# 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)}

Question
Laplace transform
Find laplace transform of each following
a) $$\displaystyle{t}^{n}$$
b) $$\displaystyle \cos{\omega}{t}$$
c) $$\displaystyle \sin{{h}}{\left({c}{t}\right)}$$
d) $$\displaystyle \cos{{h}}{\left({c}{t}\right)}$$

2020-10-29
ANSWER FOR (A)We have to find Laplace transform of $$\displaystyle{t}^{n}$$, that is, $$\displaystyle{L}{\left\lbrace{t}^{n}\right\rbrace}$$.
Now from the Laplace Transform table we know that
$$\displaystyle{L}{\left\lbrace{t}^{n}\right\rbrace}=\frac{{{n}!}}{{{s}^{{{n}+{1}}}}}$$
This shows that
$$\displaystyle{L}{\left\lbrace{t}^{n}\right\rbrace}=\frac{{{n}!}}{{{s}^{{{n}+{1}}}}}$$
ANSWER FOR (B) We have to find Laplace transform of $$\displaystyle \cos{{\left(\omega{t}\right)}}$$, that is, L $$\displaystyle{\left\lbrace \cos{{\left(\omega{t}\right)}}\right\rbrace}$$
Now from the Laplace Transform table we know that
$$\displaystyle{L}{\left\lbrace \cos{{\left({a}{t}\right)}}\right\rbrace}=\frac{s}{{{s}^{2}+{a}^{2}}}$$
Whis shows that
$$\displaystyle{L}{\left\lbrace \cos{{\left(\omega{t}\right)}}\right\rbrace}=\frac{s}{{{s}^{2}+\omega^{2}}}$$
ANSWER FOR (C) We have to find Laplace transform of $$\displaystyle \sin{{h}}{\left({c}{t}\right)}$$, that is, L $$\displaystyle{\left\lbrace \sin{{h}}{\left({c}{t}\right)}\right\rbrace}$$
Now from the Laplace Transform table we know that
$$\displaystyle{L}{\left\lbrace \sin{{h}}{\left({a}{t}\right)}\right\rbrace}=\frac{a}{{{s}^{2}-{a}^{2}}}$$
Whis shows that
$$\displaystyle{L}{\left\lbrace \sinh{{\left({c}{t}\right)}}\right\rbrace}=\frac{c}{{{s}^{2}-{c}^{2}}}$$
ANSWER FOR (D) We have to find Laplace transform of $$\displaystyle \cos{{h}}{\left({c}{t}\right)}$$, that is, L $$\displaystyle{\left\lbrace \cos{{h}}{\left({c}{t}\right)}\right\rbrace}$$
Now from the Laplace Transform table we know that $$\displaystyle{L}{\left\lbrace \cos{{h}}{\left({a}{t}\right)}\right\rbrace}=\frac{s}{{{s}^{2}-{a}^{2}}}$$
Whis shows that
$$\displaystyle{L}{\left\lbrace \cos{{h}}{\left({c}{t}\right)}\right\rbrace}=\frac{s}{{{s}^{2}-{c}^{2}}}$$

### Relevant Questions

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 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)$$f(t)=\sin(2t)+2\cos(2t)+e^{-t}\sin(2t)$$
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, $$L^{-1}[F(s)] = f(t)$$
The Laplace inverse of $$L^{-1}\left[\frac{s}{s^2+5^2}\right]$$ is
$$a) \cos(5t)$$
$$b) \sin h(5t)$$
$$c) \sin(5t)$$
$$d) \cos h(5t)$$
Find the equation by applying the Laplace transform.
$$\displaystyle{y}^{{{\left({4}\right)}}}-{y}= \sin{{h}}{t}$$
$$y(0)=y'(0)=y"(0)=0$$
$$y'''(0)=1$$
ALSO, USE PARTIAL FRACTION WHEN YOU ARRIVE
$$L(y) = \left[\frac{w}{(s^2 + a^2)(s^2+w^2)}\right]*b$$
Problem 2 Solve the differential equation
$$\frac{d^2y}{dt^2}+a^2y=b \sin(\omega t)$$ where $$y(0)=0$$
and $$y'(0)=0$$
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) $$f(t)=e^{t^2}\sin(\omega t)$$
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) $$\cos(t)+4\sin(t)$$
Let $$y(t)=\int_0^tf(t)dt$$ If the Laplace transform of y(t) is given $$Y(s)=\frac{19}{(s^2+25)}$$ , find f(t)
a) $$f(t)=19 \sin(5t)$$
c) $$f(t)=6 \sin(2t)$$
d) $$f(t)=20 \cos(6t)$$
e) $$f(t)=19 \cos(5t)$$
Find the Laplace transform of $$\displaystyle f{{\left({t}\right)}}={t}{e}^{{-{t}}} \sin{{\left({2}{t}\right)}}$$
Then you obtain $$\displaystyle{F}{\left({s}\right)}=\frac{{{4}{s}+{a}}}{{\left({\left({s}+{1}\right)}^{2}+{4}\right)}^{2}}$$