Question

In problems 1-20 find either f(S) or f(t), as indicated $\mathit{L}\left\{e^{2t}{(t-1)}^2\right\}$

Short answer

Find either f(S) or f (t), as indicated.

$$\begin{gathered} L{e}^{2t}{\left(t-1\right)}^2 \\ =\underset{s\to s-2}{\lim }\underset{s\to s-2}{L\left(t^2\right)-2\lim L\left(t\right)+}\underset{s\to s-2}{\lim }L\left(1\right) \\ =\frac{2!}{{\left(s-2\right)}^3}-\frac{2}{{\left(s-2\right)}^2}+\frac{1}{\left(s-2\right)} \end{gathered}$$

Step-by-step solution

First translation theorem

If $L\left\{f\left(t\right)\right\}=f\left(S\right)$and a is a any real number.

Then,

role="math" localid="1664211811002" $L\left\{{\mathrm{e}}^{\mathrm{at}}\mathrm{f}\left(\mathrm{t}\right)\right\}=F(\mathrm{s}-\mathrm{a})$

Step 2: definition of indicated

The definition of indicate is point to and show statement acknowledging something. When you point to Susie, you are indicating.

Give the equation:

$$\begin{gathered} L\left\{e^{2t}{\left(t-1\right)}^2\right\} \\ =\left\{L{e}^{2t}\left(t^2-2t+1\right)\right\} \end{gathered}$$

Addition and subtraction the values:

$L\left\{t^2{e}^{2t}\right\}-2L\left\{t{e}^{2t}\right\}+L\left\{e^{2t}\right\}$

Lim apply the values:

data-custom-editor="chemistry" $$\begin{gathered} =\underset{s\to s-2}{\lim }\underset{s\to s-2}{L\left(t^2\right)-2\lim L\left(t\right)+}\underset{s\to s-2}{\lim }L\left(1\right) \\ =\frac{2!}{{\left(s-2\right)}^3}-\frac{2}{{\left(s-2\right)}^2}+\frac{1}{\left(s-2\right)} \end{gathered}$$

Hence,

$=\frac{2!}{{\left(s-2\right)}^3}-\frac{2}{{\left(s-2\right)}^2}+\frac{1}{\left(s-2\right)}$