What is Laplace: Definition and 1000 Discussions

Pierre-Simon, marquis de Laplace (; French: [pjɛʁ simɔ̃ laplas]; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized and extended the work of his predecessors in his five-volume Mécanique Céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace.Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is also named after him. He restated and developed the nebular hypothesis of the origin of the Solar System and was one of the first scientists to postulate the existence of black holes and the notion of gravitational collapse.
Laplace is remembered as one of the greatest scientists of all time. Sometimes referred to as the French Newton or Newton of France, he has been described as possessing a phenomenal natural mathematical faculty superior to that of any of his contemporaries.
He was Napoleon's examiner when Napoleon attended the École Militaire in Paris in 1784.
Laplace became a count of the Empire in 1806 and was named a marquis in 1817, after the Bourbon Restoration.

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  1. S

    Verification of Laplace transform answer

    Homework Statement Find the Laplace transform of the time function using the integral definition: g(t) = e-a(t+T)*u(t+T) Homework Equations Laplace transform definition The Attempt at a Solution After working through it, I was curious to see if the answer I received is, in...
  2. W

    Laplace transform for system DFE

    dx/dt= -x+y dy/dt= 2x x(0)=0 y(0)=1 I'm not familiar how to solve a system like this. Somebody please help? Thanks
  3. C

    Finding the Potential Between Two Coaxial Cylinders Using Laplace's Equation

    Homework Statement Two coaxial cylinders, radii {a,b} where b>a. Find the potential between the two cylinder surfaces. Boundary conditions: V(a,\phi) = 2 \cos \phi V(b,\phi) = 12 \sin \phiHomework Equations Solution by separation of variables: V(r,\phi) = a_0 + b_0 \ln s + \sum_k \left[...
  4. C

    Laplace transform with step function

    Homework Statement y'+2y=f(t), y(0)=0, where f(t)=\left\{t, 0\leq t<0 \;and \; 0, t \geq 0 (I don't know how to do piecewise)Homework Equations Equation for initial conditions; step function: u(t-a) The Attempt at a Solution I know how to solve a differential equation with initial...
  5. F

    Help with E/M Potential using Laplace Solution

    I'm struggling to learn how to find the potential using laplace solution. I know that X(x) can be rewritten in terms of C1e^kx + D1e-kx OR C1 cosh kx + D1 sinh kx OR cos kx + sin kx... but when do you know how to use which form. I understand it partially but not fully. And then how do you...
  6. M

    Laplace Transformations Step Functions

    Homework Statement The attachment is the problem. Homework Equations The Attempt at a Solution I understand how to go about solving the laplace transformations but I have no idea how to start with the Heaviside functions for the 5t and the 30. What I got was 5t+30U6(t) but it turned...
  7. 1

    Proving Laplace Transform for Complimentary Error Function - Step by Step Guide

    Can anybody show me how to prove this Laplace transform which leads to a complimentary error function? Thanks!\int^\infty_0 \frac{\sqrt{a}}{\pi \sqrt{x} (x+a)} e^{-x} dx = e^a erfc(\sqrt{a})I don't know how to separate a factor of \sqrt{\pi} from the laplace transform.
  8. F

    Solving y"+9y=H(t-1) with Laplace Transform

    Hi, just like to check my answer with someone for the this question calculate y"+9y=H(t-1) y(0)=0 y'(0)=-2 using laplace method H is heaviside fuction my solution: y=cos(3t)-2sin(3t)/3 + H(t-1)/9(H(t-1) + cos (3t-3) ) thanks
  9. Y

    What is the meaning of LaPlace and Poisson's equation in English?

    \nabla^2 V = \nabla \cdot \nabla V. Let me first break this down in English from my understanding: \nabla V is the gradient of a scalar function V. \nabla V is a vector field at each point P where the vector points to the direction the maximum rate of increase and |\nabla V| is the value...
  10. N

    Finding Solution to Laplace Equation

    Usually, we use the technique of "separation of variables" as follows: In a "separable coordinate system", we assume a separable solution \Phi=A(a)B(b)C(c) Then we obtain 3 ODEs for A(a), B(b), C(c) We note that there are actually entire families of solutions to each ODE, that happen to be...
  11. Saladsamurai

    Laplace of t*x''(t) Pulling the derivative through the integral

    Homework Statement I am working on solving the DE: t*x" + x' +t*x by use of the Laplace transform. Now if we just look at the first term and its transform, we have L[tx''(t)] = \int_0^\infty tx''e^{-st}\,dt = -\int_0^\infty \frac{d}{ds}\left(x''e^{-st}\right)\,dt Now in the next...
  12. C

    Laplace Transform (First Order Differential Equation With Initial Value)

    Homework Statement Y' + 8y = e^-2t*sint, with initial condition y(0) = 0 Homework Equations L{e^(-2t)sin(t)} = 1/((s+2)²+1) The Attempt at a Solution Alright so I've been working on this one for about an hour, I really don't know why but I'm having major problems with these types...
  13. C

    Laplace Transforms (First Order DE with Initial Value)

    Y' + 8y = e^-2t*sint, with initial condition y(0) = 0 Alright so I've been working on this one for about an hour, I really don't know why but I'm having major problems with these types of problems, whether i don't understand how to set it up or I don't understand partial fraction...
  14. M

    Very Trick laplace Transform Q

    Homework Statement solve the following differential equation using laplace transform: dy/dt+yx=0 y=20 when x=0 Homework Equations The Attempt at a Solution I took the laplace for each term L(dy/dx)= s*Y(s)-y(0) L(xy)=X(s)Y(s) subtitute back to the equation...
  15. M

    LaPlace Transformations to Solve Ordinary Differential Equations

    Homework Statement Consider the initial value problem: x'' + 2x' + 5x = δ(t - 1); with: x(0) = 0 and x'(0) = 0. Using Laplace transforms, solve the initial value problem for x(t). Homework Equations L[x''] = (s^2)*L[x] - s*x(0) - x'(0) L[x'] = s*L[x] - x(0) L[δ(t - 1)] = e^(-s)...
  16. T

    Electromagnetic waves, Maxwell's Equations, Laplace?

    Homework Statement Suppose Maxwell's displacement current was left out of the Maxwell equations. Show that , in a vacuum, the magnetic field has to have the form B = grad f(r,t), where f is any function which satisfies the Laplace equation. Homework Equations curl E = - dB/dt curl B = 0...
  17. S

    Solving the Laplace Transform of an Irreducible Quadratic Factor

    Homework Statement i need to find the laplace transform of this equation by using the partial fraction. it is an irreducible quadratic factor that i don't really know how to solve it. Homework Equations [(s+1)^2 + 6]/[(s+1)^2 +4]^2 The Attempt at a Solution please help me, thanks. =)
  18. G

    Laplace Transform simpilification

    Homework Statement Is there a simplified Laplace transform of The Attempt at a Solution This is actually a three part question, I was only able to solve it when q is 0 or pi/2. but i can't seem to figure out the general solution.
  19. I

    Laplace Transform ODE Solving: y'' + 2y' + y = 0 with Initial Conditions

    Solve using laplace. The diff eq is y'' + 2*y' + *y = 0 subject to y(0)=1 and y(pi)=0 Sorry if notation isn't the norm. y'' and y' and y are time (t) based functions.
  20. D

    Laplace and Divergence theorem

    Homework Statement Use Divergence theorem to determine an alternate formula for \int\int u \nabla^2 u dx dy dz Then use this to prove laplaces equation \nabla^2 u = 0 is unique. u is given on the boundary.Homework Equations u \nabla^2 u = \nabla * (u \nabla u) -(\nabla u)^2 The Attempt at...
  21. M

    How to Find the Inverse Laplace Transform for a Given Function?

    ""Invers laplace transformation"" Homework Statement Find invers laplace transform f(t) for F(s)= s /((s+a)^2 +w^2)) ? Homework Equations The Attempt at a Solution i have the final answer but i need the steps to know how i can do it exactly the final answer is ...
  22. R

    Signal Compression: Laplace vs. Fourier

    I had posted a thread on another forum about building a 100GHz oscilloscope which resulted in the signal being chopped up by a bank of comparators and subsequently being fed into a video buffer for processing using currently developed parallel processing techniques. The subject of compression...
  23. P

    What is the Inverse Laplace Transform of Y(s) = (1/τs+1)(1/s)?

    Homework Statement Take the Inverse Laplace Transform of: Y(s)=\frac{1}{\tau s+1}\cdot \frac{1}{s} 2. The attempt at a solution I know: L^{-1}(\frac{1}{\tau s+1})=\frac{1}{\tau}e^{\frac{-t}{\tau}} and: L^{-1}({\frac{1}{s}})=1 But how to continue?
  24. R

    Can you explain the usefulness of Laplace Transformation in signal analysis?

    Please anyone tell me how laplace transformation is derived. It transform a funtion into new one. Then what we get? Any example to show how it make a function easy to solve?
  25. P

    Laplace transform physical meaning

    When comparing time and frequency domains, it is easy to imagine the meaning of the Fourier transform. In time domain, our function takes time as a parameter, and returns the value (result) of our process. When we make a Fourier transform of the same function, we take it to the frequency...
  26. D

    Laplace transforms of heaviside step functions

    Homework Statement Consider the initial value problem y'' + 1/3y' + 4y = fk(t) with y(0) = y'(0) = 0, fk(t) = 1/2k for 4 - k < t < 4 + k 0 otherwise and 0 < k < 4. (a) Write fk(t) in terms of...
  27. M

    Laplace transform with time shift

    Homework Statement I need to find the Laplace transform of t*e-t*u(t-tau) 2. Homework Equations and attempt at solution I know the general form of the transform, but my problem is in the time shift of the step function. If both parts of the expression were (t-tau), then I could just...
  28. K

    Help to stregnthen laplace and second order diff eq beckground

    Homework Statement Here is my issue. I am an EE student that has been part time for almost five years. I took diff eq and signals a long time ago. I am currently taking a feedback and controls class, and my lack of memory in these topics is killing my study time. I am looking for relevant...
  29. M

    An understanding of Laplace Transform.

    Currently enrolled in an Electrical Engineering program, i naturally had to come across Laplace Transformation. Maths and specially calculus isn't my favourite :) What I understand is that Laplace Transform is more or less a tool used to simplify matters by defining a time based quantity in...
  30. F

    Inverse Laplace Transform

    Hey everyone. I've got through most of a problem that involves finding an inverse laplace transform, but I am stuck at one part that requires algebraic manipulation. The function is 1/[s(2s2+2s+1)] So far I have modified it too look like .5/[s(s+.5)2 +.52](1/.5) I'm not sure how to...
  31. W

    Exploring Applications of Laplace Transformations in Various Systems

    Hey I'm new here. Well we're currently doing Laplace in our Maths lectures. Now the Teacher has set us a project on Laplace and we need to find some applications of Laplace Transformations. Can anyone tell me some specific areas where Laplace is applied. I remember reading somewhere it's...
  32. J

    Laplace transformation problem

    Homework Statement L{4u(t-ர)cos2t} Homework Equations I have used UNIT STEP FUNCTION but could not get the result The Attempt at a Solution
  33. S

    Laplace Transform vs. Transfer Function

    Hi, I am confused between the difference between the Laplace transform and the Transfer function. I used to think that the Transfer Function was the Laplace transform of the Differential equation representation of a system, but in my readings it seems like that is incorrect - because...
  34. D

    Laplace pressure and bubbles.

    Homework Statement The bending of the surface of any liquid creates excess pressure, known as the Laplace pressure. Consider a T-shaped pipe with two bubbles of different diameters blown at the two ends across from each other. How will the two bubbles behave? Give qualitiative reasons...
  35. C

    Laplace Transform of sin(2t)

    im new to this forum. i really need help with the steps to solve the Laplace of sin(2t). i can put it in the formula to get the answer but I am having problems getting the steps which is what i need to follow and understand it better. if anyone can please help me with the steps it would be...
  36. J

    Maple General question about Maple's capabilities with Laplace transforms

    Working through a recent assignment, I've been dealing with an inverse laplace transform \mathcal{L}^{-1}\left[\frac{e^{-kds/v}}{(s+1)^2}\right] that Maple can't solve, yet I can do this by hand and WolframAlpha will solve it too. Should Maple be able to solve this (perhaps it requires extra...
  37. H

    Master Laplace Transforms with Expert Help: Solving X' + 7X = 5cos2t Problem

    Laplace Transforms Help! Homework Statement X' + 7X = 5cos2t I.C. x(0)=0 x'(0)=0
  38. J

    Laplace transforms and resonance

    Homework Statement http://img231.imageshack.us/img231/3904/q2a.png The Attempt at a Solution Below is a screenshot of my Maple worksheet. I am currently stuck on part (a). I have taken the Laplace transform of both sides of the equation, solved for Z(s), then converted it Heaviside form, yet...
  39. K

    Laplace transform of phase-shifted sinusoid

    Homework Statement Find the laplace transform of f(t)=sin(4t+\frac{\pi}{3})u(t). http://img822.imageshack.us/img822/6996/codecogseqna.gif Uploaded with ImageShack.us Homework Equations Definition of laplace transform, and properties...
  40. L

    How to Solve a Differential Equation with Laplace Transform?

    I have a differential equation that has to be solved with Laplace. I wish someone can provide a full answer y'' + 4y = x , 0<=x<π y'' + 4y = πe^-x , π<=x Initial Conditions: y(0)=0 y'(0)=1
  41. K

    Solving Laplace Transform Homework: Step-by-Step Guide

    Homework Statement I tried extract an s at the denominator and perform complete the square at denominator.But i feel it would be way too long to solve this problem. Gimme guide on how to tackle this problem Homework Equations...
  42. K

    Laplace Transform: Get the Answers You Need

    Gimme CLue
  43. S

    Laplace Transform - Sigma+jw | Control Theory Explained

    Hi guys..:) I have a doubt regarding laplace transform. can anyone tel me... what is the physical significance of sigma+jw in it...? what interpretations we can make from sigma and jw in control theory...?
  44. R

    What is the Laplace Transform of f(x)=e^{bx}.sin ax?

    What is the Laplace Transform of f(x)=e^{bx}.sin ax?
  45. G

    Laplace transform into partial fractions, then I'm stuck

    Homework Statement It's hard to explain, I can do everything exept get the answer to what I'v pointed out below. I just don't know what order to solve it into get the correct answer, I must have tried every method except the right one! Homework Equations The Attempt at a...
  46. Y

    Verify Harmonic function and Laplace equation.

    Harmonic function satisfies Laplace equation and have continuous 1st and 2nd partial derivatives. Laplace equation is \nabla^2 u=0. Using Green's 1st identity: \int_{\Omega} v \nabla^2 u \;+\; \nabla u \;\cdot \; \nabla v \; dx\;dy \;=\; \int_{\Gamma} v\frac{\partial u}{\partial n} \; ds...
  47. C

    Help with Inverse Laplace Problems!

    Hi! Im having a little (a lot) of trouble with some inverse Laplace problems. If anyone can help, I would really appreciate it! 1. 10s/(s^2+256)^2 2. 6s+6/(s+14)(s^2+4) With 1. I have tried countless ways of rearranging etc. getting (s^2+16^2)^2 to try to work it out that way... Im...
  48. B

    Laplace equation on a semi infinite slab

    Homework Statement Could someone check my work plaese. \frac{\partial^2u}{\partial x^2}(x,y)+\frac{\partial^2u}{\partial y^2}(x,y)=0 (0<x<1, 0<y) \frac{\partial u}{\partial x}(0,y)=\frac{\partial u}{\partial y}(1,y)=0 u(x,y)\rightarrow k as y\rightarrow\infty u(x,0)=f(x) (0\leqx\leq1)...
  49. D

    Laplace Transform - what are some applications?

    Hey guys, I have to do a presentation for my class on the laplace transform and need to know some applications. But so far, all I can find is electrical circuit applications, not much else. So if you guys know of any others tell me about them! thanks in advance!
  50. T

    Partial fraction (tricky stuff for LaPlace transforms)

    Homework Statement Doing some Laplace transform stuffs and I've got Y(s) = \frac{1}{(s+1)(s^{2}+2s+2)} Using the normal method 1 = A((s+1)^{2}+1) + B(s+1) I'm not sure this method is valid though as we had a complicated term (s+1)² + 1 after A. I can find A to be 1, but I don't trust my...
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