Question Bank • QB_SUBEXP_S1_005 MSQ • 2 Marks

Section 1: Engineering Mathematics

Differential Equations

Subtopic: Laplace transforms and their inverse

Question Statement:

Let $\mathcal{L}\{f(t)\} = F(s)$ be the unilateral Laplace transform of $f(t)$ for $t \ge 0$. Which of the following properties are TRUE?

Options:

  • (A) $\mathcal{L}\{e^{at} f(t)\} = F(s - a)$
  • (B) $\mathcal{L}\{t f(t)\} = -\frac{d}{ds} F(s)$
  • (C) $\mathcal{L}\{f'(t)\} = s F(s) - f(0)$
  • (D) $\mathcal{L}\{\int_0^t f(\tau) d\tau\} = \frac{F(s)}{s}$

Answer Key:

A, B, C, D

Step-by-Step Solution & Explanation

All four properties are fundamental theorems of unilateral Laplace transforms: 1. First Shifting Theorem: $\mathcal{L}\{e^{at} f(t)\} = \int_0^\infty e^{-(s-a)t} f(t) dt = F(s-a)$. (Statement A True) 2. Frequency Differentiation: $\frac{d}{ds} F(s) = \int_0^\infty (-t) e^{-st} f(t) dt = -\mathcal{L}\{t f(t)\}$. (Statement B True) 3. Differentiation in time domain: $\mathcal{L}\{f'(t)\} = s F(s) - f(0)$. (Statement C True) 4. Integration in time domain: $\mathcal{L}\{\int_0^t f(\tau) d\tau\} = \frac{F(s)}{s}$. (Statement D True)
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