Brittney Lord

2021-10-20

Find Laplace Transform of each following
(a) ${t}^{n}$ , (b) $cos\omega t$

Nichole Watt

For A
We have to find Laplace transform of ${t}^{n}$ , that is $L\left\{{t}^{n}\right\}$
Now from the Laplace Transform table
$L\left\{{t}^{n}\right\}=\frac{n!}{{s}^{n+1}}$
It shows that:
$L\left\{{t}^{n}\right\}=\frac{n!}{{s}^{n+1}}$
For B
We have to find Laplace transform of $cos\omega t$. That is $Lcos\omega t$

From the Laplace Table:

$L\mathrm{cos}at=s/\left({s}^{2}+{a}^{2}\right)$

$L\mathrm{cos}\omega t=s/\left({s}^{2}+{\omega }^{2}\right)$

For C

We have to find Laplace transorm of $\mathrm{sinh}\left(ct\right)$ that is $L\mathrm{sinh}ct$

From the Laplace Table:

$L\mathrm{sinh}at=a/\left({s}^{2}+{a}^{2}\right)$

It shows that:

$L\mathrm{sinh}ct=c/\left({s}^{2}+{c}^{2}\right)$

For D

We have to find Laplace transorm of $\mathrm{cosh}\left(ct\right)$that is $L\mathrm{cosh}ct$

From the Laplace Table:

$L\mathrm{cosh}at=s/\left({s}^{2}+{a}^{2}\right)$

It shows that:

$L\mathrm{cosh}ct=s/\left({s}^{2}+{c}^{2}\right)$

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