Question

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

Answer 1

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