# 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)}}

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

2021-02-22
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
$$\displaystyle{L}{\left\lbrace g{{\left({t}\right)}}\right\rbrace}=\frac{{{4}{s}}}{{{s}^{2}+{16}}}-\frac{36}{{{s}^{2}+{16}}}+\frac{{{2}{s}}}{{{s}^{2}+{100}}}$$

### Relevant Questions

Find the Laplace transforms of the given functions.
$$f{{\left({t}\right)}}={6}{e}^{{-{5}{t}}}+{e}^{{{3}{t}}}+{5}{t}^{{{3}}}-{9}$$
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) $$\displaystyle{8}{e}^{{{9}{t}}}-{6} \cos{{h}}{\left({3}{t}\right)}$$
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.
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 = ?
Find laplace transform of each following
a) $$\displaystyle{t}^{n}$$
b) $$\displaystyle \cos{\omega}{t}$$
c) $$\displaystyle \sin{{h}}{\left({c}{t}\right)}$$
d) $$\displaystyle \cos{{h}}{\left({c}{t}\right)}$$
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?
Find the Laplace transforms of the functions given in problem
$$f(t)=\sin \pi t \text{ if } 2\leq t\leq3 ,$$
$$f(t)=0 \text{ if } t<2 \text{ or if } t>3$$
$$\displaystyle{y}^{{{\left({4}\right)}}}-{y}= \sin{{h}}{t}$$
$$y(0)=y'(0)=y"(0)=0$$
$$y'''(0)=1$$
$${L}{\left\lbrace{e}^{{-{t}}}\cdot{e}^{t} \cos{{\left({t}\right)}}\right\rbrace}$$
$$\displaystyle{F}{\left({s}\right)}=\frac{10}{{{s}{\left({s}^{2}+{9}\right)}}}$$