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

    Solve Laplace Transform & MacLaurin Series Problems

    I have two questions: I had to find the Laplace transform of: t \cdot sin(t) Not by definition, using a table of transforms and the properties. I did: sin(t) = i \cdot sinh(it) = i \cdot \frac{e^{t}}{2}-i \cdot \frac{e^{-t}}{2} Then t \cdot sin(t) = it \cdot \frac{e^{t}}{2}-it \cdot...
  2. K

    Engineering Circuit Analysis via Laplace Transforms :yuck:

    Hi there, I have my final in my circuit analysis class tomorrow morning(at a community college, it feels like two semesters worth of content was covered in 18 weeks). Last week, the teacher went over LaPlace Transform analysis rather quickly, and then onto basic digital logic. I do not...
  3. J

    Laplace transforms for a nontrivial solution

    Homework Statement Transform the given DE to find a nontrivial solution such that x(0)=0. tx''+(t-2)x'+x=0Homework Equations The Attempt at a Solution Using L{f(t)}=-\frac{1}{t}F'(s), I got 4sX(s)+s(s+1)X'(s)=0. I see that it is separable, but I do not know how to go about separating it...
  4. N

    Fraction decomposition for inverse Laplace

    Homework Statement Find the solution of the givien initial value problem and draw its graph y''+2y'+2y = δ(t-π) y(0) = 1, y'(0) = 0 Homework Equations A Laplace transform chart would be very useful The Attempt at a Solution I chose to solve the equation with Laplace...
  5. 1

    I don't understand how to laplace transform heaviside functions

    Homework Statement (6-t)heaviside(t-2) This is just one term of the real problem I'm working, but it will serve to help me figure this out. Homework Equations The Attempt at a Solution http://www.wolframalpha.com/input/?i=laplace+transform+%7B%286-t%29heaviside%28t-2%29%7D...
  6. A

    Circuit Analysis - Laplace notation

    Homework Statement The circuit below uses ideal components and disguises itself as a second order system when in fact it is really two first order systems. Prior to t=0 switch S is open. Then suddenly at t=0 switch S is closed. Find the peak output voltage V_{0max}, and the time taken for...
  7. M

    Laplace Transforms: Math's Elegant Solutions

    Recently in my DiffEq class, we learned how to use, and come up with, Laplace transforms. After doing my homework, I realized that Laplace Transforms are my new favorite concept in math(just beating out double/triple integrals and their applications)! The transforms just look so elegant on a...
  8. M

    Solving Partial Differential Equations with Laplace Transform

    Homework Statement \dfrac{\partial^2 \varphi }{ \partial x^2} - \dfrac{\partial ^2 \varphi }{\partial t^2} = 1 Initial Conditions: \varphi (x, 0) = 1; \varphi_t (x, 0) = 1 Boundary Condition: \varphi (0, t) = 1 On 0 \leq x < \infty, 0 \leq t < \infty...
  9. S

    Renewal Process: laplace transform approach

    Homework Statement Use the Laplace transform approach to find the renewal function for a renewal process with interrenewal p.d.f. as follows: g(x) = (c^2)xe^(-cx) , x > 0 The Attempt at a Solution M*(s) = G*(s)/(1-G*(s)) where M*(s) and G*(s) denote laplace transforms I have that G*(s) =...
  10. 1

    Can't seem to get this DE solved using laplace transforms

    Homework Statement y''-4y'+4y = 0 y(0)=1, y'(0)=1 Homework Equations The Attempt at a Solution This is annoying, because it feels very easy and I don't know why I am not getting the answer. Skipping the routine: s^2L{y} - s - 1 - 4sL{y} + 4 + 4L{y} = 0 (s+3) / (s^2 - 4s +...
  11. B

    Is there a function that equals its Laplace transform?

    Google seems to provide not much information on this. In essence, I am asking about the eigenfunctions of the Laplace transform when λ=1? Anyone have any insights on this rather unusual problem? BiP
  12. Y

    Laplace Transform (Linear Algebra)

    Homework Statement Evaluate the laplace transform of {t2e7tsinh(3t)} Homework Equations Laplace transform of {tnf(t)}=(-1)ndn/ds2 * F(s) The Attempt at a Solution I've replaced it with (-1)2d2L{e7tsinh(3t)} I'm not sure how to proceed, though, as I don't really see how to take...
  13. M

    Laplace Equation Polar Form

    Homework Statement Solve the BVP: r^{2}u_{rr} + ru_{r} + u_{ψψ} = 0 0 ≤ r ≤ 1, 0 < ψ < 2π u(1,ψ) = 0.5(π - ψ) Homework Equations The Attempt at a Solution I've derived the general solution of u(r,ψ) = C + r^{n}Ʃ_{n}a_{n}cos nψ + b_{n}sin nψ, where a,b, C are...
  14. I

    Merits of the Laplace transform

    Hey all, Learning the Laplace transform and I get the point that it is a transformation but I would like to know what are some of the merits of the Laplace transform or more general why perform transformations in the first place. Any examples would be helpful.
  15. M

    Differential Equation with unit step function and Laplace transforms

    EDIT: Nevermind I see what I did wrong near the end. Homework Statement x'' + 4x = f(t) Where f(t) is 1 if t is between 0 and π, 0 if t > π. Initial conditions are x(0) = x'(0) = 0. Homework Equations Transform of a derivative: L(f^{(n)}(t)) = s^nF(s) - s^{n-1}f(0) -...-f^{n-1}(0)...
  16. J

    LaPlace D.E. Question #9

    y"-8y'+20y=tet, y(0)=0, y'(0)=0 I need to know if I made a mistake in getting to the step below: L-1{ 1/[(s-1)2(s2-8s+20)] } because when I solve that, I get something pretty gnarly..
  17. B

    Is the inverse of the Laplace transform unique?

    I've been wondering whether the Laplace transform is injective. Suppose I have that \int^{∞}_{0}e^{-st}f(t)dt = \int^{∞}_{0}e^{-st}g(t)dt for all s for which both integrals converge. Then is it true that f(t) = g(t) ? If so, any hints on how I might prove it? Thanks! BiP
  18. J

    Understanding the Limit Step in LaPlace Transforms

    I'm not sure about the answers to parts of the limit step of the work. Can you explain what the answer to that step should be?
  19. J

    Solving Limits with L'Hopital's Rule

    1. The limit as b approaches infinity always shows up as undefined on my calc so I don't know what to put for that section of the work. 2. What pat of the work is supposed to need L'hopital's rule? The integration?
  20. M

    Calculating the Laplace Transform of t^2*u(t-a)

    Homework Statement I am trying to work out the inverse Laplace transform of t^2*u(t-a). Homework Equations The Attempt at a Solution I have been old that it starts (t-a)^2*u(t-a)... which I understand is substituting for (t-a). But I cannot seem to work out the rest can...
  21. M

    Integration of Laplace transform

    Homework Statement Find the inverse transform of the function F(s) = log\frac{s-2}{s+2} Homework Equations L(\frac{f(t)}{t}) = \int^{∞}_{s}F(x)dx f(t) = tL^{-1}(\int^{∞}_{s}F(x)dx) The Attempt at a Solution I missed the lecture on this and while I was able to figure out...
  22. A

    Transforming Inverse Laplace Equations with a Shifting Theorem

    Homework Statement L-1{\frac{s}{s^2+4s+5}} Homework Equations \frac{s-a}{(s-a)^2+k^2} \frac{k}{(s-a)^2+k^2} The Attempt at a Solution I completed the square for the denominator and got: L-1{\frac{s}{(s+2)^2+1}} (a= -2, k=1) But how do I get rid of the s in the numerator? Or do I have...
  23. U

    Stuck on Laplace Transform of Odd Trig Function

    Hey guys! I'm stuck on a Laplace transform. Following is the problematic function: [cos(t)]^3 Seems simple, but I'm having issues doing the Laplace transform on odd trigonometric functions. When I use the half-angle formula, I get this, which I can't seem to solve: 1/2cos(t) +...
  24. Z

    Solving Laplace Transforms Homework

    Homework Statement I'm given the transfer function of LTI system is \frac{1}{s^2 + 4} Homework Equations H(s) = \frac{Y(s)}{X(s)} The Attempt at a Solution first of all I had to find diff. equations of the system. I found that it's y'' + 4*y = x; Then they asked to find such...
  25. Fernando Revilla

    MHB Ggnore's question at Yahoo Answers (IVP, Laplace transform)

    Here is the question: Here is a link to the question: Consider the initial value problem for y; Laplace? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  26. R

    Does the Laplace Teansform of the function exist?

    While reading Laplace transform in my book, I came across a problem. where it evaluates the laplace transform of f(t)=1/√t => √(π/s) . But is the laplace transform of f(t) really exists? because I thought the function f(t) is not defined at t=0 and it tends to infinite. So How is it...
  27. N

    Heaviside and Laplace Transforms

    Homework Statement Rearrange f(t) using Heaviside Step Function Then Rearrange it so that the Laplace Transform can be written down Then, Write the Laplace Transform of f(t) Homework Equations The Attempt at a Solution So my first step is as follows... Using the basic Piecewise Function in...
  28. N

    Using Laplace Transforms to Solve ODE with Piecewise Forcing Function

    Homework Statement ODE: y'' + 4y' + 3y = f(t) f(t) = (?? HELP - What's the mathematical term to describe these? I can't seem t o find it in my notes :cry: ) 1, 0 ≤ t < 2 t², 2 ≤ t < 3 0, t ≥ 3 Write a brief description on how you would solve this ODE using Laplace transforms. Also use the...
  29. F

    Solve Laplace Transform of Diff. Equation: x(t) with ICs x(0)=-2, dx/dt(0)=8

    Homework Statement Differential equation: \frac{d^2x}{dt^2}+4\frac{dx}{dt}+4x=6e^{-2t} with initial conditions x(0)=-2 and \frac{dx}{dt}(0)=8 Use the laplace transform to solve for x(t) Homework Equations http://www.atp.ruhr-uni-bochum.de/rt1/syscontrol/node9.html...
  30. I

    Simple step function, Laplace transform

    Homework Statement A system is characterized by the equation y' + 3y = r' . When the input is r(t) = u(t) - u(t-1), find y(t) by taking the inverse Laplace transform of Y(s).Homework Equations The Laplace transform integral The Laplace transform of a derivative sF(s) - f(0) The transfer...
  31. M

    Engineering Laplace Step Response Circuit Analysis

    Homework Statement Shown in attachment Homework Equations The Attempt at a Solution I'm trying to analyze the circuit in the attached picture. This is a step response with a 3V input or 3u(t). What I've done so far is: 1. convert all of the components to the s-domain. R = R, L = sL, C =...
  32. O

    Laplace transform sqaured differential

    Hi I was wondering what the Laplace transform of a squared differential is. With that I mean the Laplace of (y' )^2 (this being y'*y' and not the second order derivative). So for example velocity squared.
  33. I

    Laplace transform solve integral equation

    Homework Statement Homework Equations The Attempt at a Solution The answer is y(t) = t^{4}+\frac{t^{6}}{30} Don't know what to do next any advices please
  34. schmiggy

    Laplace transforms to solve linear ODE's

    *edit* can't upload images from phone with app for some reason, finding a computer. Using the app for the first time so hopefully this works out ok.. I've attached an image with the problem written in blue, and a complete attempt in gray. I have the answer to the question however it is...
  35. M

    Solving Laplace Transforms w/ Green Functions: A Guide

    Solve... \frac{d^2}{dt^2}G(t,t') + \omega^2G(t,t') = \delta(t-t') Solve (above) if G=0 and \frac{dG}{dt}=0 at t=0 to obtain: G(t,t')=\begin{cases} 0 & 0<t<t' \\ \frac{1}{\omega}\sin\omega(t-t') & 0<t'<t \end{cases} I'm supposed to use Laplace Transforms to figure this out. (I'm going out of...
  36. K

    Finding the Laplace Transform of a Polynomial Function

    Homework Statement Find the Laplace transform of the following function t[SUP]2[SUP] - 2t The Attempt at a Solution \[\begin{gathered} f\left( t \right) = {t^2} - 2t \hfill \\ l\left( {f\left( t \right)} \right) = F\left( s \right) = \int_0^\infty {{e^{ - st}} \cdot \left( {{t^2} - 2t}...
  37. M

    Finding the Laplace Transform of sin(7t)u(t-3) with Step Response u(t)

    Homework Statement Find the laplace transform of: v(t) = sin(7t)u(t-3) Homework Equations u(t) is step responseThe Attempt at a Solution I know that taken alone the laplace of: sin(7t) is 7/(s^2+49) u(t-3) is (1/s)*(e^-3s)I don't understand how I would figure out the answer to this...
  38. M

    Laplace and Systems Control and Analysis

    Using Laplace can someone please show me in simple terms how i would solve the following function? This is a lecture example. The solution just shows the answer, nothing about how we get it or what it represents. I am finding this subject particularly difficult to come to grips with. δ(t-1)...
  39. D

    Finding the Inverse of a Circuit: Step-by-Step Guide for Basic Laplace Questions

    Homework Statement I'm stuck on finding the inverse of this circuit. Can anyone please check my work, In this circuit you are suppose to find v(t) when switch moves to point 2 ( meaning deattached from battery). Homework Equations The Attempt at a Solution...
  40. I

    Inverse Laplace of 1s: Formula-Based Solution

    This is very tricky for me. How to find the inverse laplace of 1s. I haven't been taught the integral method of inverse. Only the formula based , splitting terms kind of thing. I used MATLAB and found it was dirac delta. But how do I get to it without using the integral for inverse?
  41. I

    Unit step functions to find Laplace Transform

    Homework Statement f(t) = cos (pi*t) if 1\leq t <4 and 0 elsewhere using unit step functions to find Laplace TransformHomework Equations The Attempt at a Solution I came up with the unit step function f(t) = cos(pi*t) u(t-1) - cos(pi*t) u(t-4) in order to use the second shifting theorem f(t)...
  42. T

    Laplace and inverse laplace transformation of unit step functions u(t).

    Homework Statement So I have this laplace transformation chart and was a bit unsure about the laplace and inverse laplace of this. The unit step function, where u(t) = 0 where t < 0, u(t) = 1 where t > 0. The laplace transformation chart that I have has two columns, the column on the...
  43. I

    Finding the Inverse Laplace Transformation

    Homework Statement find the inverse laplace transformation of \frac{5s+4}{s^2} e^{-2s} Homework EquationsThe Attempt at a Solution I have tried to partial fractions \frac{5s+4}{s^2} and I got \frac{5}{s}+\frac{4}{s^2} and I know that the answer must have u(t-2) because of second shifting (...
  44. D

    Inverse Laplace Question

    Homework Statement Could anyone please check my work, The answer is wrong. Correct Answer: e^2t(-t+e^t-1) Homework Equations Inverse Laplace The Attempt at a Solution Here's my solution, please point out my mistakes. One mistake i found after taking the picture is: Its -2te^-2t ( third...
  45. X

    Factoring in Inverse Laplace Transformations

    Homework Statement see attachment Homework Equations The Attempt at a Solution I don't get the part where it says "Solving this equation for X(s), we obtain ..." Specifically jumping from 3/s(s2+3s+5) to (3/5)(1/s)-(3/5)[(s+2)/(s2+2s+5)]. How did the problem break up this...
  46. C

    Determinant using Laplace expansion

    Hello! Do somebody know where the code for calculating a determinant by Laplace theorem (in Fortran) can be found? Or maybe somebody could help with this issue. Thank you!
  47. T

    Transposition as part of a laplace transform

    I am doing a laplace transform as part of a coursework assignment. I have some example transpositions that are relevant to the question I am answering but I can't see how the author has got from one arrangement to the next. [b]2. Homework Equations He has given 1/(s^2(τs+1)) =...
  48. I

    MATLAB Inverse laplace in matlab gives iota

    I'm applying laplace transform to a spring-mass system, the most basic one. I write this code which takes initial values x(0) and v(0) as input and I'm computing x(t) in matlab. But for some values it gives me complex roots for x(t) which doesn't seem possible. If not for laplace I can solve the...
  49. V

    Laplace transform heaviside function

    I have a question asking for the inverse laplace transform of (e^(-s))/(s^2+pi^2). I split it up to (e^(-s))/s x s/(s^2+pi^2) and got u(t-1)cos(pi(t-1)),but the correct answer is (sin(pi(t-1)/pi)u(t-1). So here it was split up to (e^(-s))/pi x pi/(s^2+pi^2) and I don't understand where the...
  50. S

    Laplace and pole zero diagram.

    The following diff. equation describes the functionality of a system with respect to time. However, it is not known how the system will behave when stimulated. Apply a forward Laplace transform to determine damping ratio and pole zeros. Plot a pole zero diagram and comment on stability. d2y/dt2...
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