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

    Finding the Convolution of Two Functions Using the Laplace Transform

    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)) = [πδ(ω - 3π) + πδ(ω + 3π)] L(h(t)) = \frac{1}{s+2} Laplace Transform inverse ...
  2. D

    Calculate the Laplace for the Ramp

    Hi, i am new to Laplace transforms/Algebra. I have been given a worked example by lecture to calculate the Laplace transform for a ramped input into a single pole RC high pass filter. i have managed to calculate the Laplace for the Ramp and the Laplace for the Filter. however i can't figure out...
  3. davidbenari

    Solve system of equations using laplace transform and evaluate x(1)

    Homework Statement I keep getting the wrong answer, and wolphram seems to back me up. Here's the system of equations ##(-10+s)X(s)-7Y(s)=\frac{10}{s}## ##X(s)+(-2+s)Y(s)=0## Homework EquationsThe Attempt at a Solution Using Cramer's rule I've got...
  4. R

    Laplace transform and region of convergence

    Find the LT and specify ROC of: x(t) = e-at, 0 ≤ t ≤ T = 0, elsewhere where a > 0 Attempt: X(s) = - 1/(s+a)*e-(s+a) integrated from 0 to T => -1/(s+a)[e-(s+a) + 1] Converges to X(s) = 1/(s+a) , a ⊂ R, if Re{s} > -a for 0≤t≤T Elsewhere ROC is empty (LT doesn't exist). Is this...
  5. C

    Laplace transform ( "find x(t)" though ? )

    Mod note: Please don't tinker with SIZE tags. Things are perfectly readable without them. Homework Statement Find x(t) = L-1[(4e-4s - 3)/(s2 + 6s + 25)]. Homework Equations L(x(t)) = ∞∫∞x(t)e-stdt L-1(x(s)) = (1/2π)(σ - ∞j)∫(σ + ∞j)[x(s)est]ds, "But you want to avoid this integral." Laplace...
  6. B

    Separation of Variables in 2 Dimensions: A Laplace Equation Primer

    So I'm currently taking electricity and magnetism and I'm expected to know how to perform a separation of variables on laplace equation in 2 dimensions.I have taken Zero differntial equations courses and I literally have no freaking idea what's going on. The book I use doesn't spend any time...
  7. D

    Simple RL question with laplace

    Homework Statement I have a circuit with resistor and inductor (RL) that is placed serial and a power supply. the voltage is constant 50 V at t=0 , R=10 ohm, and L=0.2 H. Find the I using laplace transformation Homework Equations - The Attempt at a Solution my attempt is transform everything...
  8. M

    Tricky inverse Laplace transform

    << Moderator Note -- thread moved to the Homework Help forums >> I'm stuck on a problem, and I'm in serious need of help. I) Problem: Find the solution to f (t) = 2 \int^t_0 f'(u) sin 3 (t-u) \ du + 2 cos (3t) . Also find f (0) .II) Solution, so far: F(s) = 2 (s F(s) - f(0)) *...
  9. kewei chen

    Inverse Laplace transform with p^-1 and exponential

    Hello everyone, I have spend whole day but still not figure out an inverse Laplace transform. Hope someone can help me. The question is in the attachment. I'm trying to extract u^2/4D^2 out the bracket to match the standard inverse table, but it seems difficult to deal with the square root...
  10. ranju

    Solve Laplace Transform Homework: Find i(t) for t>0

    Homework Statement we have to solve the given circuit using laplace transform for i(t) for t>0 when switch k opened at t=0.. now all I wnated to make clear is the final circuit diagram of this in s-domain..[/B]Homework EquationsThe Attempt at a Solution as I solved I considered the circuit to...
  11. R

    Solving Laplace Equation in Cylindrical Coordinates - Potential Outside Cylinder

    The potential on the side and the bottom of the cylinder is zero, while the top has a potential V_0. We want to find the potential outside the cylinder. Can I use the same boundary conditions as for case of inside cylinder potential? What is different?
  12. L

    Using Laplace transforms to solve differential equations - with a twist

    I've been given this: x''+ x = 4δ(t-2π) The question asks: With initial conditions of x(0) = 1 and x'(0) = 0, find x(t) using Laplace transforms. I can easily get it to this: 4(sin(t-2π)u(t-2π)) But the question says "express your final solution without use of the unit step function". This is...
  13. I

    Laplace Transform: Find $$z(t)$$

    Homework Statement Wondering if I did this correctly.. Find the laplace transform: $$z(t)=e^{-6t}sin(\omega_{1}t)+e^{4t}cos(\omega_{2}t)$$ for ##t\geq 0## Homework Equations The Attempt at a Solution For the first part, I assume I can do this, but I'm not too sure. This is my main question...
  14. D

    MHB Finding The laplace transform

    please check my work here $\mathscr{L}[2\sin(bt)\sinh(bt)]$ I know that $\sinh(bt) = \frac{e^{bt}-e^{-bt}}{2}$$\mathscr{L}[2\sin(bt)\left(\frac{e^{bt}-e^{-bt}}{2}\right)]$...
  15. M

    Inverse laplace transform (polynomial division? Complex roots?)

    Homework Statement Decide the inverse laplace transform of the problem below: F(s)= \frac{4s-5}{s^2-4s+8} You're allowed to use s shifting. Homework Equations The Attempt at a Solution By looking at the denominator, I see that it might be factorized easily, so I try that...
  16. D

    MHB Finding laplace transform

    I came across this problem and solved it using different approach. I get a slightly different answers. here's how it goes, 1st approach $\mathscr{L}[te^{2t}\cos(3t)]$ first I get the laplace of something that's familiar to me which is $\mathscr{L}[t\cos(3t)]$ using this...
  17. Q

    Inverse Laplace transform where e^(st)F(s) is entire

    Heya folks, I'm currently pondering how to decide whether a function has an inverse Laplace transform or not. In particular I am considering the function e^(-is), which I am pretty sure does not have an inverse Laplace transform. My reasoning is that when calculating the inverse by the Bromwich...
  18. M

    So the inverse transform of \frac{3s+ 5}{s^2+ 9} is 3cos(3x)+ (5/3)sin(3x).

    Homework Statement Find the inverse Laplace transform of the expression: F(S) = \frac{3s+5}{s^2 +9} Homework Equations The Attempt at a Solution From general Laplace transforms, I see a pattern with laplace transforming sin(t) and cos(t) because: L{sin(t)+cos(t)} =...
  19. P

    Where does the Laplace Transform come from?

    I took an introduction to ODEs course this past spring semester. It always bothered me where this thing came from. I did a little bit of research and found a video of a professor explaining how it is the continuous analog of an infinite sum. He did a little bit of a derivation using that...
  20. A

    Second shifting theorem of laplace transform

    1. why do we need to use shifted unit step function in defining second shifting theorem? 2. why don't we instead calculate laplace transform of a time shifted function just by replacing t by t-a? 3. everywhere in the books as well as internet i see second shifting theorem defined for...
  21. Whovian

    Eigenfunctions of Laplace Transform

    Firstly, if this is an inappropriate forum for this thread, feel free to move it. This is a calculus-y equation related to differential equations, but I don't believe it's strictly a differential equation. The question I'm asking is which functions...
  22. N

    Help please -- inverse Laplace transform of 1/(x^2+1)^2

    Homework Statement Hi. I need help to resolve the inverse laplace transform of {1/((x^2)+1)^2}2. The attempt at a solution I have tried to do: {(1/((x^2)+1) * (1/((x^2)+1)} then, convolution, sen x But, isn't working Thanks for your help :)
  23. E

    Change of variables Laplace Equation

    Homework Statement Write the Laplace equation \dfrac {\partial ^{2}F} {\partial x^{2}}+\dfrac {\partial ^{2}F} {\partial y^{2}}=0 in terms of polar coordinates. Homework Equations r=\sqrt {x^{2}+y^{2}} \theta =\tan ^{-1}(\frac{y}{x}) \dfrac {\partial r} {\partial x}=\cos \theta \dfrac...
  24. S

    Solving Laplace Equation with Boundary Conditions: Tips and Examples

    Homework Statement I got to this while solving a physical problem, therefore it is hard for me to write the problem statement if there is none. I can write the whole physical problem here but it wouldn't make any sense. So, here is how it goes. I got to a Laplace equation ##\triangle \phi =0##...
  25. M

    Solving Laplace Transform for Arduino PID Controller

    Hi All, I'm playing around with an arduino, and have build a PID controller that controls the temperature of a light bulb, measured with an NTC. All is working fine I'm looking to get a bit more theoretical on the subject and have modeled the system in a simulink like environment. I want...
  26. K

    How to Find the Inverse Laplace Transform for Ds + E / (s^2 +1)^2?

    Homework Statement Ds + E / (s^2 +1)^2 Homework Equations The Attempt at a Solution Ds / (s^2 +1) + E / (s^2 +1) D[s/(s^2 + 1)^2] + E [1 / (s^2 + 1)^2]
  27. D

    MHB Simplifying the inverse Laplace Transform using the inverse shift formula

    before I go to bed(it's 11:30pm in my place), here is the last problem that I need help with find the inverse Laplace Transform $\frac{4s-2}{s^2-6s+18}$ the denominator is a non-factorable quadratic. I don't know what to do. thanks!
  28. D

    MHB Inverse laplace transform of a function

    find the inverse Laplace of the ff: 1. $\frac{n\pi L}{L^2s^2+n^2 \pi^{2}}$ 2. $\frac{18s-12}{9s^2-1}$ for the 2nd prob I did partial fractions $\frac{18s-12}{9s^2-1}=\frac{9}{3s+1}-\frac{3}{3s-1}$ $\mathscr{L}^{-1}\{\frac{18s-12}{9s^2-1}\} =...
  29. D

    MHB Laplace transform of a function II

    $\mathscr{L}\{\sin^{2}4t\}$ $\mathscr{L}\{\sin(3t-\frac{1}{2}\}$for the 2nd prob here's what I have tried $\mathscr{L}\{\sin(3t)\cos(0.5)-\cos(3t)\sin(0.5)}$ $\cos(0.5)\mathscr{L}\{\sin(3t)\}-\sin(0.5) \mathscr{L}\{\cos(3t)\}$ $\frac{3\cos(0.5)-s\sin(0.5)}{s^2+9}$ ---> is this correct? for...
  30. D

    MHB Laplace transform of a function

    please help me solve this problem $\mathscr{L}\{e^{3a-2bt}\}$ here's my attempt $\mathscr{L}\{e^{3a}\cdot e^{-2bt}\}$ from here I couldn't continue I looked up my table of transform but nothing matches the problem above. I'm not sure if the first shift formula would work here. please help...
  31. J

    Laplace analysis of simple LC tank (no resistance)

    I am stumped by an exercise in using Laplace transforms to analyze the voltage and current in simple LC tank. My issue is with the correct sign of the voltage across the capacitor ...let me pose the problem. A circuit consists of a voltage source V, 2 switches, a cap C and an inductor L. The...
  32. R

    Power series and Laplace transform

    As it can be read here, http://en.wikipedia.org/wiki/Laplace_transform#Relation_to_power_series the Laplace transform is a continuous analog of a power series in which the discrete parameter n is replaced by the continuous parameter t, and x is replaced by exp(-s). Therefore, computing a...
  33. P

    Laplace transform with Heaviside function

    Hello, I am searching for the Laplace transform of this function u_a(y)\frac{\partial c(t)}{\partial t} where u_a(y) is the Heaviside step function (a>0). Can anyone help me? Thanks in advance! Paolo
  34. dwn

    S-domain transformations (Laplace)

    Homework Statement Image attached. Homework Equations S-domain transformations The Attempt at a Solution Solving this using mesh analysis. I1 is straightforward : V = I1R I1 = 1.8∠75° / 2 = 0.9∠75° I2 I'm having a little trouble with. i = C dv/dt = C*V*s...
  35. DreamWeaver

    MHB Laplace Transforms (proofs of)

    I should state, from the outset, that this tutorial is NOT going to go into any great detail about the theory and applications of Laplace transforms. Some of the aforementioned will be discussed in a cursory way, but the aim here is merely to provide a selection of proofs for common transforms...
  36. A

    Laplace Transform. Is this correct?

    Homework Statement ƒ(t) = 1.15t Homework Equations Standard Laplace Transform Table The Attempt at a Solution L(f(t)) = e(t * ln(1.15) = 1 / (s - ln(1.15))
  37. S

    Inverse Laplace Transforms Problem 2

    Homework Statement f(s) = 6/s^2-9 Homework Equations I think f(t) = (1/b-a)(e^-at-e^-bt) The Attempt at a Solution Replace 6/s^2-9 with 6/(s-3)(s+3) a=-3 b=3 Plug in (1(6)/3-(-3))(e^-(-3)t-e^-3t) Final Result e^3t-e^-3t
  38. S

    Applying Inverse Laplace Transforms to f(s) = -5s/S^2+9

    Homework Statement f(s) = -5s/S^2+9 Homework Equations I think f(t) cosωt = f(s) s/s^2+ω^2 The Attempt at a Solution ω=3 Answer -5cos(3t) Can anyone tell me if I did this correctly? I think I did but just want to make sure, if not can you tell me what I did wrong? Thanks
  39. A

    Conditions for Laplace and its inverse transform to exist

    I usually see that Laplace transform is used a lot in circuit analysis. I am wondering why can we know for sure that the Laplace and its inverse transform always exists in these cases. Thank you.
  40. J

    MHB Laplace Transform, Finding solution: y′′+4y′+4y=f(t)

    y′′+4y′+4y=f(t) where f(t)=cos(ωt) if 0<t<π and f(t)=0 if t>π? The initial conditions are y(0) = 0 , y'(0) = 1 I know that f(t)=cos(ωt)−uπ(t)cos(ωt), the heaviside equation. AND ω is allowed to vary, supposed to find the general solution, i.e. f(t) in terms of ω I think that after...
  41. J

    Laplace Transform to Find Solution

    Use Laplace transfer to find the solution of the following initial value problem: y''+4y'+4y=f(t) where f(t) = cos(ωt) if 0<t<π and f(t)=0 if t>π ? Also, y(0) = 0, y'(0) = 1 Currently, I have gotten to here, but not sure how to perform inverse Laplace: (s+2)² * F(s) − 1 = [s/(s²+w²)]...
  42. S

    Find power series if you know its laplace transformation

    Homework Statement a) Determine power series ##\sum _{n=0}^{\infty }a_nt^n## if you know that its laplace transformation is ##-s^{-1}e^{-s^{-1}}## b) Determine function ##g## that this power series will be equal to ##J_0(g(t))##Homework Equations The Attempt at a Solution Hmmm, I am having...
  43. S

    Laplace transformation: system of DE

    Homework Statement Let ##y_1^{'}+y_1=y_2##, ##y_2^{'}+5y_2=y_3##, ##y_3^{'}+y_3=f## and ##y_1(0)=y_2(0)=y_3(0)=0##. Find ##Y_1(s)## in terms of ##F(s)##. Homework Equations The Attempt at a Solution I am completely lost here. I tried to rewrite the system so that I would...
  44. M

    Inverse Laplace Transform and Court

    Homework Statement I had a question in my midterm, it was to find inverse laplace tansform of: (4s+5) / (s^2 + 5s + 18.5) Where ^ denotes power. Homework Equations The Attempt at a Solution My answer was to find the complex roots of equation (s^2 + 5s + 18.5) , by them...
  45. S

    Laplace Transform for Functions: 5cos(7t+π/4) and e^(3t)sintcost

    Homework Statement Find Laplace transformation for functions ##f(t)##: a) ##5cos(7t+\pi /4)## b) ##e^{3t}sintcost##Homework Equations The Attempt at a Solution a) I know that for ##cos(\omega t)## the laplace is ##\frac{s}{s^2+\omega ^2}## but what can I do with that ##\pi /4## ? I believe I...
  46. A

    Laplace transform for set of differential equations

    I have a set of differential equations with the basic form: dy_n/dt = t*(a_(n-1)*y_(n-1)+b(n+1)*y_(n+1)-2c_n*y_n) So the time depence is a simple factor in front of the coefficient matrix. Does this set of differential equations have closed form solutions?
  47. C

    Laplace Equation in a square

    Homework Statement i need to solve the laplace equation in square with length side 1 i tried to solve by superposition and i got infinite sum enen thouth i know that the answer should be finite Homework Equations 1.ψ(x=0,0≤y≤1)=0 2.ψ(y=0,0≤x≤1)=0 3.ψ(x=1,0≤y≤1)=10sin(∏*y)+3x...
  48. K

    Laplace Transform solution for 2nd order differential equation

    Homework Statement d^2x/dt^2 - 3 dx/dt + 2x= 2e^3t give that at t=0, x=5, and dx/dt=7 Homework Equations i can't figure out how to derive the values of A, B, and C from the attempted equation solution. please help me out here. thanks The Attempt at a Solution
  49. M

    Solving a Laplace Transform Problem: Where Am I Going Wrong?

    I'm out of college and am brushing up on Laplace Transforms. I have a problem I've solved, but I believe the solution I got is wrong and can't find my error. The problem is 2x''-x'=t*sin(t) x(0)=5,x'(0)=3 My solution... Take the Laplace Transform 2(s^2x-5s-3)-(sx-5)=2s/(s^2+1)^2...
  50. F

    Problems with Laplace Transforms

    Homework Statement The coordinates ##(x,y)## of a particle moving along a plane curve at any time t, are given by \frac{dy}{dt} + 2x=\sin 2t, \frac{dx}{dt} - 2y=\cos 2t. If at ##t=0##, ##x=1## and ##y=0##, using Lapace transform show that the particle moves along the curve 4x^2+4xy+5y^2=4...
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