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Question #73828

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = y(x)$ be a continuous function on $\mathbb{R}$ satisfying the differential equation $\frac{dy}{dx} - a y = e^{2ax} (a |x| + 2 \text{sgn}(x))$ for all $x \in \mathbb{R} \setminus \{0\}$ , where $a > 0$ is a constant. If $y(-b) = 0$ for a given constant $b > 0$ , then $y(b)$ is equal to:

Options

A.
$b(e^{2ab} - 1) + \frac{1}{a} (e^{ab} - 1)^2$
B.
$b(e^{2ab} - 1) + \frac{1}{a} (e^{ab} + 1)^2$
C.
$b(e^{2ab} - 1) + \frac{1}{a} (e^{2ab} + 1)$
D.
$b(e^{2ab} + 1) + \frac{1}{a} (e^{ab} - 1)^2$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Exact Differential Recognition|Integration by Parts (LIATE Rule)|Modulus Function Breakdown

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the 'Boundary Consistency' trap in piecewise integration.

The exact differential $d(|x|e^{ax})$ simplifies the integral, but leaves a $\text{sgn}(x)e^{ax}$ term.

Students commonly integrate this as $\frac{1}{a}\text{sgn}(x)e^{ax} + C$ globally, ignoring the jump discontinuity at $x=0$ .

Because the antiderivative must be continuous, a constant shift ( $C_1 = C_2 - \frac{2}{a}$ ) is required at the boundary.

Failing to reconcile this shift leads directly to the trap option C, missing the $-2e^{ab}/a$ cross-term.

Quick Facts About This Question

Question Information

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