# Find the inverse laplace transforms for the following functions F(s)=frac{e^{-3s}}{s^2} F(s)=frac{2e^{-5s}}{s^2+4}

Question
Laplace transform
Find the inverse laplace transforms for the following functions
$$F(s)=\frac{e^{-3s}}{s^2}$$
$$F(s)=\frac{2e^{-5s}}{s^2+4}$$

2021-02-14
Step 1
Given
the given functions are
$$F(s)=\frac{e^{-3s}}{s^2}$$
$$F(s)=\frac{2e^{-5s}}{s^2+4}$$
Apply formula $$L^{-1}\left\{G(s)e^{-as}\right\}=g(t-a)u(t-a)$$
Step 2
$$F(s)=\frac{e^{-3s}}{s^2}$$
$$L^{-1}(\frac{1}{s^2})=t$$
Apply formula.
$$L^{-1}\left\{\left(\frac{1}{s^2}\right)e^{-3s}\right\}=(t−3)u(t−3)$$
Hence inverse Laplace transform of the function is $$F(t)=(t−3)u(t−3)$$
$$F(s)=\frac{2e^{-5s}}{(s^2+4)}$$
$$L^{-1}\left(\frac{2}{(s^2+4)}\right)=2L^{-1}\left(\frac{1}{(s^2+4)}\right)=2 \cdot \frac{\sin(2t)}{2}=\sin(2t)$$
Apply formula.
$$L^{-1}\left\{\left(\frac{2}{s^2+4}\right)e^{-5s}\right\}=\sin[2(t-5)]u(t-5)=\sin[2t-10]u(t-5)$$
Hence inverse Laplace transform of the function is $$F(t)=\sin[2t-10]u(t-5)$$

### Relevant Questions

Find the inverse Laplace transforms of the functions given. Accurately sketch the time functions.
a) $$F(s)=\frac{3e^{-2s}}{s(s+3)}$$
b) $$F(s)=\frac{e^{-2s}}{s(s+1)}$$
c) $$F(s)=\frac{e^{-2s}-e^{-3s}}{2}$$
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, $$L^{-1}[F(s)] = f(t)$$
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.
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 inverse Laplace transform of the given function
$$F(s)=\frac{e^{-2}+e^{-2s}-e^{-3s}-e^{-4s}}{s}$$
Use Theorem 7.4.3 to find the Laplace transform F(s) of the given periodic function.
F(s)=?
Find the Laplace transform $$L\left\{u_3(t)(t^2-5t+6)\right\}$$
$$a) F(s)=e^{-3s}\left(\frac{2}{s^4}-\frac{5}{s^3}+\frac{6}{s^2}\right)$$
$$b) F(s)=e^{-3s}\left(\frac{2}{s^3}-\frac{5}{s^2}+\frac{6}{s}\right)$$
$$c) F(s)=e^{-3s}\frac{2+s}{s^4}$$
$$d) F(s)=e^{-3s}\frac{2+s}{s^3}$$
$$e) F(s)=e^{-3s}\frac{2-11s+30s^2}{s^3}$$
$$\text{Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by }$$
$$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}$$
Find inverse Laplace transform $$L^{-1}\left\{\frac{s-5}{s^2+5s-24}\right\}$$ Please provide supporting details for your answer