Question

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

8.$\mathcal{L}\left\{t{e}^{-7t}\right\}$…….

Short answer

$\mathcal{L}\left\{t{e}^{-7t}\right\}=\frac{1}{{(s+7)}^2}$

Step-by-step solution

To Find the Laplace transform

$\begin{array}{l}LaplaceTransformofgivenexpression\\ \mathcal{L}\left\{t{e}^{-7t}\right\}\\ AccordingtoFirstTranslationTheorem\\ \mathcal{L}\left\{e^{at}f\left(t\right)\right\}=F(s-a)\mathcal{L}\left\{t^n\right\}=\frac{n!}{s^{n+1}}\\ Where,F\left(s\right)=\mathcal{L}\left\{f\right(t\left)\right\}Thus,\\ \mathcal{L}\left\{t{e}^{-7t}\right\}={\left.\frac{1}{s^2}\right|}_{s\to s-(-7)}\\ =\frac{1}{{(s+7)}^2}\end{array}$

Final proof

The value is,

$\mathcal{L}\left\{t{e}^{-7t}\right\}=\frac{1}{{(s+7)}^2}$