=0$ , incorrectly applying the $x \ge 0$ solution globally, which leads to the trap answer $\cosh(2)$ .

Another common error is correctly splitting the domain but failing to equate the left-hand and right-hand limits of $y(x)$ at $x=0$ , thereby using an incorrect constant of integration (like $C_2 = C_1$ ) for the $x < 0$ branch.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Integrating Factor (IF) Sign Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free