I struggle to find an appropriate inverse Laplace transform of the following
$$F(p)= 2^n a^n \frac{p^{n-1}}{(p+a)^{2n}}, \quad a>0.$$
WolframAlpha gives as an answer
$$f(t)= 2^n a^n t^n \frac{_1F_1 (2n;n+1;-at)}{\Gamma(n+1)}, \quad (_1F_1 - \text{confluent hypergeometric function})$$
which...
Hey all,
I hope this is the correct forum section to post this in.
I heard about this problem from a youtube video but I've not been able to simulate it because the video was meant only for an introduction into PID control.
Here's the problem:
A remote control helicopter is hovering just...
Homework Statement
Solve the following partial differential equation , using Fourier Transform:
Given the following:
And a initial condition:
Homework Equations
The Attempt at a Solution
First , i associate spectral variables to the x and t variables:
## k ## is the spectral variable...
Homework Statement
I’m being asked to prove if and why (what instances in which) T<0 for the Laplace transform property of time shifting doesn’t hold.
Homework Equations
L{f(t-T)}=e^-aT* F(s)
The Attempt at a Solution
I know that for T<0 there are instances where the property cannot hold, but...
Homework Statement
I am given this equation:
and asked to solve using Laplace transforms
The Attempt at a Solution
This is what I did:
This seemed logical to me, I used partial fractions and it stayed pretty simple.
This is what the solutions my prof posted do:
Is my answer equivalent...
Homework Statement
Homework Equations
If i solve the wave equation using separation of variable and laplace tranform. Will i get the same answer ?
The Attempt at a Solution
Homework Statement
L-1{[(2s-1)]/[(s^2)(s+1)^3]}
Homework Equations
L{f(t)e^(at)}=F(s-a)
The Attempt at a Solution
I have tried million ways but the different exponents in the denominator are throwing me off.
The other problem is that I cannot use partial fractions, the homework instructions...
Prelude
Consider the convolution h(t) of two function f(t) and g(t):
$$h(t) = f(t) \ast g(t)=\int_0^t f(t-\tau) g(\tau) d \tau$$
then we know that by the properties of convolution
$$\frac{d h(t)}{d t} = \frac{d f(t)}{d t} \ast g(t) = f(t) \ast \frac{d g(t)}{d t}$$
Intermezzo
We also know that...
Homework Statement
The Attempt at a Solution
At this point, usually I would replace the values and sometimes separate into partial fractions, and then take the inverse Laplace transformation. So I know that the inverse Laplace needs to give me 6+12e^-2t.
I am given the answers in my...
Homework Statement
A beam is supported at one end, as shown in the diagram (PROBLEM 11 page 281 of Lea, 159 of the course pack). A block of mass M and length l is placed on the beam, as shown. Write down the known conditions at x = 0. Use the Laplace transform to solve for the beam...
I am trying to come up with an analytical solution (even as a infinte series etc.) for the following diffusion-convection problem.
A thin layer of gel (assumed rectangular) is in direct contact with a liquid layer (perfusate) flowing with velocity v in the x direction (left to right) just...
I understand the conditions for the existence of the inverse Laplace transforms are
$$\lim_{s\to\infty}F(s) = 0$$
and
$$
\lim_{s\to\infty}(sF(s))<\infty.
$$
I am interested in finding the inverse Laplace transform of a piecewise defined function defined, such as
$$F(s) =\begin{cases} 1-s...
This is mostly a procedural question regarding how to evaluate a Bromwich integral in a case that should be simple.
I'm looking at determining the inverse Laplace transform of a simple exponential F(s)=exp(-as), a>0. It is known that in this case f(t) = delta(t-a). Using the Bromwich formula...
Not homework question, just need clarification and explanation. How did the person get from the left equation to the right side. I know he's just simplifying. But he didn't include steps and I've been trying to work out how to no avail. Any help on how this person simplified the LHS to RHS? Thanks!!
Hello - I'm not sure this is where this should go, but I'm working with Laplace Transforms and differential equations, so this seems as good a place as any. Also, I doubt this is graduate level math strictly speaking, but I went about as high as you can go in calculus and linear algebra during...
Use laplace Transform to solve this ode:
So I got:
sV(s)-V(0)-12V(s)=U(s+5)
V(s)(s-12)=U(s+5)+1
V(s)=[U(s+5)+1]/(s-12)
Now to go back to time domain with Inverse Laplace Transform...My question is, how to transform U(s+5)/(s-12)?
Any help?
Thanks guys
Homework Statement
Differential equation: ##Ay''+By'+Cy=f(t)## with ##y_{0}=y'_{0}=0##
Write the solution as a convolution (##a \neq b##). Let ##f(t)= n## for ##t_{0} < t < t_{0}+\frac{1}{n}##. Find y and then let ##n \rightarrow \infty##.
Then solve the differential equation with...
Homework Statement
Given the Laplace transform
$$F_L(s) = \frac{1}{(s+2)(s^2+4)},$$
by using the complex inversion formula compute the inverse Laplace transform, ##f(t),## for the following regions of convergence:
(i) ##Re(s)<-2;##
(ii) ##-2<Re(s)<0;##
(iii) ##Re(s)>0.##
Homework Equations...
Homework Statement
[/B]
http://www.dartmouth.edu/~sullivan/22files/New Laplace Transform Table.pdf
(see item 26a)
homogenous solution to underdamped in amplitude phase form: (see attached image)
2. Relevant info
- non zero initial conditions: x(t=0) = xo AND dx/dt(t=0) = vo
- unforced...
Homework Statement
Diagram for a vehicle suspension is given. Displacement of wheel is given by 'x' and and displacement of body is 'y'.
Spring constant, k = (7*10^4) Nm
Damping coefficient, c = (3*10^3) N/m/s
mass,m = 250kg
a) Make a Laplace Transform of system and utilize it to predict 'y'...
Homework Statement
Solve the following initial value problem using Laplace transforms: y' + 4y = 3t3 e−4t ; y(0) = 0 . Useful information: Recall that the Laplace transform of y 0 is pY − y(0), where Y is the Laplace Transform of y. The Laplace transform of tk e−at is k!/(p + a)k+1 . Confirm...
Homework Statement
[/B]
I know for t[u(t)-u(t-2)], we can simplify that to tu(t)-((t-2)+2)u(t-2) which then gives us tu(t)-(t-2)u(t-2)-u(t-2). Now, the laplace transform seems trivial but I am having problems with this equation:
sin(t)[u(t)-u(t-pi)]
Homework...
The ordinary differential equation, with initial values,shall be solved using Laplace transform. The ODE looks like this
\begin{equation}
y''(t')+2y''(t)-2y(t)=0
\end{equation}
And the initial conditions are
\begin{equation}
y(0)=y'(0)=0, y''(0)=0
\end{equation}
The problem is with the first...
Homework Statement
y'' + 3y' + 2y = r(t),
r(t) = u(t - 1) - u(t - 2),
y(0) = y'(0) = 0.
I need to solve this by convolution, which I know is commutative. The problem is that my calculation gives (f * g) =/= (g * f). Could someone please tell me where my mistake is?
Homework Equations...
Homework Statement
y'' + 4y = 8t^2 if 0 < t < 5, and 0 if t > 5; y(1) = 1 + cos(2), y'(1) = 4 - 2sin(2). Use the Laplace transform to find y.
Homework Equations
t-shift, s-shift, unit step function.
The Attempt at a Solution
I have been trying to solve it for hours, but keep getting the wrong...
Homework Statement
Here is an imgur link to my assignment: http://imgur.com/N0l2Buk
I also uploaded it as a picture and attached it to this post.
Homework Equations
u_c (t) =
\begin{cases}
1 & \text{if } t \geq c \\
0 & \text{if } t < c
\end{cases}
The Attempt at a Solution
Question 1.1 -...
i have read many of the answers and explanations about the similarities and differences between laplace and fourier transform.
Laplace can be used to analyze unstable systems.
Fourier is a subset of laplace.
Some signals have fourier but laplace is not defined , for instance cosine or sine...