I'm having trouble with the laplace transform: L\{\sqrt{\frac{t}{\pi}}\cos(2t)\} The problem gives

Stacie Worsley 2021-12-11 Answered
I'm having trouble with the laplace transform: \(\displaystyle{L}{\left\lbrace\sqrt{{{\frac{{{t}}}{{\pi}}}}}{\cos{{\left({2}{t}\right)}}}\right\rbrace}\)
The problem gives me the transform identity \(\displaystyle{L}{\left\lbrace{\frac{{{\cos{{\left({2}{t}\right)}}}}}{{\sqrt{{\pi{t}}}}}}\right\rbrace}={\frac{{{e}^{{-{\frac{{{2}}}{{{s}}}}}}}}{{\sqrt{{s}}}}}\) but i'm not sure/confused as to why that would help me

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Expert Answer

Neil Dismukes
Answered 2021-12-12 Author has 3622 answers
Assuming t>0 (which is a usual assumption with Laplace transforms),
\(\displaystyle\sqrt{{{\frac{{{t}}}{{\pi}}}}}{\cos{{2}}}{t}={t}{\frac{{{\cos{{2}}}{t}}}{{\sqrt{{\pi{t}}}}}}\)
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Daniel Cormack
Answered 2021-12-13 Author has 1076 answers
\(\displaystyle{L}{\left[\sqrt{{{\frac{{{t}}}{{\pi}}}}}{\cos{{\left({2}{t}\right)}}}\right]}\)
If you define the function: \(\displaystyle{f{{\left({t}\right)}}}=\sqrt{{{\frac{{{t}}}{{\pi}}}}}{\cos{{\left({2}{t}\right)}}}\)
If you multiply it by (\(\displaystyle{\frac{{{t}}}{{{t}}}}\)) You can rewrite: \(\displaystyle{f{{\left({t}\right)}}}={t}{\frac{{{1}}}{{\sqrt{{{t}\pi}}}}}{\cos{{\left({2}{t}\right)}}}={t}{\frac{{{\cos{{\left({2}{t}\right)}}}}}{{\sqrt{{{t}\pi}}}}}\)
At this point, I have to remind you: \(\displaystyle\text{L[t^n f(t)]=-(1)^n \frac{d}{ds^n}(L[f(t)])}\)
Finally, if you have \(\displaystyle{L}{\left[{\frac{{{\cos{{\left({2}{t}\right)}}}}}{{\sqrt{{{t}\pi}}}}}\right]}={\frac{{{e}^{{-{\frac{{{2}}}{{{s}}}}}}}}{{\sqrt{{s}}}}}\) you have to calculate
\(\displaystyle-{\left({1}\right)}^{{n}}{\frac{{{d}}}{{{d}{s}}}}{\left({\frac{{{e}^{{-{\frac{{{2}}}{{{s}}}}}}}}{{\sqrt{{s}}}}}\right)}\)
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