Question

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

Short answer

$$\begin{gathered} L\left\{t{\left(e^t+e^{2t}\right)}^2\right\} \\ =L\left\{t\left(e^{2t}+2e^{3t}+e^{4t}\right)\right\} \\ \underset{S\to S-2}{\lim }L\left\{t\right\}+2\underset{S\to S-3}{\lim }L\left\{t\right\}+\underset{S\to S-4}{\lim }L\left\{t\right\} \\ =\frac{1}{{\left(S-2\right)}^2}+\frac{1}{{\left(S-3\right)}^2}+\frac{1}{{\left(S-4\right)}^2} \end{gathered}$$

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

$L\left\{e^{at}f\left(t\right)\right\}=F\left(s-a\right)$

Step 2: definition of indicated

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

Give the equation:

$$\begin{gathered} L\left\{t{\left(e^t+e^{2t}\right)}^2\right\} \\ =L\left\{t\left(e^{2t}+2e^{3t}+e^{4t}\right)\right\} \end{gathered}$$

Addition the values:

$L\left\{t{e}^{2t}\right\}+\left\{L2e^{3t}\right\}+\left\{L{e}^{4t}\right\}$

Lim the equation:

$$\begin{gathered} =\underset{S\to S-2}{\lim }L\left\{t\right\}+2\underset{S\to S-3}{\lim }L\left\{t\right\}+\underset{S\to S-4}{\lim }L\left\{t\right\} \\ =\frac{1}{{\left(S-2\right)}^2}+\frac{1}{{\left(S-3\right)}^2}+\frac{1}{{\left(S-4\right)}^2} \end{gathered}$$

Hence,

$=\frac{1}{{\left(S-2\right)}^2}+\frac{1}{{\left(S-3\right)}^2}+\frac{1}{{\left(S-4\right)}^2}$