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Question #41415

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be the continuous solution to the differential equation $\frac{dy}{dx} + \text{sgn}(x) y = |x| \cos(x) e^{-|x|}$ for all $x \in \mathbb{R}$ , subject to the condition $y(\pi) = e^{-\pi}$ . Find the value of $y(-\pi)$ .

Options

A.
$5 e^{-\pi}$
B.
$3 e^{-\pi}$
C.
$e^{-\pi}$
D.
$-e^{-\pi}$

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

Integration by Parts (LIATE Rule)|Integrating Factor (IF) Sign Sensitivity|Signum Function Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle linear differential equations with discontinuous coefficients, specifically the signum function.

The primary trap lies in the integration step after multiplying by the integrating factor $e^{|x|}$ .

Students often incorrectly assume a single global constant of integration $C$ for the entire domain.

However, because the antiderivative of $|x|\cos(x)$ behaves differently for $x>0$ and $x<0$ , the constant must be defined piecewise ( $C_1$ and $C_2$ ).

Failing to enforce continuity at $x=0$ leads to the incorrect assumption that $C_1 = C_2$ , trapping them into choosing $3e^{-\pi}$ instead of the correct $5e^{-\pi}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Integration by Parts (LIATE Rule)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free