To find: The Laplace transform of the function {L}{leftlbrace{t}^{4}-{t}^{2}-{t}+ sin{sqrt{{{2}{t}}}}rightrbrace}

To find: The Laplace transform of the function {L}{leftlbrace{t}^{4}-{t}^{2}-{t}+ sin{sqrt{{{2}{t}}}}rightrbrace}

Question
Transformation properties
asked 2021-02-12
To find: The Laplace transform of the function
\({L}{\left\lbrace{t}^{4}-{t}^{2}-{t}+ \sin{\sqrt{{{2}{t}}}}\right\rbrace}\)

Answers (1)

2021-02-13
Approach:
With the help of Laplace Transform table and linearity property of Laplace Transformation calculation is carried out as follows:
Calculation:
\({L}{\left\lbrace{t}^{4}-{t}^{2}-{t}+ \sin{\sqrt{{{2}{t}}}}\right\rbrace}\)
Consider \(f,\ f_{1}\ \text{and}\ f_{2}\ \text{be the functions whose Laplace Transform exists for}\ s\ >\ \alpha\) and c be a constant.
Then, for \(s\ >\ \alpha\)
\({L}{\left\lbrace{{f}_{{1}}+}{f}_{{2}}\right\rbrace}={L}{\left\lbrace{f}_{{1}}\right\rbrace}+{L}{\left\lbrace{f}_{{2}}\right\rbrace}\)
Also,
\({L}{\left\lbrace{c}{f}\right\rbrace}={c}{L}{\left\lbrace{f}\right\rbrace}\)
Using above properties:
\({L}{\left\lbrace{t}^{4}-{t}^{2}-{t}+ \sin{\sqrt{{{2}{t}}}}\right\rbrace}={L}{\left\lbrace{t}^{4}\right\rbrace}+{L}{\left\lbrace-{t}^{2}\right\rbrace}+{L}{\left\lbrace-{t}\right\rbrace}+{L}{\left\lbrace \sin{\sqrt{{{2}{t}}}}\right\rbrace}\)
\(={L}{\left\lbrace{t}^{4}\right\rbrace}-{L}{\left\lbrace{t}^{2}\right\rbrace}-{L}{\left\lbrace{t}\right\rbrace}+{L}{\left\lbrace \sin{\sqrt{{{2}{t}}}}\right\rbrace}\)
Since the Laplace Transform from the Laplace Transform table:
\({L}{\left\lbrace{t}^{n}\right\rbrace}=\frac{{{n}!}}{{s}^{{{n}+{1}}}},\forall{s}>{0},{n}={1},{2},{3},\ldots\)
\({L}{\left\lbrace \sin{{a}}{t}\right\rbrace}=\frac{a}{{{s}^{2}+{a}^{2}}},\forall{s}>{0}\)
Using above formulae:
\({L}{\left\lbrace{t}^{4}\right\rbrace}-{L}{\left\lbrace{t}^{2}\right\rbrace}-{L}{\left\lbrace{t}\right\rbrace}+{L}{\left\lbrace \sin{\sqrt{{{2}{t}}}}\right\rbrace}=\frac{{{4}!}}{{s}^{{{4}+{1}}}}-\frac{{{2}!}}{{s}^{{{2}+{1}}}}-\frac{{{1}!}}{{s}^{{{1}+{1}}}}+\frac{\sqrt{{2}}}{{{s}^{2}+{\left(\sqrt{{2}}\right)}^{2}}},\forall{s}>{0}\)
\(=\frac{24}{{s}^{5}}-\frac{2}{{s}^{3}}-\frac{1}{{s}^{2}}+\sqrt{/}{\left({s}^{2}+{2}\right)},\forall{s}>{0}\)
Gives, the Laplace Transform is
\({F}{\left({s}\right)}=\frac{24}{{s}^{5}}-\frac{2}{{s}^{3}}-\frac{1}{{s}^{2}}+\sqrt{/}{\left({s}^{2}+{2}\right)},\forall{s}>{0}\)
Conclusion:
Therefore, the required Laplace Transform for the given function is
\({F}{\left({s}\right)}=\frac{24}{{s}^{5}}-\frac{2}{{s}^{3}}-\frac{1}{{s}^{2}}+\sqrt{/}{\left({s}^{2}+{2}\right)},\forall{s}>{0}\)
0

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