Question

In Problems 3 - 24 fill in the blanks or answer true or false.

13. $\mathit{L}{}^{\mathbf{-}\mathbf{1}}\left\{\frac{20}{s^6}\right\}$=….

Short answer

$L{}^{-1}\left\{\frac{20}{s^6}\right\}=\frac{t^5}{6}$

Step-by-step solution

Consider the function

The function is,

$\begin{array}{l}L{}^{-1}\left\{\frac{20}{s^6}\right\}\\ \Rightarrow L{}^{-1}\left\{\frac{20}{s^6}\right\}=L^{-1}\left\{\frac{1}{6}·\frac{120}{s^6}\right\}\end{array}$

Compute the Laplace inverse

$\begin{array}{l}\Rightarrow L{}^{-1}\left\{\frac{20}{s^6}\right\}\\ Thes\tan dardformulais,\\ L^{-1}\left\{\frac{\left(n\right)!}{s^{n+1}}\right\}=t^n\\ Here,\\ \mathrm{For} \mathrm{function} f\left(t\right) \mathrm{and} \mathrm{constant} a:L^{-1}\left\{a·f\left(t\right)\right\}=a·L^{-1}\left\{f\left(t\right)\right\}\\ Thus,L^{-1}\left\{\frac{1}{6}·\frac{120}{s^6}\right\}=\frac{1}{6}{L}^{-1}\left\{\frac{120}{s^6}\right\}\\ Substituethevalueandfindtheinverse,\\ L^{-1}\left\{\frac{\left(n\right)!}{s^{n+1}}\right\}=t^n\\ \Rightarrow L^{-1}\left\{\frac{5!}{s^{5+1}}\right\}=t^5\\ \Rightarrow L^{-1}\left\{\frac{120}{s^6}\right\}=t^5\\ Here,\\ \frac{1}{6}{L}^{-1}\left\{\frac{120}{s^6}\right\}=\frac{1}{6}L{}^{-1}\left\{\frac{20}{s^6}\right\}\\ \Rightarrow L{}^{-1}\left\{\frac{20}{s^6}\right\}=\frac{t^5}{6},\\ Therefore,theLaplaceinverseofL{}^{-1}\left\{\frac{20}{s^6}\right\}is,\\ \therefore L{}^{-1}\left\{\frac{20}{s^6}\right\}=\frac{t^5}{6}\end{array}$