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# determine a function f(t) that has the given Laplace transform F(s)=frac{4s+5}{s^{2}+9} # determine a function f(t) that has the given Laplace transform F(s)=frac{4s+5}{s^{2}+9}

Question
Laplace transform asked 2020-12-17
determine a function f(t) that has the given Laplace transform
$$F(s)=\frac{4s+5}{s^{2}+9}$$

## Answers (1) 2020-12-18
Step 1
Given Laplace transform is $$F(s)=\frac{4s+5}{s^{2}+9} The function can be obtained as follows. \(f(t)=L^{-1}\left\{F(s)\right\} \(=L^{-1}\left\{\frac{4s+5}{s^{2}+9}\right\} \(=L^{-1}\left\{\frac{4s}{s^2+9}\right\}+L^{-1}\left\{\frac{5}{s^2+9}\right\} Step 2 On further simplifications, \(f(t)=4L^{-1}\left\{\frac{s}{s^2+3^{2}}\right\}+\frac{5}{3}L^{-1}\left\{\frac{3}{s^2+3^{2}}\right\} \(=4\cos(3t)+\frac{5}{3}\sin(3t)$$

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Part II
29.[Poles] (a) For each of the pole diagrams below:
(i) Describe common features of all functions f(t) whose Laplace transforms have the given pole diagram.
(ii) Write down two examples of such f(t) and F(s).
The diagrams are: $$(1) {1,i,-i}. (2) {-1+4i,-1-4i}. (3) {-1}. (4)$$ The empty diagram.
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(i) What is the characteristic polynomial of the system? What is the transfer function W(s)?
(ii) Sketch the pole diagram of the system.
(ii) The team wants to transport this artifact to a museum. They know that vibrations from the truck that moves it result in vibrations of the system. They hope to avoid circular frequencies to which the system response has the greatest amplitude. What frequency should they avoid? asked 2021-03-02
$$\text{Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by }$$
$$F(s)=\int_0^\infty e^{-st} f(t)dt \(\text{where we assume s is a positive real number. For example, to find the Laplace transform of } f(t)=e^{-t} \text{ , the following improper integral is evaluated using integration by parts:} \(F(s)=\int_0^\infty e^{-st}e^{-t}dt=\int_0^\infty e^{-(s+1)t}dt=\frac{1}{s+1}$$
$$\text{ Verify the following Laplace transforms, where u is a real number. }$$
$$f(t)=t \rightarrow F(s)=\frac{1}{s^2}$$ asked 2020-10-18
Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by
$$F(s)=\int_0^\infty e^{-st}f(t)dt$$
where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^{-t}, the following improper integral is evaluated using integration by parts:
$$F(s)=\int_0^\infty e^{-st}e^{-t}dt=\int_0^\infty e^{-(s+1)t}dt=\frac{1}{(s+1)}$$
Verify the following Laplace transforms, where u is a real number.
$$f(t)=1 \rightarrow F(s)=\frac{1}{s}$$ asked 2020-10-25
Please provide steps
The inverse Laplace transform for
$$\displaystyle{F}{\left({s}\right)}=\frac{8}{{{s}+{9}}}-\frac{6}{{{s}^{2}-\sqrt{{3}}}}$$ is
a) $$\displaystyle{8}{e}^{{-{9}{t}}}-{6} \sin{{h}}{{\left({3}{t}\right)}}$$
b) $$\displaystyle{8}{e}^{{-{9}{t}}}-{6} \cos{{h}}{\left({3}{t}\right)}$$
c) $$\displaystyle{8}{e}^{{{9}{t}}}-{6} \sin{{h}}{\left({3}{t}\right)}$$
d) $$\displaystyle{8}{e}^{{{9}{t}}}-{6} \cos{{h}}{\left({3}{t}\right)}$$ asked 2021-02-08
Find the inverse Laplace transform of $$F(s)=\frac{(s+4)}{(s^2+9)}$$
a)$$\cos(t)+\frac{4}{3}\sin(t)$$
b)non of the above
c) $$\cos(3t)+\sin(3t)$$
d) $$\cos(3t)+\frac{4}{3} \sin(3t)$$
e)$$\cos(3t)+\frac{2}{3} \sin(3t)$$
f) $$\cos(t)+4\sin(t)$$ asked 2021-01-31
Find Laplace transform for
$$\frac{s+8}{s^2+4s+5}$$ asked 2021-01-08
$$f(t)=3e^{2t}$$
Determine L[f]
Let f be a function defined on an interval $$[0,\infty)$$
The Laplace transform of f is the function F(s) defined by
$$F(s) =\int_0^\infty e^{-st}f(t)dt$$
provided that the improper integral converges. We will usually denote the Laplace transform of f by L[f]. asked 2021-02-25
Use properties of the Laplace transform to answer the following
(a) If $$f(t)=(t+5)^2+t^2e^{5t}$$, find the Laplace transform,$$L[f(t)] = F(s)$$.
(b) If $$f(t) = 2e^{-t}\cos(3t+\frac{\pi}{4})$$, find the Laplace transform, $$L[f(t)] = F(s)$$. HINT:
$$\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha) \sin(\beta)$$
(c) If $$F(s) = \frac{7s^2-37s+64}{s(s^2-8s+16)}$$ find the inverse Laplace transform, $$L^{-1}|F(s)| = f(t)$$
(d) If $$F(s) = e^{-7s}(\frac{1}{s}+\frac{s}{s^2+1})$$ , find the inverse Laplace transform, $$L^{-1}[F(s)] = f(t)$$
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