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).

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).