# Find the Laplace transforms of the following time functions. Solve problem 1(a) and 1 (b) using the Laplace transform definition i.e. integration. For problem 1(c) and 1(d) you can use the Laplace Transform Tables. a)f(t)=1+2t b)f(t) =sin omega t text{Hint: Use Euler’s relationship, } sinomega t = frac{e^(jomega t)-e^(-jomega t)}{2j} c)f(t)=sin(2t)+2cos(2t)+e^{-t}sin(2t)

Question
Laplace transform
Find the Laplace transforms of the following time functions.
Solve problem 1(a) and 1 (b) using the Laplace transform definition i.e. integration. For problem 1(c) and 1(d) you can use the Laplace Transform Tables.
a)$$f(t)=1+2t$$ b)$$f(t) =\sin \omega t \text{Hint: Use Euler’s relationship, } \sin\omega t = \frac{e^(j\omega t)-e^(-j\omega t)}{2j}$$
c)$$f(t)=\sin(2t)+2\cos(2t)+e^{-t}\sin(2t)$$

2021-02-22
Step 1 To find the Laplace transform of the following functions.
Laplace transform of a function f(t) can be defined as $$L(f(t))=\int_0^\infty f(t)e^{-st}dt$$
Step 2
a) To find the Laplace transform of $$f(t)=1+2t$$
$$L(f(t))=L(1+2t)$$
$$=\int_0^\infty (1+2t)e^{-st}dt$$
$$=\int_0^\infty e^{-st}dt+\int_0^\infty(2t)e^{-st}dt$$
$$=\left[\frac{e^{-st}}{-s}\right]_0^\infty+2\left[(t)\cdot\frac{e^{-st}}{-s}\right]-\left[1\cdot \frac{e^{-st}}{s^2}\right]_0^\infty$$
$$=\left[\frac{0-1}{-s}\right]+2\left[(0-0)-(0-\frac{1}{s^2}\right]$$
$$=\frac{1}{s}+\frac{2}{s^2}$$
$$=\frac{s+2}{s^2}$$ Thus, $$L(f(t))=\frac{s+2}{s^2}$$
Step 3
b) To find the Laplace transform of $$f(t)=\sin \omega t = \frac{e^{j\omega t}-e^{-j\omega t}}{2j}$$
$$L(f(t))=L(\sin\omega t)$$
$$=\int_0^\infty\sin\omega t \cdot e^{-st}dt$$
$$=\int_0^\infty\frac{e^{j\omega t}-e^{-j\omega t}}{2j} \cdot e^{-st}dt$$ $$=\frac{1}{2j}\int_0^\infty e^{(j\omega-s)t}-e^{-(j\omega+s)t}dt$$
$$=\frac{1}{2j}\left[\left(\frac{e^{(j\omega-s)t)}}{j\omega-s}\right)-\left(\frac{e^{-(j\omega+s)t}}{-(j\omega+s)}\right)\right]_0^\infty$$
$$=\frac{1}{2j}\left[\left(\frac{e^{-(s-j\omega)t}}{-(s-j\omega)}\right)-\left(\frac{e^{-(j\omega+s)t}}{-(j\omega+s)}\right)\right]_0^\infty$$
$$=\frac{1}{2j}\left[\frac{0-1}{-(s-j\omega)}-\frac{0-1}{-(j\omega+s)}\right]$$
$$=\frac{1}{2j}\left[\frac{1}{(s-j\omega)}-\frac{1}{(j\omega+s)}\right]$$
$$=\frac{1}{2j}\left[\frac{1}{(s-j\omega)}-\frac{1}{(s+j\omega)}\right]$$ $$=\frac{1}{2j}\left[\frac{(s+j\omega)-(s-j\omega)}{(s-j\omega)(s+j\omega)}\right]$$
$$=\frac{\omega}{s^2-(j\omega)^2}$$
$$=\frac{\omega}{s^2+\omega^2}$$ Since $$\omega^2=-1$$
$$L(f(t))=\frac{\omega}{(s^2+omega^2)}$$
Step 4
c) To find the Laplace transform of $$f(t)=\sin(2t)+2\cos(2t)+e^{-t}\sin(2t)$$
We have from the Laplace transform table,
$$L(\sin at)=\frac{a}{(s^2+a^2)}$$
$$L(\cos at)=\frac{s}{(s^2+a^2)}$$
$$\text{If } L(f(t))=F(t) \text{ then } L(f(t))=F(s-a)$$
Consider,
$$L(f(t))=L(\sin(2t)+2\cos(2t)+e^{-t}\sin(2t))$$
$$=\frac{2}{(s^2+4)}+2\cdot \frac{s}{(s^2+4)}+\frac{2}{(s+1)^2+4}$$
$$=\frac{2}{(s^2+4)}+\frac{2s}{s^2+4}+\frac{2}{(s^2+2s+5)}$$
Thus, Laplace transform of $$f(t)=\sin(2t)+2\cos(2t)+e^{-t}\sin(2t)$$ is $$\frac{2}{s^2+4}+\frac{2s}{s^2+4}+\frac{2}{(s^2+2s+5)}$$

### Relevant Questions

Existence of Laplace Transform
Do the Laplace transforms for the following functions exist? Explain your answers. (You do not need to find the transforms , just show if they exist or not)
a) $$f(t)=t^2\sin(\omega t)$$
b) $$f(t)=e^{t^2}\sin(\omega t)$$
$$\text{Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by }$$
$$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}$$
Use properties of the Laplace transform to answer the following
(a) If $$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)$$
Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by
$$F(s)=\int_0^\infty e^{-st}f(t)dt$$
where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^{-t}, the following improper integral is evaluated using integration by parts:
$$F(s)=\int_0^\infty e^{-st}e^{-t}dt=\int_0^\infty e^{-(s+1)t}dt=\frac{1}{(s+1)}$$
Verify the following Laplace transforms, where u is a real number.
$$f(t)=1 \rightarrow F(s)=\frac{1}{s}$$
Find the inverse Laplace transform $$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)}}}$$
Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms.
In an integro-differential equation, the unknown dependent variable y appears within an integral, and its derivative $$\frac{dy}{dt}$$ also appears. Consider the following initial value problem, defined for t > 0:
$$\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}+{4}{\int_{{0}}^{{t}}}{y}{\left({t}-{w}\right)}{e}^{{-{4}{w}}}{d}{w}={3},{y}{\left({0}\right)}={0}$$
a) Use convolution and Laplace transforms to find the Laplace transform of the solution.
$${Y}{\left({s}\right)}={L}{\left\lbrace{y}{\left({t}\right)}\right)}{\rbrace}-?$$
b) Obtain the solution y(t).
y(t) - ?
Solve both
a)using the integral definition , find the convolution
$$f*g \text{ of } f(t)=\cos 2t , g(t)=e^t$$
b) Using above answer , find the Laplace Transform of f*g
Given that $$f{{\left({t}\right)}}={4}{e}^{{-{3}{\left({t}-{4}\right)}}}$$
a) Find $${L}{\left[\frac{{{d} f{{\left({t}\right)}}}}{{{\left.{d}{t}\right.}}}\right]}$$ by differentiating f(t) and then using the Laplace transform tables in lecture notes.
b) Find $${L}{\left[\frac{{{d} f{{\left({t}\right)}}}}{{{\left.{d}{t}\right.}}}\right]}$$ using the theorem for differentiation
c) Repeat a) and b) for the case that $$f{{\left({t}\right)}}={4}{e}^{{-{3}{\left({t}-{4}\right)}}}{u}{\left({t}-{4}\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})$$
(i) What is the characteristic polynomial of the system? What is the transfer function W(s)?
(ii) Sketch the pole diagram of the system.
(ii) The team wants to transport this artifact to a museum. They know that vibrations from the truck that moves it result in vibrations of the system. They hope to avoid circular frequencies to which the system response has the greatest amplitude. What frequency should they avoid?
$$L(y) = \left[\frac{w}{(s^2 + a^2)(s^2+w^2)}\right]*b$$
$$\frac{d^2y}{dt^2}+a^2y=b \sin(\omega t)$$ where $$y(0)=0$$
and $$y'(0)=0$$