1. ### I Inverse Laplace transform of a rational function

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...
2. ### I Transfer Function relating momentum and force

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...
3. ### Solving a 2D PDE using the Fourier Transform

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...
4. ### Laplace Transform Time Shift Property

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...
5. ### Laplace & Inverse Laplace transforms

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...
6. ### Wave equation, separation of variables and the Laplace transform

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
7. ### Laplace transform (translation on the s-axis)

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...
8. ### Laplace transform of derivative of convolution

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...
9. ### Laplace transforms in circuit analysis, finding missing values

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...
10. ### Inverting Shifted Laplace function

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...
11. ### A Modeling diffusion and convection in a complex system

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...

19. ### I How to find Y(s)/X(s)

y(t) = u(t - a) . x(t) u(t) is a unit step function. I have to find Y(s)/X(s). Do I have to do convolution in frequency domain?
20. ### Dartmouth Extended Laplace Tables -- Not general enough? item26.a

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...
21. ### Help with Laplace Transform

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'...
22. ### Explicit check for Laplace transform?

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...
23. ### How would I take the laplace transform of f(t)= te^tsin^2(t)?

How would I take the laplace transform of f(t)= te^tsin^2(t)?
24. ### Laplace tranform of unitstep (heaviside) function

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...
25. ### Laplace transform of y''(t')

The ordinary differential equation, with initial values,shall be solved using Laplace transform. The ODE looks like this $$y''(t')+2y''(t)-2y(t)=0$$ And the initial conditions are $$y(0)=y'(0)=0, y''(0)=0$$ The problem is with the first...
26. ### Is there a mistake in my calculation or in my reasoning?

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...
27. ### Keep getting the wrong answer in this lengthy Laplace problem

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...
28. ### Laplace Transforms Involving: Unit-Step, and Ramp Functions

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 -...
29. ### Prove a Laplace Transform Equality

Homework Statement Prove that http://www4f.wolframalpha.com/Calculate/MSP/MSP26931c1531g07285beh7000062h7f6g1ggd95eea?MSPStoreType=image/gif&s=5&w=98.&h=38. [Broken] =http://www4f.wolframalpha.com/Calculate/MSP/MSP6901c153574d0bdbh20000048829f0g4d1fi1d0?MSPStoreType=image/gif&s=5&w=69.&h=35...
30. ### A discussion about Fourier and Laplace transforms and calculus

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...