# Find the inverse Laplace transform of (any two) i) frac{(s^2+3)}{s(s^2+9)} ii) logleft(frac{(s+1)}{(s-1)}right)

Question
Laplace transform
Find the inverse Laplace transform of (any two)
i) $$\frac{(s^2+3)}{s(s^2+9)}$$
ii) $$\log\left(\frac{(s+1)}{(s-1)}\right)$$

2021-02-21
(i) The given Laplace transformation is,$$\frac{(s^2+3)}{s(s^2+9)}$$
We can write it,
$$\frac{s^2+3}{s(s^2+9)}=\frac{s^2}{s(s^2+9)}+\frac{3}{s(s^2+9)}$$
$$=\frac{s}{(s^2+9)+\frac{s^2-(s^2+9)}{s(s^2+9))} \left(\frac{3}{-9}\right)}$$
$$=\frac{s}{(s^2+9)+\frac{1}{3} \cdot \frac{s}{s^2+9}+\frac{1}{3} \cdot \frac{1}{s}}$$
$$=\frac{2}{3} \cdot \frac{s}{s^2+9} + \frac{1}{3} \cdot \frac{1}{s}$$
$$=\frac{2}{3} \cdot \frac{s}{s^2+3^2}+ \frac{1}{3} \cdot \frac{1}{s}$$
Now taking inverse Laplace of both sides we have,
$$L^{-1}\left[\frac{s^2+3}{s(s^2+9)}\right]=\frac{2}{3}L^{-1}\left[\frac{s}{(s^2+3^2)}\right]+\frac{1}{3}L^{-1}\left[\frac{1}{s}\right]$$
$$=\frac{2}{3}\cos3t+\frac{1}{3}$$
ANSWER:$$L^{-1}\left[\frac{s^2+3}{s(s^2+9)}\right]=\frac{2}{3}\cos3t+\frac{1}{3}$$
Step 3
(ii)
The given Laplace transformation is,
$$\log\left(\frac{s+1}{s-1}\right)$$
We can write it,
$$\log\left(\frac{s+1}{s-1}\right)=\log(s+1)-\log(s-1)$$
Now,the property of Laplace transformation is,
$$tf(t) \leftrightarrow -\frac{d}{ds}[F(s)]$$
$$\text{Therefore, }$$
$$tf(t) \leftrightarrow -\frac{d}{ds}\left[\log(s+1)-\log(s-1)\right]$$
$$\Rightarrow tf(t) \leftrightarrow -\frac{1}{s+1}+\frac{1}{s-1}$$
$$\Rightarrow tf(t) \leftrightarrow -e^{-t}+e^t$$
$$\Rightarrow f(t) \leftrightarrow \frac{e^t-e^{-t}}{t}=2\left(\frac{e^t-e^{-t}} {2t}\right)=2\left(\frac{\sin ht}{t}\right)$$
Therefore,
$$L^{-1}\left[\log\frac{s+1}{s-1}\right]=2\left(\frac{\sin ht}{t}\right)$$
Answer: $$L^{-1}\left[\log\frac{s+1}{s-1}\right]=2\left(\frac{\sin ht}{t}\right)$$

### Relevant Questions

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.
Use the appropriate algebra and Table of Laplace's Transform to find the given inverse Laplace transform. $$L^{-1}\left\{\frac{1}{(s-1)^2}-\frac{120}{(s+3)^6}\right\}$$
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 of the given function.
$${F}{\left({s}\right)}=\frac{{{2}{s}-{3}}}{{{s}^{2}-{4}}}$$
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) $$\cos(t)+4\sin(t)$$
Find the inverse Laplace transform of the given function by using the convolution theorem. $${F}{\left({s}\right)}=\frac{s}{{{\left({s}+{1}\right)}{\left({s}^{2}+{4}\right)}}}$$
Solution of I.V.P for harmonic oscillator with driving force is given by Inverse Laplace transform
$$y"+\omega^{2}y=\sin \gamma t , y(0)=0,y'(0)=0$$
$$1)\ y(t)=L^{-1}\bigg(\frac{\gamma}{(s^{2}+\omega^{2})^{2}}\bigg) \(2)\ y(t)=L^{-1}\bigg(\frac{\gamma}{s^{2}+\omega^{2}}\bigg) \(3)\ y(t)=L^{-1}\bigg(\frac{\gamma}{(s^{2}+\gamma^{2})^{2}}\bigg) \(4)\ y(t)=L^{-1}\bigg(\frac{\gamma}{(s^{2}+\gamma^{2})(s^{2}+\omega^{2})}\bigg)$$
Part II
29.[Poles] (a) For each of the pole diagrams below:
(i) Describe common features of all functions f(t) whose Laplace transforms have the given pole diagram.
(ii) Write down two examples of such f(t) and F(s).
The diagrams are: $$(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})$$
(i) What is the characteristic polynomial of the system? What is the transfer function W(s)?
(ii) Sketch the pole diagram of the system.
(ii) The team wants to transport this artifact to a museum. They know that vibrations from the truck that moves it result in vibrations of the system. They hope to avoid circular frequencies to which the system response has the greatest amplitude. What frequency should they avoid?
Find inverse Laplace transform $$L^{-1}\left\{\frac{s-5}{s^2+5s-24}\right\}$$ Please provide supporting details for your answer
$$\frac{6s+9}{s^2+17} s>0$$
$$y(t)=\dots$$