# Find the laplace transform of the following: MULTIPLICATION BY POWER OF t g(t)=(t^2-3t+2)sin(3t)

Question
Laplace transform
Find the laplace transform of the following:
MULTIPLICATION BY POWER OF t
$$g(t)=(t^2-3t+2)\sin(3t)$$

2020-12-25
Step 1
To find Laplace transform of $$g(t)=(t^2-3t+2)\sin(3t)$$, we simplify the expression and we get
$$g(t)=t^2\sin(3t)-3t\sin(3t)+2\sin(3t)$$
Using the following results:
$$1. L(f(t))=F(s)$$
$$2. L(t^nf(t))=(-1)^n\frac{d^nF(s)}{ds^n}$$
$$3. L(\sin(at))=\frac{a}{s^2+a^2}$$
Step 2
Using the above results and finding the required transforms,
$$L(\sin(3t))=\frac{3}{s^2+3^2}=\frac{3}{s^2+9}$$
$$L(t^2\sin(3t))=(-1)\frac{2}{d}$$
Now,
$$L(-3t \sin(3t))=-3(-1)\frac{d}{ds}(L(\sin(3t)))$$
$$=3\frac{d}{ds}\left(\frac{3}{s^2+9}\right)$$
$$=3\left(\frac{-3 \cdot 2s}{(s^2+9)^2}\right)$$
$$=\frac{-18s}{(s^2+9)^2}$$
Now,
$$L(2\sin(3t))=2\frac{3}{(s^2+9)}$$
$$=\frac{6}{(s^2+9)}$$
Step 3
Therefore,
$$L(g(t))=L(t^2\sin(3t)-3t\sin(3t)+2\sin(3t))$$
$$=L[t^2\sin(3t)]+L[-3t\sin(3t)]+L[2\sin(3t)]$$
Substituting the values of Laplace transforms from above step, we get
$$G(s)=\frac{6[3s^2-9]}{(s^2+9)^3}-\frac{18s}{(s^2+9)^2}+\frac{6}{(s^2+9)}$$

### Relevant Questions

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 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) $$\cos(t)+4\sin(t)$$
Solve the following differential equations using the Laplace transform and the unit step function
$$y"+4y=g(t)$$
$$y(0)=-1$$
$$y'(0)=0 , \text{ where } g(t)=\begin{cases}t &, t\leq 2\\5 & ,t > 2\end{cases}$$
$$y"-y=g(t)$$
$$y(0)=1$$
$$y'(0)=2 , \text{ where } g(t)=\begin{cases}1 &, t\leq 3\\t & ,t > 3\end{cases}$$
Use the Laplace transform table and the linearity of the Laplace transform to determine the following transform.
$$L\left\{e^{3t}\sin(4t)-t^{4}+e^{t}\right\}$$
$$\text{Find the Laplace transform }\ F(s)=L\left\{f(t)\right\}\ \text{of the function }\ f(t)=6+\sin(3t) \ \text{defined on the interval }\ t\geq0$$
Find the laplace transform of the following:
$$a) t^2 \sin kt$$
$$b) t\sin kt$$
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) $$f(t)=e^{t^2}\sin(\omega t)$$
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?
$$\text{(i) }\ f(t)=\cos(3t)$$
$$\text{(ii) }\ f(t)=t^{\frac{1}{2}}$$
$$\text{Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by }$$
$$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}$$