Question

in problems 1-20 find either f(S)or f(t),as indicated $\mathit{L}\left\{\left(1-e^{-4t}+3e^{-4t}\right)cos5t\right\}$

Short answer

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

$\begin{array}{c}L\left\{\left(1-e^t+3e^{-4t}\right)\cos 5t\right\}\\ =L\left\{\cos 5t-\cos \left(5t\right)e^t+3\cos \left(5t\right)e^{-4t}\right\}\\ =\frac{s}{s^2+25}-\frac{s}{{\left(s-1\right)}^2+25}+\frac{s\left(S+4\right)}{{\left(s+4\right)}^2+25}\end{array}$

Step-by-step solution

Step1: 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.

$\begin{array}{l}L\left\{\left(1-e^t+3e^{-4t}\right)\cos 5t\right\}\\ =L\left\{\cos 5t-\cos \left(5t\right)e^t+3\cos \left(5t\right)e^{-4t}\right\}\end{array}$

Now,

$\begin{array}{l}L\left\{\left(1-e^t+3e^{-4t}\right)\cos 5t\right\}\\ =L\left\{\cos 5t-\cos \left(5t\right)e^t+3\cos \left(5t\right)e^{-4t}\right\}\\ =L\left\{\cos 5t\right\}-L\left\{\cos \left(5t\right)e^t\right\}+3L\left\{\cos \left(5t\right)e^{-4t}\right\}\end{array}$

First translation theorem

We move on

$$\begin{gathered} =\frac{s}{s^2-25}-\underset{S\to S-1}{\lim }\frac{s}{s^2+25}+3\underset{S\to S+4}{\lim }\frac{s}{s^2+25} \\ =\frac{s}{s^2+25}-\frac{s}{{\left(s-1\right)}^2+25}+\frac{s\left(S+4\right)}{{\left(s+4\right)}^2+25} \end{gathered}$$

First translation theorem

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

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

Hence, the solution is $\frac{s}{s^2+25}-\frac{s}{{\left(s-1\right)}^2+25}+\frac{s\left(S+4\right)}{{\left(s+4\right)}^2+25}$