Find the Laplace transform of (at).

GATE 2019 CE (CE2) – Question 12

Engineering Mathematics · ODE: First order and higher order linear equations, Euler-Cauchy equations, initial and boundary value problems · 1 mark · Multiple choice

Find the Laplace transform of $\sinh(at)$.

$a/(s^2-a^2)$
$a/(s^2+a^2)$
$s/(s^2-a^2)$
$s/(s^2+a^2)$

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Correct answer: (A) $a/(s^2-a^2)$

Explanation

Write $\sinh(at)=(e^{at}-e^{-at})/2$. Thus $$\mathcal L\{\sinh(at)\}=\tfrac12\left(\frac1{s-a}-\frac1{s+a}\right)=\frac a{s^2-a^2}.$$ Answer **A**, for $\operatorname{Re}s>|a|$.