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GATE 2021 ME (ME1) – Question 13

Engineering Mathematics · Differential Equations: Initial and boundary value problems, Laplace transforms · 1 mark · Multiple choice

The Dirac-delta function $\delta(t-t_0)$ for $t,t_0\in\mathbb R$ has the following property: $\int_a^b\delta(t-t_0)\,dt=1$ if $a<t_0<b$, and $0$ otherwise. The Laplace transform of the Dirac-delta function $\delta(t-a)$ for $a>0$, $\mathcal L\{\delta(t-a)\}=F(s)$ is

  1. 0
  2. $\infty$
  3. $e^{sa}$
  4. $e^{-sa}$

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Correct answer: (D) $e^{-sa}$

Explanation

By definition the transform is $\int_0^\infty e^{-st}\delta(t-a)dt$. The impulse at a lies inside the interval and samples the smooth factor there.

Thus the transform is **e⁻ˢᵃ (D)**. The negative sign comes from the Laplace kernel.