Question Bank • QB_EM_ADV_063
MSQ • 2 Marks
Section 1: Engineering Mathematics
Differential Equations
Subtopic: Laplace transforms and their inverse
Question Statement:
Which of the following Laplace transform operational properties is/are CORRECT for sufficiently smooth functions $f(t)$ and $g(t)$?
Options:
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(A) $\mathcal{L}\{f'(t)\} = s F(s) - f(0)$
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(B) $\mathcal{L}\{f''(t)\} = s^2 F(s) - s f(0) - f'(0)$
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(C) $\mathcal{L}\{t f(t)\} = -\frac{d}{ds} F(s)$
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(D) $\mathcal{L}\{(f * g)(t)\} = F(s) \cdot G(s)$
Step-by-Step Solution & Explanation
All four statements are foundational theorems in Laplace operational calculus:
1. **First Derivative Rule (Option A)**: $\int_0^\infty e^{-st} f'(t)\,dt = [e^{-st} f(t)]_0^\infty + s\int_0^\infty e^{-st} f(t)\,dt = sF(s) - f(0)$ (True).
2. **Second Derivative Rule (Option B)**: Applying the derivative rule twice gives $s[sF(s) - f(0)] - f'(0) = s^2 F(s) - sf(0) - f'(0)$ (True).
3. **Multiplication by $t$ (Option C)**: $\frac{d}{ds} F(s) = \int_0^\infty (-t) e^{-st} f(t)\,dt = -\mathcal{L}\{t f(t)\}$ (True).
4. **Convolution Theorem (Option D)**: The Laplace transform of the convolution $(f * g)(t) = \int_0^t f(\tau) g(t - \tau)\,d\tau$ is the algebraic product $F(s) \cdot G(s)$ (True).
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