# Write down the qualitative form of the inverse Laplace transform of the following function. For each question first write down the poles of the function , X(s) a) X(s)=frac{s+1}{(s+2)(s^2+2s+2)(s^2+4)} b) X(s)=frac{1}{(2s^2+8s+20)(s^2+2s+2)(s+8)} c) X(s)=frac{1}{s^2(s^2+2s+5)(s+3)}

Question
Laplace transform
Write down the qualitative form of the inverse Laplace transform of the following function. For each question first write down the poles of the function , X(s)
a) $$X(s)=\frac{s+1}{(s+2)(s^2+2s+2)(s^2+4)}$$
b) $$X(s)=\frac{1}{(2s^2+8s+20)(s^2+2s+2)(s+8)}$$
c) $$X(s)=\frac{1}{s^2(s^2+2s+5)(s+3)}$$

2020-11-08
The given equation of Laplace transformation is:
$$X(s)=\frac{s+1}{(s+2)(s^2+2s+2)(s^2+4)}$$
Poles are the zeros of the denominator the equation of Laplace transformation
$$\Rightarrow (s+2)(s^2+2s+2)(s^2+4)=0$$
$$\Rightarrow s+2=0, s^2+2s+2=0 , s^2+4=0$$
So , the poles are : $$s=-2, -1+-i , \pm2i$$
To find the inverse Laplace transformation, use the partial fraction
$$X(s)=-\frac{1}{16(s+2)}+\frac{-7s-2}{80(s^2+4)}+\frac{3s+4}{20(s^2+2s+2)}$$
$$L^{-1}\left[X(s)\right]=L^{-1}\left[-\frac{1}{16(s+2)}\right]+L^{-1}\left[\frac{-7s-2}{80(s^2+4)}\right]+L^{-1}\left[\frac{3s+4}{20(s^2+2s+2)}\right]$$
Using the property of inverse Laplace transformation, we get
$$=\frac{-1}{16}L^{-1}\left[\frac{1}{(s+2)}\right]+\frac{-7}{80}L^-1\left[\frac {s}{(s^2+4)}\right]-\frac{1}{80}L^-1\left[\frac{2}{(s^2+4)}\right]+\frac{3}{20}L^-1\left[\frac {s}{(s+1)^2+1}\right]+\frac{1}{20}L^-1\left[\frac{4}{(s+1)^2+1}\right]$$
Using the formula of inverse Laplace transformation, we get
$$L^{-1}[\frac{1}{(s-a)}]=e^{at}$$
$$L^{-1}[\frac{s}{s^2+a^2}]=\cos(at)$$
$$L^{-1}[\frac{1}{s^2+a^2}]=\frac{1}{a \sin(at)}$$
$$L^{-1}[\frac{A(s-\lambda)+B}{(s-\lambda)^2+\mu^2}]=e^{\lambda t}(A\cos (\mu t)+B\sin (\mu t))$$
Part(b)
The given equation is:
$$X(s)=\frac{1}{(2s^2+8s+20)(s^2+2s+2)(s+8)}$$
So to find the poles set denominator =0
$$(2s^2+8s+20)(s^2+2s+2)(s+8)=0$$ poles are:
$$s=-8,-4\pm2i,-1\pmi$$
To find the inverse Laplace transformation use the partial fraction, we get
$$X(s)=\frac{3s+20}{600(s^2+8s+20)}+\frac{-9s+4}{1500(s^2+2s+2)}+\frac{1}{1000(s+8)}$$
Using the inverse Laplace properties and formula, we get
$$L^{-1}\left[X(s)\right]=L^{-1}\left[\frac{3s+20}{600(s^2+8s+20)}\right]+L^{-1}\left[\frac{-9s+4}{1500(s^2+2s+2)}\right]+L^{-1}\left[\frac{1}{1000(s+8)}\right]$$
$$L^{-1}[X(s)]=\frac{1}{200}L^{-1}\left[\frac{s+4}{(s+4)^2+4}\right]+\frac{1}{75}L^{-1}\left[\frac{1}{(s+4)^2+4}\right]+L^{-1}\left[\frac{-9s+4}{1500((s+1)^2+1)}\right]+L^-1\left[\frac{1}{1000(s+8)}\right]$$
Therefore, the required Laplace transformation is:
$$L^{-1}[X(s)]=\frac{1}{200}e^{-4t}\cos(2t)+\frac{1}{150}e^{-4t}\sin(2t)-\frac{3}{500}e^{-t}\cos(t)+\frac{13}{1500}e^{-t}\sin(t)+\frac{1}{1000}e^{-8t}$$

### Relevant Questions

Use the table of Laplace transform and properties to obtain the Laplace transform of the following functions. Specify which transform pair or property is used and write in the simplest form.
a) $$x(t)=\cos(3t)$$
b)$$y(t)=t \cos(3t)$$
c) $$z(t)=e^{-2t}\left[t \cos (3t)\right]$$
d) $$x(t)=3 \cos(2t)+5 \sin(8t)$$
e) $$y(t)=t^3+3t^2$$
f) $$z(t)=t^4e^{-2t}$$
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, $$L^{-1}[F(s)] = f(t)$$
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?
Use Theorem 7.4.3 to find the Laplace transform F(s) of the given periodic function.
F(s)=?

Inverse Laplace transformation

$$(s^2 + s)/(s^2 +1)(s^2 + 2s + 2)$$

Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
Solid NaBr is slowly added to a solution that is 0.010 M inCu+ and 0.010 M in Ag+. (a) Which compoundwill begin to precipitate first? (b) Calculate [Ag+] when CuBr justbegins to precipitate. (c) What percent of Ag+ remains in solutionat this point?
a) AgBr: $$\displaystyle{\left({0.010}+{s}\right)}{s}={4.2}\cdot{10}^{{-{8}}}$$ $$\displaystyle{s}={4.2}\cdot{10}^{{-{9}}}{M}{B}{r}$$ needed form PPT
CuBr: $$\displaystyle{\left({0.010}+{s}\right)}{s}={7.7}\cdot{\left({0.010}+{s}\right)}{s}={7.7}\cdot{10}^{{-{13}}}$$ Ag+=$$\displaystyle{1.8}\cdot{10}^{{-{7}}}$$
b) $$\displaystyle{4.2}\cdot{10}^{{-{6}}}{\left[{A}{g}+\right]}={7.7}\cdot{10}^{{-{13}}}$$ [Ag+]$$\displaystyle={1.8}\cdot{10}^{{-{7}}}$$
c) $$\displaystyle{\frac{{{1.8}\cdot{10}^{{-{7}}}}}{{{0.010}{M}}}}\cdot{100}\%={0.18}\%$$
$$\frac{3}{(s+2)^2}-\frac{2s+6}{(s^2+4)}$$
$$\displaystyle{F}{\left({s}\right)}=\frac{8}{{{s}+{9}}}-\frac{6}{{{s}^{2}-\sqrt{{3}}}}$$ 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 Laplace transform of the function $$L\left\{f^{(9)}(t)\right\}$$