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Question #41604

Difficulty: 8/10 | 🟢 Free

The Question

Let $Q(x) = \sum_{k=0}^{3} \binom{3}{k} (-1)^k x^{2k} (\text{sgn}(x))^k$ , where $\text{sgn}(x)$ denotes the signum function. Let $y = y(x)$ be a differentiable function satisfying the differential equation $(1 - x|x|) \frac{dy}{dx} + 6 |x| y = x (Q(x))^3$ for all $x \in \mathbb{R}$ , with the initial condition $y(0) = 0$ . Find the value of $y(-1)$ .

Options

A.
42
B.
0
C.
-42
D.
84

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Linear Differential Equation Structure|Coefficient of x^r in Expansion|Sign Flipping Rule

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize a piecewise binomial expansion and correctly compute the Integrating Factor (IF) across a sign-flip boundary.

The trap lies in the term $x^{2k}(\text{sgn}(x))^k$ , which simplifies to $(x|x|)^k$ , making $Q(x) = (1-x|x|)^3$ .

Students often ignore the absolute value or signum function, assuming $x>0$ globally, which leads to $Q(x) = (1-x^2)^3$ and an incorrect symmetric solution yielding $y(-1)=0$ .

Furthermore, integrating the piecewise LDE requires matching the constant of integration at $x=0$ , penalizing those who blindly apply standard formulas without respecting domain boundaries.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Linear Differential Equation Structure
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free