Allied & ICAR Exam Bank • ALLIED_1010 MCQ • 1 Mark

Section 1: Engineering Mathematics

Differential Equations

Subtopic: Higher order linear equations

Question Statement:

The particular integral of the differential equation $\frac{d^2 y}{dx^2} + 4y = \sin(2x)$ is:

Options:

  • (A) -\frac{x}{4} \cos(2x)
  • (B) \frac{x}{4} \cos(2x)
  • (C) -\frac{1}{4} \sin(2x)
  • (D) \frac{x}{2} \sin(2x)

Answer Key:

A

Step-by-Step Solution & Explanation

$y_p = \frac{1}{D^2 + 4} \sin(2x)$. Since $D^2 = -2^2 = -4$ makes the denominator zero, $y_p = x \frac{1}{2D} \sin(2x) = \frac{x}{2} \left(-\frac{\cos(2x)}{2}\right) = -\frac{x}{4}\cos(2x)$.
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