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. R

    Exploring Laplace Transforms: Understanding the Division by t Theorem

    Okay so I am brushing up my Laplace transforms as an independent study and I come across this proof for the "division by t theorem". The idea proof it self I have no problems with except for the limits of the first integration, It feels like they just arbitrarily choose the limits to be from...
  2. R

    Physical interpretation of the Laplace operator

    "physical" interpretation of the Laplace operator Is there a "physical" interpretation of the Laplace operator, much like there exists a physical interpretation of the divergence and curl?
  3. noowutah

    De Moivre Laplace theorem question

    Homework Statement According to the de Moivre Laplace theorem \binom{n}{k}p^{k}q^{n-k}\approx\frac{1}{\sqrt{2\pi{}npq}}e^{-\frac{(k-np)^{2}}{2npq}} For p=q=1/2 this translates nicely into an approximation for the binomial distribution by the normal distribution (the +1/2 is a...
  4. T

    Help!applications of Laplace method

    Homework Statement Anyone knows the applications of Laplace method (not Laplace transform) in physics? I'm writing a project about the Laplace method. The given project area is Mechanics/Mathematical Physics, but any application I could find is about statistics. Homework Equations The...
  5. F

    Solving y''-6y+9y=2 with Laplace Transforms

    Homework Statement y''-6y+9y=2 y(0)=y'(0)=0 *Note* Professor will NOT allow use of partial fractions, so please don't use it.Homework Equations Laplace transform table Y=[y'(0)+sy(0)+ay(0)+R]/[(s^2)+as+b]The Attempt at a Solution Y=L(2)/(s-3)^2 L(2)=2/s Y=(2/s)[1/(s-3)^2]...
  6. P

    Find Real Values of K in Laplace Transform Homework

    Homework Statement If f(t)=K + 2cost and F(s) = L{f(t)}, find all the real values of K such that \int_{1}^{2}F(s)ds = 2ln5 The attempt at a solution So L{f(t)} = L{K} + L{2cost} = (K/s) + [2/(s2 + 1)] So \int_{1}^{2}\frac{K}{s}ds + \int_{1}^{2}\frac{2s}{s^{s}+1}ds = 2ln5 After...
  7. S

    Laplace Transform Initial Value Problem

    1. y'' -5y' + 6y = 0, y(0) = 1, y'(0) =2 3. [s^2 F(s) - s f(0) - f'(0)] -5 [F(s) - f(0)] + 6[F(s)] = 0 (s^2 +1)F(s) - (s -5)f(0) - f'(0) = 0 (s^2 + 1)F(s) - (s-5)(1) - 2 = 0 (s^2 + 1)F(s) = s -3 F(s) = (s-3)/(s^2 + 1) Here's where I'm stuck. I can't factor the denominator to...
  8. D

    Inverse Laplace Transform, can't use partial fractions

    Hi, this is my first post, and I'm not sure if I'm posting this in the right place. Homework Statement I have a 2nd order diff. equation: 3y''(t) + 2y'(t) + 5y(t) = 3 with initial values: y'(0) = 1, y(0) = 0 Homework Equations After using Laplace transform I get: Y(s) = (3 +...
  9. I

    Laplace heaviside step function question

    Homework Statement (quick sketch of part i. yes i know the graph should specifically end at t=4 lol) Homework Equations u(t-a) = 0 for t>a = 1/2 for t=a = 1 for t>aThe Attempt at a Solution i'm having trouble solving part (ii). i remember a friend of mine saying something like "start with...
  10. I

    How Do You Apply Time Shift Properties in Laplace Transforms?

    Hi guys, needing a bit of help understanding laplace transformations. Homework Statement 1. f(t) = (t-4)u(t-2) 2. g(t) = (2e^-4t)u(t-1) 3. h(t) = 5 cos(2t-1)u(t) Homework Equations Laplace transform table. The Attempt at a Solution So basically I am given the laplace...
  11. P

    Inverse Laplace transform for small 's', Taylor expansion

    Dear all, This question is close to the post "Laplace transform of a Taylor series expansion" in PhysicsForums.com, dated Jul06-09. This is my problem: Consider the Laplace transform F(s) = 1 / ( s - K(s) ) , where K(s) = -1/2 + i/(2*Pi) * ln[ ( Lambda - (b+i*s) )/( b + i*s...
  12. X

    Solving ODE Step function with Laplace

    Homework Statement Hi all came across this problem whilst doing some revision and i can't work out the answer Solve the following equation with laplace transformation Homework Equations y''+16y = f(t) = { 1 t < pi ] with y(0) = 0 and y'(0) = 0 _____________{ 0 t >= pi ]...
  13. A

    Calculators How can I perform Laplace transforms and inversions on a TI-89 Titanium?

    Hello I have a final coming up in ODE and I would like to check my answers during the exam. Does anybody know how to how to do laplace transforms/ inversions on a ti 89 titanium? I tried just typing lap(function) in my ti89 but that doesn't work. I tried searching google but couldn't find...
  14. C

    Working Examples RLC Circuit with 5V Ramp Voltage Laplace Transforms

    can anyone give me the working examples of current expressions of series RLC circuit using ramp voltage by Laplace transforms. thanks R=15, L= 0.4H, C=12uF ...voltage 5v
  15. A

    Laplace Transform: Time Scaling Property

    Hi all According to the textbook Signal and Systems by Oppenheim (2nd edition) pages 685 and 686, if the Laplace transform of x(t) is X(s) with ROC (region of convergence) R, then the Laplace transform of x(at) is (1/|a|)X(s/a) with ROC R/a. Consequently, for a>1, there is a compression...
  16. E

    Problem with Laplace transform unit step function questions

    Homework Statement L{2t u(t-1)} Homework Equations L{g(t) u(t-c)} = e^-cs L{g(t+c)} The Attempt at a Solution L{2t u(t-1)}=e^-s L{2(t+1)} L{2(t+1)}=2/s^2+2/s L{2t u(t-1)} = e^-s {2/s^2 + 2/s} i think the whole attempt is wrong , I'm getting confused in this type of...
  17. K

    Initial Value Problem using Laplace

    Is this problem possible? Solve the initial value problem x''(t) + 6x'(t) + 9x(t) = f(t); x(0) = N, x'(0) = M I get to X(s)=(F(t)+Ns+6N+M)/(s^2+15) and don't know where to go from here. Any help would be appreciated.
  18. T

    How Do You Transform f(t) = cos(t) for t>=2 Using Step Functions?

    Homework Statement just trying to get the below equation in terms of step functions, just not sure how to get the cos(t) with the same shifted variable f(t) = 0 for t<2 f(t)=cos(t) t>=2 The Attempt at a Solution this is what i got cos(t)u(t-2) just having troubles working out the...
  19. P

    Solving Laplace Transform Homework: Equations 1 & 2

    Homework Statement Im having trouble finding ways to manipulate equations to fit something from the table The two I'm stuck on are these 1. \frac{1}{s^{2}- 2s + 3} (\frac{1+(s^{2}+1)e^{-3\Pi S}}{(s^{2}+1)}) = Y(s) 2.\frac{1}{s^{2}- 2s + 2} (\frac{s}{s^{2}+1} + s - 2) = Y(s)...
  20. C

    Solve Piecewise Equation with Laplace

    Solve Piecewise Equation with Laplace (Problem Solved ty) Homework Statement Solve the initial value problem: v''=(1/3)v-| sin t for t in [0,pi/2] | 0 for t in [pi/2,infinity) v(0)=v'(0)=0 Homework Equations N/A The Attempt at a Solution First I...
  21. C

    Understanding the Laplace System of Equations for Initial Value Problems

    Homework Statement Solve the initial value problem | x'=x+y | y'=-x+y x(0)=y(0)=1 Homework Equations N/A The Attempt at a Solution This is actually a solved problem with a too many details missing. You have to set up the equation as | sX(s)-x(0)=X(s)+Y(s) |...
  22. P

    How Do You Solve a Second Order Differential Equation Using Laplace Transforms?

    [Solved] Laplace Transform Homework Statement \frac{d^{2}y}{dt^{2}}+4y=sin(t),\quad y(0)=0,\quad\frac{dy}{dt}(0)=0 Homework Equations Laplace transform is defined as: \mathcal{L}\{f(t)\} = \int_{-\infty}^{\infty}f(t)e^{st}dt The Attempt at a Solution...
  23. R

    Solving a Laplace Transform Integral

    \documentclass[12pt,a4paper,twoside]{article} \usepackage{amssymb,amsmath,amsbsy,amscd,fancyhdr} \usepackage[mathscr]{eucal} \usepackage{tikz} \usepackage{graphicx,cite,xcolor,comment,soul,ulem} \begin{document} I met some problem in the following integration; does anyone know how to...
  24. T

    Can you use inverse Laplace transforms to solve problems with repeated roots?

    I am looking for inverse Laplace transform techniques with repeated roots. Example (10-4s)/(s-2)^2
  25. M

    Laplace by parts -> series

    Homework Statement Find analytic solution of some kind: 0&=Y''(y)-\frac{\alpha^2 [u(y) + U]}{\epsilon}Y(y) U eigenvalue, u(y) known, epsilon & alpha paramertres,and& Y thing to be found Homework Equations u(y) is a parallel flow of some kind and laplace transform given by...
  26. I

    What is the inverse Laplace transform of F(s) = (s + 1)/(s^2 + 1)^2

    Homework Statement F(s) = (s + 1)/(s2 + 1)2 Homework Equations The Attempt at a Solution I used partial fractions but I still end up with a term that includes a (s^2 + 1)^2 in the denominator. I'm pretty much lost at this point.
  27. B

    How to apply Laplace transform on a non-homogeneous equation to find y

    Homework Statement do a laplace transform on a given non homogeneous equation. to eventually find out y Homework Equations i don't think the equation is needed. however this non homogeneous equation is equal to Ae^kt The Attempt at a Solution well. i applied the laplace to the...
  28. D

    Solve Laplace Transform DE: y''+4y'+4y=delta_3(t)

    Homework Statement Solve the differential equation: y'' + 4y' + 4y = delta_3 (t) y(0) = 0 and y'(0) = 1Homework Equations 1) lapace [delta_a] = e^-as 2) laplace inverse [ 1 / (s+a)^n] = [(t^n-1) / (n-1)! ] *e^-atThe Attempt at a Solution so I began by taking the laplace transform of both...
  29. S

    Finding Inverse Laplace Transform with Fractional Powers and Convolution

    Homework Statement Find the inverse laplace transform of: 1/(s-2)3 + 25/(s+1)(s-2)2 + s/(s-2)2 The attempt at a solution I get (1/2)e2tt2 + (25/7)e-t-(25/7)e2t+(75/7)te2t for the first two, but I'm not even sure where to start for s/(s-2)2. I was thinking it might use...
  30. R

    Inverse laplace of this beheemoth

    Homework Statement I had to solve an IVP using laplace transforms. the answer should be the laplace inverse of: (1-2e^(-s/2)+2e^(-s)+se^(-s))/((s^2)(1-e^(-s))(s^2+10s+14)) Homework Equations The Attempt at a Solution i tried partial frac. decomp. and failed miserably. I also...
  31. B

    Can someone explain Laplace transformations

    Can someone explain Laplace transformations! i don't understand ittttt. [edit] sorry. i hadnt even heard of laplace transformations until i found it in my assignment. basically i want to know how to use them so, some simple, well explained examples perhaps?
  32. M

    Solving LaPlace Equation: Water Potential=-KH=-x^3y+xy^3+5

    Homework Statement water potential=-KH=-x^3y+xy^3+5. Is this a solution to the Laplace equation and is yes, show how Homework Equations LaPlace equation= Partial derivitive^2 of H/partial deriv x^2+partial deriv^2H/partial deriv y^2=0 The Attempt at a Solution I have absolutely no...
  33. S

    Laplace of a Unit Step Function

    Homework Statement f(t) = {0, if t<4 and (t-3)^3 if t\geq4 The Attempt at a Solution I feel like its pretty basic but i can't get it down I have u(t-4)(t-4)^3 Can i change it to u(t-4)^4? Then do i multiply it out and take the laplace? If someone can work it out for me that would...
  34. F

    Solv Laplace Equation: Finite-Integral-Transform Method

    Hi everybody, I just want to know, anybody has any information or sources about the method of Finite-Integral-Transform method in order to solve the Laplace Equations. I couldn't find this topic in any texts, mostly they just introduce the method of SOV or Fourier Integral Transform. I...
  35. C

    Help finding inverse laplace transform?

    http://img215.imageshack.us/i/unledc.jpg/ Using the residue method, I found K1 and K2 to be -2 and 4 respectively. However, where do I go from there?
  36. J

    Laplace Transform of f(t)*u(t)

    Homework Statement I need to know what the laplace transform is for f(t) * u(t) More specifically, h(t) = cos(3t)*u(t) Homework Equations I know that f(t-to) * u(t-to) = e-tos * F(s)The Attempt at a Solution So, f(t) = cos(3t) F(s) = s/(s2 + 9) In the formula I gave in (2), I would...
  37. H

    Second Shifting theorem to find Laplace transform

    Can some one help me to understand this: I need to use the second shifting theorem to get the Laplace transform, given: f(t) = { 4 - t2 , t < 2 ...... 1 { 0 , t >= 2 They got this: f(t) = (4 - t2)(1 - u(t - 2)) .... 2 I know that the...
  38. L

    Inverse LaPlace Transform help

    Homework Statement Use the First Shifting Theorem (translation on the s axis) to find f(t). L-1[(2s-1)/(s2(s+1)3] Homework Equations The Attempt at a Solution This is going to take forever.. Can we assume that I am putting the L^-1 all over the place? [(2s-1)/(s^2(s+1)^3)] =...
  39. Q

    Laplace Transform and Convolution

    Homework Statement The signal x(t) = u(t-1) - u(t-3) is the input to an LTI system with the impulse response h(t) = u(t-5) - u(t-8). the system is initially at rest. a) Compute the output y(t) of this system using convolution. b) Compute the output y(t) of this system using the Laplace...
  40. A

    Signals & Systems - Laplace - Oscillatory Component

    Homework Statement What is that oscillatory component? And is my answer for the following correct? x(t) = u(t) H(s) = \frac{R}{R + sL} y(t) will contain oscillatory component if R > \frac{1}{L} True or False? Homework Equations Basic Laplace Transform: u(t) \longrightarrow...
  41. P

    Transfer function, Laplace Transform

    Homework Statement [PLAIN]http://img340.imageshack.us/img340/8508/circuit.png Find V2(s)/V1(s) (Laplace transform domain) Homework Equations KCL, capacitor equation. The Attempt at a Solution \frac {-V_2}{R} + \frac {V_1-V_2}{R} + C \frac {d(V_1-V_2)}{dt}=0 But in Laplace domain it follows...
  42. Y

    Ampere's Law and Laplace equations

    Homework Statement I need to write ampere's law (differential form) in the form of Laplace equations / Poisson equations and then solve them numerically using Matlab. Homework Equations Del x H = J The Attempt at a Solution pls see my attachment I need help from someone who is...
  43. M

    What is the Inverse Laplace Transform of (2s^2 + 20s + 36) / (s^3 + 6s^2 + 9s)?

    Homework Statement F(s) = (2s2 + 20s + 36) / (s3 + 6s2 + 9s) Homework Equations L[e-at] = 1 / (s + a) L-1[(s+a) / ((s+a)2 + w2)] = e-atcos(wt) L-1[w / ((s+a)2 + w2)] = e-atsin(wt) The Attempt at a Solution Factored the top by 2 and the bottom by s, but the rest of the top seems...
  44. R

    Is this the correct inverse laplace transform

    Homework Statement Well i did all my work but the forum logged me out and when i submitted it I lost it all because it wanted me to login again >_< anyways i need to find the inverse laplace transform of F(s)=[12*s^2 - (28+2*sqrt(3))*s + 26 + 6*sqrt(3)]/[(s-3)(s^2-2s+2)] Homework...
  45. L

    Laplace Transform of DE with Discontinuous Forcing Function

    This is my first time posting on these boards, so forgive me if I am posting incorrectly or in the wrong forum. I have been able to successfully work through problems with forcing functions of constant value "steps", but am having trouble working with those with linear portions of the...
  46. X

    Laplace transform of piecewise functions

    I do not have trouble doing these types of problems if the domains are not equalities. However for this type of problem.. f(t)= 0 , t=2 t , t\neq 2 (I do not know how to do piecewise function in latex) I figured this is on the right track... t-tu_{2}(t)+2tu_{3} I have a problem...
  47. K

    Differential equations using Laplace Transform

    Homework Statement ((d^2)x)/(d(t^2)) + 5(dx/dt) + 6x = f(t) f(t)=3 for (0<=t<6) f(t)=0 for t>=6 Homework Equations This I am not sure of, this is my question. Would I use the Second Order Differential equation to do this problem: L{((d^2)f)/(d(t^2))} = (s^2)F(s) - sf(0") -...
  48. D

    Laplace transform of cos2(t-1/8π), help?

    Homework Statement I'm working on some Differential Equations homework and I'm stuck. The question is apply the translation theorem to find the Laplace transform. f(t) = e^(-t/2)cos2(t-1/8π) I know how to apply the method, I just need to figure out how to transform the cosine...
  49. D

    Solving IVP using Laplace Transforms

    Homework Statement "Use Laplace transforms to solve the initial value problem" from section 4.2 in Elementary Differential Equations (6th ed.) Edwards & Penny x'' + x = cos3t; x(0) = 1 & x'(0) = 0 Homework Equations L{coskt} = s/(s²+k²) Apparently the answer is...
  50. B

    Laplace equation on square (using fourier transform)

    Hello, I'm trying to solve the following problem: \nabla^2 p = 0 \frac{\partial p}{\partial y}(x, y_{bot}) = \frac{\partial p}{\partial y}(x, y_{top}) = 0 \frac{\partial p}{\partial x}(x_{left}, y) = \frac{\partial p}{\partial x}(x_{right}, y) = C_0 which is the laplace equation...
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