Laplace Definition and 1000 Threads

Hello PF, maybe you can help with this one! I need to show that the Laplace transform of J0(at) is (s^2 + a^2)^-1/2. My prof told me to start with the form: x2y'' + x y' + (x2 + p2)y = 0, where p = 0 ITC. What have I got so far?... Doing the Laplace transform on both sides, where Y...

Homework Statement Calculate the inverse Laplace transform of exp(-as)/s (a is a constant). Homework Equations The Attempt at a Solution I need to calculate the integral (1/2*pi*i)int(c-i(inf), c+i(inf))(exp(s(t-a))/s). I'm guessing I need to integrate around a circular...

Okay, I'm stumped at what seems like a very simple mathematical step I start with Then, the next step is I see what changed, but I don't understand exactly what happened. Can someone please explain? Thanks in advance!

How can I solve this Laplace transform (or how can I find series solution)? http://www.forkosh.dreamhost.com/mathtex.cgi?M=\int[/URL] _0^{\infty }e^{-\text{kR}}J_1^2(\text{ka})dk[/PLAIN] ;R>2a

Homework Statement ...find the inverse transform of the given function. F(s)= (2s+2)/(s2+4s+5) Homework Equations On another page, I see these 2: f(t)=eatsin(bt)--------->F(s)=b/[(s-a)2+b2) f(t)=eatcos(bt)--------->F(s)=(s-a)/[(s-a)2+b2) The Attempt at a Solution This is my first time...

Hey guys! I just have a question regarding some notes I've been looking at and trying to understand... Homework Statement Taking the transform: Just wondering, can someone explain to me why the e-3s/s is positive, and not negative because it is originally -u3(t)... Another quick...

Homework Statement y'' -6y' + 9y = f(t) f(t)= 0, 0<t<1 1, 1<t<3 0, t>3 Homework Equations The Attempt at a Solution Turning it to heaviside functions I get: y'' -6y' + 9y = u1 - u3 and I solve. in the answers it should be: y(t) = 2tet + u1(1/9 - 1/9e3(t-1)+...

I am wondering if there is a physical interpretation of the Laplace operator (also known as Laplacian, Δ, ∇2, or ∇·∇). From my impression a gradient of a function is the vector field in the direction with the greatest change. Also a divergence is volume density of flux from a point source...

Laplace Transform of unit step function HELP! Homework Statement f(t)= e^t on 0<=t<1 . . . . t on 1<=t<2 . . . . sin(t) on 2<=t<infinity Homework Equations Unit Step Function The Attempt at a Solution Here is my attempt at a solution...

Homework Statement Solve the Laplace equation: delta u = d2u/dx2+d2u/dy2 inside the half disk 0<r<R, 0<phi<pi Temperature on the bottom side of the disk is zero, u(x,y=0)=0. Temperature on the upper side of the disk is u(r=R, theta) = u0(phi), 0<phi<pi Homework Equations I'm...

Homework Statement Find the inverse Laplace of: X(s) = \frac{3}{s^2 - 6} I am kind of stuck on this one. I am pretty sure this is not sinusoidal. Can I even use partial fractions on this? Just a hint here :smile:

I'm trying to use the Laplace transform to work out another capacitor problem, the voltage as a function of time on a capacitor that's discharging into another capacitor through a resistor. It's the classic two capacitor problem, but I'd like to actually find an expression for the voltage as a...

Homework Statement I'm trying to use the Laplace transform to find the charging equation of a capacitor with an initial voltage already on the capacitor.Homework Equations V_i = RC\frac{dVc}{dt} + V_c The initial condition of Vc will be \lambda. The Attempt at a Solution V_i(s) = sRC*V_c(s)...

Homework Statement Given: dx/dt + 3*x = exp(-3*t) and all initial conditions are zero. Homework Equations Laplace The Attempt at a Solution L[dx/dt + 3*x = exp(-3*t)] s*X(s) + 3*X(s) = 1 / (s + 3) X(s) = 1 / (s + 3)^2 So here is where I get mixed up. For some reason, I thought I was...

In relativity, the scalar wave equation in the coordinate system (x,y,z,ict) is \frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}+\frac{\partial^2\phi}{\partial (ict)^2}=0 In 3D classical mechanics, the Laplace equation is:{when...

Hello, its been a pleasure finding you:smile: I have an asignment due to the end of this week and due to some problems, i hadn't found time to get to it so far. I have to calculate the exact solution of the Laplace equation in polar coordinates, in a hollow disk in the domain Ω where...

Homework Statement I just want to verify that I am dong this correctly and if not, where my misconceptions are. I keep getting a different answer so I am going to go through this step by step: After a some steps in a controls problem, I end up with a function in the Laplace domain...

Homework Statement Given: \frac{d^2y}{dt^2} + 12\frac{dy}{dt} + 32y = 32u(t) and that all initial conditions are zero, use the Laplace transform to find y(t) This is an example problem from my text. They start the solution by taking the Laplace of both sides assuing y'(0) = 0 & y(0)...

Homework Statement t y'' + (3 t-1) y' + 3 y = 6 e^{-3t} y(0) = 1 y(5) = 2 Homework Equations The Attempt at a Solution I tried applying the Laplace transform to the equation but I was having a little trouble... L [t·y''] = -dL[y'']/ds = s2·Y' + 2·s·Y - y(0) = s2·Y' + 2·s·Y - 1 L...

I am asked to find inverse laplace of (3s+5)/(s^2-6s+25) I do partial frac and get (3s)/((s-3)^2+16) + (5/(s-3)^2+16) I do the translation and get [(3s)/s^2+16 + 5/(s^2+16)]*e^3t My final answer is e^3t[3cos(4t) + 5/4sin(4t)] but the book says the sin term should be 7/2sin(4t)? I say...

Homework Statement Could \[F(s) = \frac{1}{{s(1 - {e^{ - s}})}}\] be the Laplace transform of some periodic function? Why? If so, find that periodic function The Attempt at a Solution If it was the Laplace transform of some periodic function, the the Laplace transform of the first wave...

Homework Statement If f(t) transforms into F(s), so that \[ F(s) = \frac{{s + 1}}{{s^2 + as + 1}},a \in \] , prove that if a < 0, the function f(t) isn't bounded, and if a >= 0, it is bounded. Prove that if -2 < a < 2, f(t) oscilates. The Attempt at a Solution I honestly have...

Hi, I've been doing Laplace transforms lately...the sums are pretty simple (as much as there is in our syllabus),and I've been practising hard...but the problem is, I don't really understand what I'm doing or why what I am dong is justified at all... I want to know certain things like...

I was looking through some tables of Laplace Transforms on f(t) the other day, and I noticed that in all cases, as s \to \infty, F(s) \to 0. A question that I have been trying to prove is that if \lim_{s\to\infty}F(s) = 0, then does that necessitate whether F(s) can undergo an inverse Laplace...

I have this DE: \[y'' - 4y' + 8y = f(t) = \left\{ \begin{array}{l} t{\rm{ }},t \in [0,2) \\ t + 1{\rm{ }},t \in [2,4) \\ 0{\rm{ }},t \ge 4 \\ \end{array} \right.\] I have problems transforming f(t). I know that when I have a function defined by parts, I...

Homework Statement The capacitor is assumed to consist of two parallel circular disc electrodes of radius R. The electrodes are of infinite small thickness, placed a distance 2H apart, and are equally and oppositely charged to potentials +U and -U. A metal cylinder is placed near the two...

Hello, I am trying to take the Laplace transform of floor(f(t)) in order to solve the differential equation f'(t)=floor(f(t)). I know that http://functions.wolfram.com/IntegerFunctions/Floor/22/04/" gives L(floor(t)) = (e^(-s))/(s(1-e^(-s))) instead -- are these equivalent?) and L(f(t)) =...

Hello, I am trying to take the Laplace transform of floor(f(t)) in order to solve the differential equation f'=floor(f(t)). I know that L(floor(t)) = (e^(-s))/(s(1-e^(-s))) and that L(f(t)) = F(t) (of course), but I realized that I have no idea how to take the Laplace transform of a...

it's konwn to all that laplace equation in 3 dimentions have clearly physical meaning,how about that in higher dimentions? I first thought that had something to do with relativity or quantum mechenics with higher dimentions ,but it is hard to figure it out . can anybody explain that ,thanks a lot

Hi, I want to inverse this laplace transform, -(s^(1/2)), seems that the inverse is in complex plane. Where should i start to find this inverse... Thank you.

Hey, I have been studying differential equations a bit and was wanting some help on some proofs. There are 3 laplace transforms I would like proofs for. Not really sure where to get started or if someone could lead me to place that has these proofs I would greatly appreciate it. Thank you.

I was asked to use a laplace transform to solve d^2x/dt^2 - 3(dx/dt) - 10x = e^-t with the initial conditions x(0) = 0 and x'(0) = 0 I got down to (s^2)x(s) - sx(0) - x'(0) - 3(sX(0) - x(0)) - 10X =1/(s+1) I tried plugging in the initail conditions and didnt get the answer I was supposed...

"easy" Laplace Transform...help! Homework Statement Evaluate L{(e^-2t)*t*sinht} Homework Equations translation theorem The Attempt at a Solution Just to clarify: the contents of the bracket is the product of 3 terms: e^-2t (e to the power of -2t) t sinht all...

Homework Statement The solution to the ODE y''(t) + 4y(t) = 1 + u(t − 2), y(0) = 0, y'(0) = 0 is given by... The Attempt at a Solution OK well I figured this one is good to solve with Laplace transforms. So I took the Laplace of both sides to obtain (s2 + 4)Y(s) = [e-2s/s] + 1/s, which...

Homework Statement This problem is killing me. I need to bust this thing up using partial fractions. Homework EquationsThe Attempt at a Solution I'm leaning towards it being separated like this. Is this correct? If it is, I'm not exactly sure what I'm supposed to do next.

Laplace inital value help! Homework Statement y''+4y'+4y=(t-2)u(t-2)exp-(t-2) y(0)=1 y'(0)=-1 what is the use of this u i have worked the problem to the following The Attempt at a Solution (s+2)^2*L(y)(s)= 5s-1 ... i have no idea how to approach the laplace ot the right side...

"Generalized" Laplace transform Hello, I'm having trouble proving injectivity of what might be called a "generalized" Laplace transform (not the one by Varma). Let f be a rational function and C be a fixed closed contour in the complex plane, (such that C contains not pole of f): The operator...

Homework Statement Find the inverse Laplace Transform of (9s+5)/(s2-9) given s>3 Homework Equations I have a table of Laplace Transforms. it says "sinh at" goes with a/(s2-a2) and "cosh at" goes with s/(s2-a2) The Attempt at a Solution I know how to split it up and find the...

Homework Statement Find the Laplace Transform of: f(t) = {0, t < 5 t2 - 10t + 31, t\geq5Homework Equations The Attempt at a Solution The answer I got was: F(s) = e-5s (2/s3 + 6/s) Is this correct?

Laplace Transformations... help me please? 1. Homework Statement . Find the laplace transformations of the following: a. \sin\, {\sqrt\,{x}} b. \frac{\cos\,{\sqrt{x}}}{{\sqrt{x}}} c. \ erf\,{(t)}^\frac{1}{2}} d. \int_{t}^\infty\;\frac{\cos\,x}{x}\ e...

Homework Statement Use the Laplace transform to solve the following initial value problem: x' = 7 x + 5 y, y'= -2 x + e5t, x(0)=0, y(0)=0 Find the expressions you obtain by taking the Laplace transform of both differential equations and solving for Y(s) and X(s) Homework Equations...

Homework Statement Find the Laplace transform of L{texp(9t)sin(2t)} Homework Equations (-1)^n F^n (s) The Attempt at a Solution I am not sure how to start it. The exp(9t) term throws me off. Usually I would just do (-1) F'(s) where F(s) is the laplace of sin(2t). But do I take...

I'm doing lapace transforms and I'm to the stage where I am finding inverse transforms. A problem I am stuck on involves doing partial fractions of a complex equation. I have attempted to do it, and now I have a question about my answer. My answer is -2i+1exp((1-i)*t) +...

I have to solve and IVP we were given an 2x2 matrix and i used the determinant method to solve for L=s^2 +2s+2, Lx =s-2 and Ly =s. so now i did this LX=Lx so s-2/(s^2+2s+2) and LY=Ly so s/(s^2+2s+2) now i have to Laplace them and I am stuck. I think i have to make the (s^2+2s+2) => ((s+1)^2 +1)...

Homework Statement supposed to prove informally that f(x)=Integral 0-->infinity (cosxt/(t^2+1))dt == Pi/2*Exp[-x] Homework Equations The Attempt at a Solution no clue, need some hints to get started

Homework Statement Just a quick question concerning a Laplace transformation... Find the Laplace transform of the following function: f(t)=10t3/2-e(-7t) Homework Equations The Attempt at a Solution I wasn't sure what to do with the t3/2 so I just followed the formula for t1/2...

Homework Statement Find the Laplace transform of f(t). f(t) = -9t + 7 + u5(-7 + 9t + et) Homework Equations The Attempt at a Solution The Laplace transform of -9t is just -9/t2 The Laplace transform of 7 is 7/s But I am not sure how to find it for u5(-7 + 9t + et). In the...

Homework Statement F(s)=(−2s−2)/((s−1)2+9) Find f(x) by computing L-1[F(s)] = f(x). Homework Equations The Attempt at a Solution I made put (s-1) in place of s in the numerator, then compensated for that by subtracting 2: F(s) = (-2(s-1) - 2 -2 ) / ((s-1)2 + 9) Then s-1...

I have some problems where I am to take the Laplace transform of some functions, for example te^{xt} not by using the integral definition but instead by using a table of transforms and the fact that the Laplace transform is a linear operator. However, the instructions say that "A preliminary...

Homework Statement Determine the inverse Laplace transform of the given function F(s) = (2s+1) / (s^2 + 16 ) Homework Equations the LaPlace transform of different functions The Attempt at a Solution I divide the above equation into 2 fractions one with the 2s in the numerator...