banganX

2020-10-28

Find laplace transform of each following
a) ${t}^{n}$
b) $\mathrm{cos}\omega t$
c) $\mathrm{sin}h\left(ct\right)$
d) $\mathrm{cos}h\left(ct\right)$

Arham Warner

ANSWER FOR (A)We have to find Laplace transform of ${t}^{n}$, that is, $L\left\{{t}^{n}\right\}$.
Now from the Laplace Transform table we know that
$L\left\{{t}^{n}\right\}=\frac{n!}{{s}^{n+1}}$
This shows that
$L\left\{{t}^{n}\right\}=\frac{n!}{{s}^{n+1}}$
ANSWER FOR (B) We have to find Laplace transform of $\mathrm{cos}\left(\omega t\right)$, that is, L $\left\{\mathrm{cos}\left(\omega t\right)\right\}$
Now from the Laplace Transform table we know that
$L\left\{\mathrm{cos}\left(at\right)\right\}=\frac{s}{{s}^{2}+{a}^{2}}$
Whis shows that
$L\left\{\mathrm{cos}\left(\omega t\right)\right\}=\frac{s}{{s}^{2}+{\omega }^{2}}$
ANSWER FOR (C) We have to find Laplace transform of $\mathrm{sin}h\left(ct\right)$, that is, L $\left\{\mathrm{sin}h\left(ct\right)\right\}$
Now from the Laplace Transform table we know that
$L\left\{\mathrm{sin}h\left(at\right)\right\}=\frac{a}{{s}^{2}-{a}^{2}}$
Whis shows that
$L\left\{\mathrm{sinh}\left(ct\right)\right\}=\frac{c}{{s}^{2}-{c}^{2}}$
ANSWER FOR (D) We have to find Laplace transform of $\mathrm{cos}h\left(ct\right)$, that is, L $\left\{\mathrm{cos}h\left(ct\right)\right\}$
Now from the Laplace Transform table we know that $L\left\{\mathrm{cos}h\left(at\right)\right\}=\frac{s}{{s}^{2}-{a}^{2}}$
Whis shows that
$L\left\{\mathrm{cos}h\left(ct\right)\right\}=\frac{s}{{s}^{2}-{c}^{2}}$

Do you have a similar question?