Question

in problems find either f(S)or f(t), as indicated

$\mathit{L}\left\{e^{3t}\left(9-4t+10sin\frac{t}{2}\right)\right\}$

Short answer

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

$$\begin{gathered} L{e}^{3t}\left(9-4t+10\sin \left(\frac{t}{2}\right)\right) \\ =9\underset{s\to s-3}{\lim }L\left\{1\right\}-4\underset{s\to s-3}{\lim }L\left\{t\right\}+10\underset{s\to s-3}{\lim }L\sin \left(\frac{t}{2}\right) \\ =9\underset{s\to s-3}{\lim }\frac{1}{s}-4\underset{s\to s-3}{\lim }\frac{1}{s^2}+10\underset{s\to s-3}{\lim }\frac{1}{2\left(S^2+\frac{1}{4}\right)} \end{gathered}$$

Step-by-step solution

Step 1: definition of indicated

The definition of indicate is to refer to, point to, or show something, or to make a brief statement acknowledging something. When you point to Susie, you are indicating

$$\begin{gathered} L{e}^{3t}\left(9-4t+10\sin \left(\frac{t}{2}\right)\right) \\ =9L{e}^{3t}-4Lt{e}^{3t}+10L{e}^{3t}\left(\frac{t}{2}\right) \\ =9\underset{s\to s-3}{\lim }L\left\{1\right\}-4\underset{s\to s-3}{\lim }L\left\{t\right\}+10\underset{s\to s-3}{\lim }L\sin \left(\frac{t}{2}\right) \end{gathered}$$

First translation theorem

We move on

$$\begin{gathered} =9\underset{s\to s-3}{\lim }L\left\{1\right\}-4\underset{s\to s-3}{\lim }L\left\{t\right\}+10\underset{s\to s-3}{\lim }L\sin \left(\frac{t}{2}\right) \\ =\frac{9}{S-4}-\frac{9}{{\left(S-3\right)}^2}+\frac{5}{{\left(S-3\right)}^2+\frac{1}{4}} \end{gathered}$$

Hence, the solution is$=\frac{9}{S-4}-\frac{9}{{\left(S-3\right)}^2}+\frac{5}{{\left(S-3\right)}^2+\frac{1}{4}}$