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.

View More On Wikipedia.org
Recent contents Top users
  1. A

    Laplace transform of anti causal systems

    The unilateral laplace transform integrates from 0 to infinity.Then how can i take laplace transform of exp(t)u(-t)?PS:I don't want to flip it and use time scaling property.
  2. S

    Inverse Discrete Laplace Transform

    Hi, I have an idea which when tested looks like its clearly flawed. I am hoping someone can tell me where my procedure is flawed, or point me to some other theory that has already done something similar. The first two are the laplace transform. The third line is the Fourier Transform. The...
  3. P

    MHB Johnsy's question over Facebook about an Inverse Laplace Transform

    To start with, apply partial fractions... $\displaystyle \begin{align*} \frac{A\,s + B}{ s^2 + 9} + \frac{C\,s + D}{ \left( s^2 + 9 \right) ^2 } &\equiv \frac{ 4s^3 + 5s^2 + 57s + 45}{ \left( s^2 + 9 \right) ^2 } \\ \left( A\,s + B \right) \left( s^2 + 9 \right) + C\,s + D &\equiv 4s^3 + 5s^2 +...
  4. kostoglotov

    Laplace in Cyl. form: one step I'm not sure about

    I have completed the exercise, but I did something weird in one step to make it work, and I'd like to know more about what I did...or if what I did was at all valid. 1. Homework Statement Show that Laplace's equation \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial...
  5. J

    Laplace Transform question with unit step function

    Homework Statement Homework Equations Laplace Trasformations The Attempt at a Solution a. done b. f(t)= t -3*t*u(t-1) + 4*u(t-1) -3*u(t-2) -2*t*(t-2) c. 1/(s^2) - (3e^-s -2e^-2s)/(s^3) + (4e^-s -3e^-2s)/s d. 1/(s-1) * (1/(s^2) - (3e^-s -2e^-2s)/(s^3) + (4e^-s -3e^-2s)/s) These are the...
  6. Feodalherren

    What is the issue with applying the Laplace transform to tcos(4t)?

    Homework Statement f(t)=tcos(4t)Homework Equations tnf(t)=(-1)n dF(s)/dsn The Attempt at a Solution I don't understand why this formula is giving me the oppiste sign of the answer. If I apply the formula I get (16-s2)/(s2+16)2 Because n=1 I need to multiply by a negative but this yields...
  7. M

    Convolution theorem for Laplace Transform proof

    Hi all. Sorry about creating this new threat despite existing some others on the same topic. I have a problem in understanding a very specific step in the mentioned proof. Let me take the proof given in this link as our guide. My problem is just at the ending. When it says: "The region...
  8. J

    Laplace Transforms Homework: Initial Displacements & Velocities

    Homework Statement Homework Equations Laplace Transforms The Attempt at a Solution Using basic physics knowledge I got m1a1=-k1x1+k2(x2-x1) and m2a2=-k3x2-k2(x2-x1) Sub in values and use laplace transforms and rearrange partial fraction and I found that By doing this I am assuming...
  9. END

    Inverse Laplace Transformation

    Homework Statement Solve the following: $$\mathscr{L}_s^{-1} \left\{ \frac{s}{s^2-s+\frac{17}{4}} \right\}$$ Homework Equations Table of Laplace Transforms.The Attempt at a Solution The solution is $$f(t) = (1/4 )e^{t/2} (\sin(2 t)+4 \cos(2 t))$$ I know I need to break up ##F(s)## into...
  10. S

    Laplace transformation question

    Trying to answer the question: x"+3x'+2x=u(t-1)+2(t-2)u(t-2), x(0)=1, x'(0)=-1 My book has the answer but I need to see how to solve a problem like this. Professor didn't have time to cover this section but he said one of these might be on the test. Any help is greatly appreciated.
  11. D

    LaPlace transform method to find the equation of motion

    < Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown > So this is the problem Here is the question: A 32lb weight strecthes a spring 2ft.The weight is released from rest at the equilibrium position. beginning at t=0, a force equal to f(t)= sint acts on...
  12. I

    Laplace Transform of A Second Order ODE

    23.) y'' + 2y' + y = 4e-t; y(0) = 2, y'(0) = -1 Y(s) = [(as + b) y(0) + a y'(0) + F(s)]/(as2 + bs + c) My attempt: a = 1, b = 2, c = 1 F(s) = 4 L{ e-t } = 4/(s+1) (From Laplace Transform Table) Plugging and simplifying: Y(s) = (2s2 + 5s + 7)/[(s + 1)(s2 + 2s + 1) Here is where I get...
  13. T

    General Solution of a Poisson Equation of a magnetic array

    Hi, I'll give some background, say you've got a planar structure of thickness 'd', lying on the z plane. Also say the upper and lower surfaces are y = 0 and y = -d, respectively. The structure has scalar potentials inside it as so: As you can see the vector fields cancel out on one side, As it...
  14. Feodalherren

    Laplace transform of a piecewise function

    Homework Statement f(t) = e^t when 0≤t<1 and 0 when t≥1 Homework Equations Laplace transformations The Attempt at a Solution so the Laplace integral becomesfrom 0 to 1 ∫e^(st^2)dt + 0 how do I integrate this?
  15. R

    Inverse Laplace Transform with e^{a s}

    Homework Statement How can I take the Inverse Laplace Transform of $F(s) = \frac{d}{ds}\left(\frac{1-e^{5s}}{s}\right)$? I have tried going with inverse of the derivative and convolution (even tried evaluating the derivative and go from there) but although I can get to some results none of them...
  16. S

    MHB How can I solve this LaPlace Transform using Laplace Transforms?

    Solve by Laplace Transforms. So I'm stuck on how to find this \mathcal{L}^{-1} $( \frac{\frac{5s}{4} + \frac{13}{4}}{s^2+5s+8} ) $ I'm not sure what t odo. I was thinking I need to use the $\cos(at)$ and $\sin(at)$ formulas but I'm not sure... Any help would be great
  17. S

    MHB Yes, that is correct! Good job!

    Solev by Laplace Transforms $y'' - 5y' + 6y = 1$ $y(0) = 1$, $y'(0) = 0$So I am getting stuck. Here's my work $s^2Y - 5sY + 6Y = \frac{1}{s} + s - 5$ multiplied through by $s$ to get $s^3Y - 5s^2Y + 6sY = 1 + s^2 - 5s$ so: $Y = \frac{1+s^2-5s}{s^3-5s^2+6s}$ so: $1+s^2-5s = \frac{A}{s} +...
  18. S

    MHB Having trouble with the Laplace Transform

    Solve by Laplace Transforms. $y'' + 4y' + 4y = e^t$ $y(0) = 1$, $y'(0) = 0$So I've got $s^2Y - s + 4sY - 1 + 4Y = \frac{1}{s+1}$ then I got: $ Y = \frac{s^2+2s+2}{(s+2)(s+2)}$ Now here is where I am getting lost on the partial fraction decomposition.. I've got $s^2+2s+2 = A(s+2) + B$ I...
  19. Chacabucogod

    Can the Laplace Inverse be Applied to Divided Transfer Functions?

    I was wondering whether this can be done: Let's say you have transfer function that goes like this: \frac{Y(s)}{U(s)}= \frac{N(s)}{D(s)} Now let's say I divide my transfer into two: \frac{Y(s)}{Z(s)}= N(s) \frac{Z(s)}{U(s)}= \frac{1}{D(s)} Can I apply the Laplace Inverse to these two...
  20. I

    Simple Laplace transform question

    in an example in my text i don't see how they got the "sU" for the transform. actually, i don't even see it in my table of transforms.
  21. MexChemE

    Integrating factor vs. Laplace. Engineering problems

    Hello PF! We were doing mass balances on mixing tanks in one of my ChemE courses, and in one of the problems we arrived at the following DE: \frac{dC_B}{d \theta} + 0.025C_B=0.0125 e^{-0.025 \theta} Where CB is the concetration of salt in the tank and θ is time. The professor made us solve the...
  22. B

    How to get rid of laplace of pressure in an equation

    Hello! I have a question about an equation. I have an equation which is a boundary condition to a problem I have concerning fluid flow in a layered porous medium. I have a equation where my only variable is the temperature T, and I have 10 boundary conditions with it. In order to solve the...
  23. K

    MHB Initial Conditions in Laplace Transform of Second Order Differential Equations

    And also: y`+2y=2(1-e^-2t) Y(0)=0 y¨-2y`+y = t+e^t y(0)=1 and y`(0)=0 Please help me out here folks ;)
  24. K

    MHB Confused about Laplace and Inverse Laplace Transform of Various Functions?

    Hi. I`m new here and I need some help with Inverse Laplace Transform: f(t)=5+3t+e^3t g(t)=(t+1)u(t-2) g(t)=(t^2-9t+20)u(t-5) and Laplace Transform: F(s)=1/(s+2)^5 F(s)= 2s^2+10/s(s^2+2s+10) G(S)=2s/s^2+4e^-sso if anywone can please help me:)
  25. R

    MHB Laplace transform of the integral of a difference equation

    Hi, Please I need some help, how can I get the Laplace transform of the integration of a difference equation?? $\int _{ 0 }^{ \infty }{ { e }^{ -st } } \int _{ -\tau }^{ 0 }{ G(\theta )x(t+\theta )d\theta } dt$ Many thanks in advanced.
  26. J

    Laplace Equation with Charged Conductor

    I know the following works, but I cannot explain why. Say you have an infinite conducting cylinder (radius r=a) with charge Q/(unit length) in an electric field E. To find the potential, you just match the boundary conditions V=0 at a, and V→ -Er cosφ far away. Then you add in the potential of...
  27. J

    Fourier COSINE Transform (solving PDE - Laplace Equation)

    I'm trying to solve Laplace equation using Fourier COSINE Transform (I have to use that), but I don't know if I'm doing everything OK (if I'm doing everything OK, the exercise is wrong and I don't think so). NOTE: U(..) is the Fourier Transform of u(..) This are the equations (Laplace...
  28. J

    Residue Theorem for Laplace Transform

    I need to know what's the Residue Theorem for a Laplace Transform. Does anyone know the name or something, so I can search it? I couldn't find anything. For example, if I have this two equations: X(s).(s-1) = -Y(s)+5 Y(s).(s-4) = 2.X(s)+7 I know how to solve them using Simple Fractions, but...
  29. Brunoboy

    Help with an easy Laplace transform

    Hi everyone! :) Have a problem here I can't solve atm. Solve the Laplace transform, when: My try: Would really appreciate some help!
  30. J

    Integration in Laplace Transform

    Hello everyone, I have a question about integrating in Laplace Transform. For example, if I have: f(t)=e^{i.t} I have to solve this equation: \int_{0}^{\infty}e^{i.t}.e^{-s.t}dt If I do like this, it's very simple...
  31. W

    Laplace transform for differential equation

    Homework Statement use laplace transforms to solve the differential equation y"+2y'+17y = 1 Homework Equations Initial conditions are y(0) = 0 y'(0) = 0 The Attempt at a Solution so it converts to Y(s) (s^2+2s+17) = 1/s which then ends up as; Y(s) = 1/s*1/(s^2+2s+17) i know i need to invert...
  32. C

    Non-differential Form of Biot-Savart Law: Comparison and Confusion Clarified

    What is the non-differential form of the Biot-Savart law? Is it: B=mi*I/(4R*pi)*(cos(a)-cos(b)) or B=mi*I/(4R*pi)*(cos(a)+cos(b))? For a infinitely long conductor, the law is: B=mi*I/(2R*pi) because a=0 and b=pi. So I would say that the correct expression is the one where the cos are...
  33. E

    Solving differential equation using Laplace Transform

    Homework Statement solve the following differential equation using Laplace transforms: y'' + 4y' + 4y = t^2 e^{-2t}, y_0 = 0, y'_0 = 0 y_0 and y'_0 are initial conditions. Homework Equations Using L to represent the Laplace transform, we have that L(y) = Y L(y') = pY - y_0 L(y'') =...
  34. R

    Inverse Laplace involving heaviside function

    Homework Statement Give the inverse Laplace transform of F(s) = (-3/s) + (e^-4s)/(s^2) + (3e^-4s)/s Homework Equations Inverse Laplace [e^(-cs) F(s)] = f(x-c)u(x-c) The Attempt at a Solution I'll break this into 3 parts. Part 1 - (-3/s) -3/s = -3(1/s) -> inverse laplace of -3(1/s) = -3...
  35. S

    Inverse laplace transform without partial fractions

    Homework Statement take inverse laplace of: 6/[s^4(s-2)^2] Homework Equations 6/[s^4(s-2)^2] The Attempt at a Solution I used partial fractions. I was wondering if It could be manipulated to where I could use the laplace table?
  36. R

    How Do You Find the Inverse Laplace Transform for a Given Differential Equation?

    Homework Statement Find H(s) = \frac{Y(s)}{X(s)} \frac {d^2y(t)}{dt^2} + a\frac {dy(t)}{dt} = x(t) + by(t) Homework EquationsThe Attempt at a Solution [s^2 + as - b] Y(s) = X(s) H(s) = \frac{1}{s^2+as-b} I assume the inverse is a sign or a cosine but unsure which one.
  37. S

    How do I take this inverse laplace?

    Homework Statement find the inverse laplace Homework Equations L^(-1)[(s+1)^2/(s+2)^4] The Attempt at a Solution included in attachment. the top problem, #20.
  38. S

    How do I take the laplace of this?

    Homework Statement Find F(S) Homework Equations L[e^(2-t)U(t-2)] The Attempt at a Solution a = 2 and using the laplace table I got: e^(-2s)/(s-2) answer should be: e^(-2s)/(s+1)
  39. Sneakatone

    Differential equation with laplace transform and springs

    Homework Statement I do not know how to find f(t) with the given Ampliture 40 and a=pi Homework EquationsThe Attempt at a Solution I have the solution above. my set up was 1/2y''+y'+5=f(t) 1/2S^2* Y(s) + Y(s)+5=f(t)
  40. V

    Why don't my two approaches to finding c(t) match in Laplace transform problem?

    Homework Statement I'm given a transfer function C(s)=10R(s)/(s+4) And I have to find c(t) for r(t)=6u(t) The Attempt at a Solution First I did this problem by taking inverse laplace of the transfer function, and inserting the value of r(t) in it. Next I did the same problem by first...
  41. C

    Finding potential for a sphere, when to use laplace equation

    Homework Statement This is example 3.9 in Griffiths Electrodynamics. "A specified charge density σ(θ) (inclination angle) is glued over the surface of a spherical shell of radius R. Find the resulting potential inside and outside the sphere." The problem suggests that although it is possible to...
  42. E

    Laplace transform - solve integral

    Homework Statement Solve the integral y(t) + \int_0^t (t-u)y(u) \, du = 3sin(2t) Homework EquationsThe Attempt at a Solution Rewrite the equation: y(t) = 3sin(2t) - \int_0^t (t-u)y(u) \, du I assume the integral to be the convolution: f(t) * y(t) = t * y(t) as f(t-u) = f(t) = t...
  43. T

    Laplace Transform for Solving a First Order Linear IVP

    Homework Statement Solve the IVP : dy/dt + y = f(t) y(0) = -5 where f(t) = -1, 0 <= t < 7 -5, t >= 7 y(t) for 0 <= t < 7 = ? y(t) for t >= 7 = ? Homework EquationsThe Attempt at a Solution So I have never seen a problem of this type, excuse my silly mistakes if I'm interpreting this question...
  44. _N3WTON_

    Laplace Transform of Heaviside Function

    Homework Statement Determine the Laplace transform of the given function: f(t) = sin(t) for 0 <= t < \pi and f(t) = 0 for \pi <= t Homework EquationsThe Attempt at a Solution Ok, I've been having some trouble figuring out how I should write the above branched function (sorry for the...
  45. L

    Inverse Laplace transform. Bromwitch integral

    Inverse Laplace transform \mathcal{L}^{-1}[F(p)]=\frac{1}{2\pi i}\int^{c+i\infty}_{c-i\infty}F(s)e^{st}dp=f(t) Question if we integrate along a straight line in complex plane where axis are Re(p), Im(p), why we integrate from c-i \ínfty to c+\infty? So my question is, because Im(p) are also...
  46. C

    Another Laplace Transform problem, need region of convergence help

    Homework Statement Find L[x(t)], where $$ x(t) = tu(t) + 3e^{-1}u(-t) $$ Also determine the region of convergenceHomework EquationsLaplace properties, Laplace table: L[te-at = 1/(s+a)2 L[u(t)] = 1/s L[t] = 1/s2 The Attempt at a Solution I don't really know what to do with this as my table...
  47. _N3WTON_

    Inverse Laplace Transform with Division and Integration

    Homework Statement Division by s Equals integration by t: For this problem use the following property (see relevant equations) to find the inverse transform of the given function: F(s) = \frac{1}{s(s-1)} Homework Equations L^{-1}(\frac{F(s)}{s}) = \int_{0}^{t} f(\tau)\,d \tau The Attempt...
  48. C

    Laplace Transform L[x(t)] given, find L[tx(t)]

    Homework Statement If L[x(t)] = (s + 4)/(s2 + 1), find L[tx(t)] Homework Equations Laplace transform: F(s) = 0∫ f(t)e-stdtLaplace table The Attempt at a Solution Clearly it's not just asking for a Laplace transform. Not sure what it's specifically asking to be honest. t multiplied by...
  49. R

    Applying Partial Fractions to Solve Laplace Transform Convolution

    Homework Statement x(t) = cos(3πt) h(t) = e-2tu(t) Find y(t) = x(t) * h(t) (ie convolution) Homework Equations Y(s) = X(s)H(s) and then take inverse laplace tranform of Y(s) The Attempt at a Solution L(x(t)) = \frac{s}{s^2+9π^2} L(h(t)) = \frac{1}{s+2} I then try to find the partial...
  50. Sneakatone

    Damped spring problem with laplace transform

    Homework Statement A 4-pound weight stretches a spring 2 feet. The weight is released from rest 15 inches above the equilibrium position, and the resulting motion takes place in a medium offering a damping force numerically equal to 7/8 times the instantaneous velocity. Use the Laplace...
Back
Top