 # Find the Laplace transforms of the given functions.f{{({t})}}=6e^(-5t)+e^(3t)+5t^(3)-9 babeeb0oL 2021-03-11 Answered

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

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