Question

Find the Laplace transform F(s)=L\left\{f(t)\right\} of the function f(t)=2-7t^5+\sin(3t)

Laplace transform
ANSWERED
asked 2021-09-12

Find the Laplace transform \(F(s)=L\left\{f(t)\right\}\) of the function \(\displaystyle{f{{\left({t}\right)}}}={2}-{7}{t}^{{5}}+{\sin{{\left({3}{t}\right)}}}\)

Expert Answers (1)

2021-09-13

Step 1
Given function is,
\(\displaystyle{f{{\left({t}\right)}}}={2}-{7}{t}^{{5}}+{\sin{{\left({3}{t}\right)}}}\)
The objective is to find the laplace transformation
Step 2
\(L\left\{f(t)\right\}=L\left\{2-7t^5+\sin 3t\right\}\)
\(=L\left\{2\right\}-7L\left\{t^5\right\}+L\left\{\sin 3t\right\}\)
Since,
\(L\left\{a\right\}=\frac{a}{s} , L\left\{t^n\right\}=\frac{n!}{s^{n+1}} \text{ and } L\left\{\sin(at)\right\}=\frac{a}{s^2+a^2}\)
Hence,
\(L\left\{f(t)\right\}=L\left\{2\right\}-7L\left\{t^5\right\}+L\left\{\sin 3t\right\}\)
\(\displaystyle={\frac{{{2}}}{{{s}}}}-{7}{\left({\frac{{{5}!}}{{{s}^{{{5}+{1}}}}}}\right)}+{\frac{{{3}}}{{{s}^{{2}}+{3}^{{2}}}}}\)
\(\displaystyle={\frac{{{2}}}{{{s}}}}-{\frac{{{7}\times{120}}}{{{s}^{{{5}+{1}}}}}}+{\frac{{{3}}}{{{s}^{{2}}+{9}}}}\)
\(\displaystyle={\frac{{{2}}}{{{s}}}}-{\frac{{{840}}}{{{s}^{{{5}+{1}}}}}}+{\frac{{{3}}}{{{s}^{{2}}+{9}}}}\)
Therefore, \(L\left\{2-7t^5+\sin 3t\right\}=\frac{2}{s}-\frac{840}{s^{5+1}}+\frac{3}{s^2+9}\)

37
 
Best answer

expert advice

Have a similar question?
We can deal with it in 3 hours
...