Find out the answer to "Find the Laplace transforms of the given functions...."

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babeeb0oL

Answered question

2021-03-11

Find the Laplace transforms of the given functions.
$f (t) = 6 e^{- 5 t} + e^{3 t} + 5 t^{3} - 9$

Answer & Explanation

okomgcae

Skilled2021-03-12Added 93 answers

Step 1
It can be solve using Laplace transformation table
We know that
$L {e^{a t}} = \frac{1}{s - a}, L {t^{n}} = \frac{n!}{s^{n + 1}}$
$f (t) = 6 e^{- 5 t} + e^{3 t} + 5 t^{3} - 9$
$L {f (t)} = 6 L {e^{- 5 t}} + L {e^{3 t} + 5 L {t^{3}} - 9 L {1}$
$= \frac{6}{s + 5} + \frac{1}{s - 3} + 5 \cdot \frac{6}{s^{3}} - 9 H (t)$
$= \frac{7 s - 13}{(s + 5) (s - 3)} + \frac{30}{s^{3}} - 9 H (t)$ where
$H (t) = {\begin{matrix} 1 & t \geq 0 \\ 0 & t < 0 \end{matrix}$

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