Find the Laplace transforms of the given functions. displaystyle g{{left({t}right)}}={4} cos{{left({4}{t}right)}}-{9} sin{{left({4}{t}right)}}+{2} cos{{left({10}{t}right)}}

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asked 2021-03-11

Find the Laplace transforms of the given functions.
\(f{{\left({t}\right)}}={6}{e}^{{-{5}{t}}}+{e}^{{{3}{t}}}+{5}{t}^{{{3}}}-{9}\)

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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) \(\displaystyle{8}{e}^{{{9}{t}}}-{6} \cos{{h}}{\left({3}{t}\right)}\)

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asked 2020-11-01

Find the inverse Laplace transform \(f{{\left({t}\right)}}={L}^{ -{{1}}}{\left\lbrace{F}{\left({s}\right)}\right\rbrace}\) of each of the following functions.
\({\left({i}\right)}{F}{\left({s}\right)}=\frac{{{2}{s}+{1}}}{{{s}^{2}-{2}{s}+{1}}}\)
Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms.
\({\left({i}{i}\right)}{F}{\left({s}\right)}=\frac{{{3}{s}+{2}}}{{{s}^{2}-{3}{s}+{2}}}\)
Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms.
\({\left({i}{i}{i}\right)}{F}{\left({s}\right)}=\frac{{{3}{s}^{2}+{4}}}{{{\left({s}^{2}+{1}\right)}{\left({s}-{1}\right)}}}\)
Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms.

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asked 2020-11-29

Find the Laplace transform of \(\displaystyle f{{\left({t}\right)}}={t}{e}^{{-{t}}} \sin{{\left({2}{t}\right)}}\)
Then you obtain \(\displaystyle{F}{\left({s}\right)}=\frac{{{4}{s}+{a}}}{{\left({\left({s}+{1}\right)}^{2}+{4}\right)}^{2}}\)
Please type in a = ?

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asked 2020-10-28

Find laplace transform of each following
a) \(\displaystyle{t}^{n}\)
b) \(\displaystyle \cos{\omega}{t}\)
c) \(\displaystyle \sin{{h}}{\left({c}{t}\right)}\)
d) \(\displaystyle \cos{{h}}{\left({c}{t}\right)}\)

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asked 2020-11-29

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.
(b) A mechanical system is discovered during an archaeological dig in Ethiopia. Rather than break it open, the investigators subjected it to a unit impulse. It was found that the motion of the system in response to the unit impulse is given by \(w(t) = u(t)e^{-\frac{t}{2}} \sin(\frac{3t}{2})\)
(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?

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asked 2020-11-23

Find the Laplace transforms of the functions given in problem
\(f(t)=\sin \pi t \text{ if } 2\leq t\leq3 ,\)
\(f(t)=0 \text{ if } t<2 \text{ or if } t>3\)

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asked 2021-02-10

Find the equation by applying the Laplace transform.
\(\displaystyle{y}^{{{\left({4}\right)}}}-{y}= \sin{{h}}{t}\)
\(y(0)=y'(0)=y"(0)=0\)
\(y'''(0)=1\)

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asked 2021-01-05

Use Theorem 7.4.2 to evaluate the given Laplace transform. Do not evaluate the convolution integral before transforming.(Write your answer as a function of s.)
\({L}{\left\lbrace{e}^{{-{t}}}\cdot{e}^{t} \cos{{\left({t}\right)}}\right\rbrace}\)

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asked 2021-02-19

Find the inverse laplace trans.
\(\displaystyle{F}{\left({s}\right)}=\frac{10}{{{s}{\left({s}^{2}+{9}\right)}}}\)

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Answer 1

Step 1 : Introduction
Given function is,
\(\displaystyle g{{\left({t}\right)}}={4} \cos{{\left({4}{t}\right)}}-{9} \sin{{\left({4}{t}\right)}}+{2} \cos{{\left({10}{t}\right)}}\)
We have to evaluate the Laplace transform of given function g(t)
step 2
\(\displaystyle g{{\left({t}\right)}}={4} \cos{{\left({4}{t}\right)}}-{9} \sin{{\left({4}{t}\right)}}+{2} \cos{{\left({10}{t}\right)}}\)
using Laplace transform properties
\(\displaystyle{L}{\left\lbrace \sin{{a}}{t}\right\rbrace}=\frac{a}{{{s}^{2}+{a}^{2}}}\)
\(\displaystyle{L}{\left\lbrace \cos{{a}}{t}\right\rbrace}=\frac{s}{{{s}^{2}+{a}^{2}}}\)
Applying Laplace transform on g(t) function
\(\displaystyle\therefore{L}{\left\lbrace g{{\left({t}\right)}}\right\rbrace}={L}{\left\lbrace{4} \cos{{\left({4}{t}\right)}}-{9} \sin{{\left({4}{t}\right)}}+{2} \cos{{\left({10}{t}\right)}}\right\rbrace}\)
\(\displaystyle{L}{\left\lbrace g{{\left({t}\right)}}\right\rbrace}={4}{L}{\left\lbrace \cos{{\left({4}{t}\right)}}\right\rbrace}-{9}{L}{\left\lbrace \sin{{\left({4}{t}\right)}}\right\rbrace}+{2}{L}{\left\lbrace \cos{{\left({10}{t}\right)}}\right\rbrace}\)
\(\displaystyle={4}{\left(\frac{s}{{{s}^{2}+{4}^{2}}}\right)}-{9}{\left(\frac{4}{{{s}^{2}+{4}^{2}}}\right)}+{2}{\left(\frac{s}{{{s}^{2}+{10}^{2}}}\right)}\)
\(\displaystyle=\frac{{{4}{s}}}{{{s}^{2}+{16}}}-\frac{36}{{{s}^{2}+{16}}}+\frac{{{2}{s}}}{{{s}^{2}+{100}}}\)
Thus , Laplace transform of given function g(t) is
\(\displaystyle{L}{\left\lbrace g{{\left({t}\right)}}\right\rbrace}=\frac{{{4}{s}}}{{{s}^{2}+{16}}}-\frac{36}{{{s}^{2}+{16}}}+\frac{{{2}{s}}}{{{s}^{2}+{100}}}\)

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Find the Laplace transforms of the given functions. displaystyle g{{left({t}right)}}={4} cos{{left({4}{t}right)}}-{9} sin{{left({4}{t}right)}}+{2} cos{{left({10}{t}right)}}

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