Find the Laplace transformation (evaluating the improper integral that defines this transformation) of the real valued function f(t) of the real variable t>0. (Assume the parameter s appearing in the Laplace transformation, as a real variable). f{{left({t}right)}}={2}{t}^{2}-{4} cosh{{left({3}{t}right)}}+{e}^{{{t}^{2}}}

Find the Laplace transformation (evaluating the improper integral that defines this transformation) of the real valued function f(t) of the real variable t>0. (Assume the parameter s appearing in the Laplace transformation, as a real variable). f{{left({t}right)}}={2}{t}^{2}-{4} cosh{{left({3}{t}right)}}+{e}^{{{t}^{2}}}

Question
Laplace transform
asked 2021-03-04
Find the Laplace transformation (evaluating the improper integral that defines this transformation) of the real valued function f(t) of the real variable t>0. (Assume the parameter s appearing in the Laplace transformation, as a real variable).
\(f{{\left({t}\right)}}={2}{t}^{2}-{4} \cosh{{\left({3}{t}\right)}}+{e}^{{{t}^{2}}}\)

Answers (1)

2021-03-05
here \(f{{\left({t}\right)}}={2}{t}^{2}-{4} \cosh{{\left({3}{t}\right)}}+{e}^{{{t}^{2}}}\)
we know that
\({L}{\left\lbrace{t}^{n}\right\rbrace}=\frac{{{n}!}}{{{s}^{{{n}+{1}}}}},{L}{\left\lbrace \cosh{{\left({a}{t}\right)}}\right\rbrace}=\frac{s}{{{s}^{2}-{a}^{2}}}\)
\({L}{\left\lbrace{e}^{{{t}^{2}}}\right\rbrace}={F}{\left(\frac{s}{{2}}\right)}-\frac{1}{{2}}{i}\sqrt{\pi}\cdot{e}^{{-\frac{{s}^{2}}{{4}}}}\)
Also , \({L}{\left\lbrace f{{\left({t}\right)}}+ g{{\left({t}\right)}}+{h}{\left({t}\right)}\right\rbrace}={L}{\left\lbrace f{{\left({t}\right)}}\right\rbrace}+{L}{\left\lbrace g{{\left({t}\right)}}\right\rbrace}+{L}{\left\lbrace{h}{\left({t}\right)}\right\rbrace}\)
Taking Laplace transform of eq
\({L}{\left\lbrace f{{\left({t}\right)}}\right\rbrace}={L}{\left\lbrace{2}{t}^{2}-{4} \cosh{{\left({3}{t}\right)}}+{e}^{{{t}^{2}}}\right\rbrace}\)
\(={2}{L}{\left\lbrace{t}^{2}\right\rbrace}-{4}{L}{\left\lbrace \cosh{{\left({3}{t}\right)}}\right\rbrace}+{L}{\left\lbrace{e}^{{{t}^{2}}}\right\rbrace}\)
\(={2}\cdot\frac{{{2}!}}{{{s}^{{{2}+{1}}}}}-{4}\frac{{{s}}}{{{s}^{2}-{3}^{2}}}+{F}{\left(\frac{s}{{2}}\right)}-\frac{1}{{2}}{i}\sqrt{\pi}\cdot{e}^{{-\frac{{s}^{2}}{{4}}}}\)
\({L}{\left\lbrace f{{\left({t}\right)}}\right\rbrace}=\frac{4}{{s}^{3}}-\frac{{{4}{s}}}{{{s}^{2}-{9}}}+{F}{\left(\frac{s}{{2}}\right)}-\frac{1}{{2}}{i}\sqrt{\pi}\cdot{e}^{{-\frac{{s}^{2}}{{4}}}}\)
Step 2
This is required Laplace transformation of given f(t).
0

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Please type in a = ?
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