Given the function begin{cases}e^{-t}& text{if } 0leq t<2 0&text{if } 2leq tend{cases} Express f(t) in terms of the shifted unit step function u(t -a) F(t) - ? Now find the Laplace transform F(s) of f(t) F(s) - ?

Question
Laplace transform
asked 2020-11-08
Given the function \(\begin{cases}e^{-t}& \text{if } 0\leq t<2\\ 0&\text{if } 2\leq t\end{cases}\)</span>
Express f(t) in terms of the shifted unit step function u(t -a)
F(t) - ?
Now find the Laplace transform F(s) of f(t)
F(s) - ?

Answers (1)

2020-11-09
Step 1
Since f(t)=0 for \(2\leq t \text{ and } t<0\)</span> so create step function: \(u(t)-u(t-2)\) which will give those zero values.
Thus, \(f(t)=e^{-t}[u(t)-u(t-2)] \text{ or } f(t)=e^{-t}u(t)-e^{-t}u(t-2)\)
Step 2
Calculate Laplace transform:
\(F(s)=\int_{-\infty}^\infty f(t)e^{-st}dt\)
\(=\int_{-\infty}^\infty(e^{-t}u(t)-e^{-t}u(t-2))e^{-st}dt\)
\(=\int_{-\infty}^\infty(e^{-t}u(t))e^{-st}dt - \int_{-\infty}^\infty(e^{-t}u(t-2))e^{-st}dt\)
\(=\int_0^\infty e^{-(s+1)t}dt - \int_2^\infty e^{-(s+1)t}dt\)
\(=\frac{1}{(s+1)}- \frac{e^{-2(s+1)}}{(s+1)}\)
\(=\frac{1-e^{-2(s+1)}}{(s+1)}\)
Step 3
Thus, Laplace transform is \(F(s)=\frac{1-e^{-2(s+1)}}{(s+1)}\)
0

Relevant Questions

asked 2020-12-25
Let x(t) be the solution of the initial-value problem
(a) Find the Laplace transform F(s) of the forcing f(t).
(b) Find the Laplace transform X(s) of the solution x(t).
\(x"+8x'+20x=f(t)\)
\(x(0)=-3\)
\(x'(0)=5\)
\(\text{where the forcing } f(t) \text{ is given by }\)
\(f(t) = \begin{cases} t^2 & \quad \text{for } 0\leq t<2 ,\\ 4e^{2-t} & \quad \text{for } 2\leq t < \infty . \end{cases}\)
asked 2021-02-02
Solve the following differential equations using the Laplace transform and the unit step function
\(y"+4y=g(t)\)
\(y(0)=-1\)
\(y'(0)=0 , \text{ where } g(t)=\begin{cases}t &, t\leq 2\\5 & ,t > 2\end{cases}\)
\(y"-y=g(t)\)
\(y(0)=1\)
\(y'(0)=2 , \text{ where } g(t)=\begin{cases}1 &, t\leq 3\\t & ,t > 3\end{cases}\)
asked 2021-01-13
The function
\(\begin{cases}t & 0\leq t<1\\ e^t & t\geq1 \end{cases}\)
has the following Laplace transform,
\(L(f(t))=\int_0^1te^{-st}dt+\int_1^\infty e^{-(s+1)t}dt\)
True or False
asked 2021-03-02
\(\text{Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by }\)
\(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:}
\(F(s)=\int_0^\infty e^{-st}e^{-t}dt=\int_0^\infty e^{-(s+1)t}dt=\frac{1}{s+1}\)
\(\text{ Verify the following Laplace transforms, where u is a real number. }\)
\(f(t)=t \rightarrow F(s)=\frac{1}{s^2}\)
asked 2021-02-12
Use the definition of Laplace Transforms to find \(L\left\{f(t)\right\}\)
\(f(t)=\begin{cases}-1 & 0\leq t <1\\1 & t\geq 1\end{cases}\)
Then rewrite f(t) as a sum of step functions, \(u_c(t)\), and show that by taking Laplace transforms, this yields the same answer as your direct computation.
asked 2020-11-29
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?
asked 2020-10-18
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.
\(f(t)=1 \rightarrow F(s)=\frac{1}{s}\)
asked 2020-11-22
Find the Laplace transform of the given function
\(\begin{cases}t & 0,4\leq t<\infty \\0 & 4\leq t<\infty \end{cases}\)
\(L\left\{f(t)\right\} - ?\)
asked 2020-11-23
Find the Laplace transforms of the functions given in problem
\(f(t)=\sin \pi t \text{ if } 2\leq t\leq3 ,\)
\(f(t)=0 \text{ if } t<2 \text{ or if } t>3\)
asked 2021-01-24
\(g(t)=\begin{cases}0 & 0 \(g(t)=\prod_{1,3}(t)+6\prod_{3,5}(t)+4u(t-5)\)
compute the Laplace transform of g(t)
...