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. V

    Transfer function with laplace transform

    Hi, I am trying to derive one transfer function for a system, but got stuck at a point. I want the solution to the ratio V_o(s)/V_g(s), the transfer function. V_o(s) = [a*L{(V_g)^2} - b*L{V_o * (V_g)^2} - c*V_g(s) - d/s]*z(s) In the following equation, I have: s-> lapace transform...
  2. C

    Heat Equation (Non Homogeneous BCs) - Difficult Laplace Transform help ;)

    Heat Equation (Non Homogeneous BCs) - Difficult Laplace Transform... help! ;) Hi I'm trying to model the temperature profile of an inertia friction welding during and after welding. I have the welding outputs and have come up with a net heat flow wrt time during the process. I now want to...
  3. C

    Inverse Laplace Transfrom - Taylor Series/Asymptotic Series?

    Homework Statement W(x,s)=(1/s)*(sinh(x*s^0.5))/(sinh(s^0.5)) Find the inverse laplace transform of W(x,s), i.e. find w(x,t). Answer: w(x,t)= x + Ʃ ((-1)^n)/n) * e^(-t*(n*pi)^2) * sin(n*pi*x) summing between from n=1 to ∞ Homework Equations An asymptotic series..?! The...
  4. N

    Laplace Transform Time Shift problem

    Homework Statement Determine the Laplace transform: g(t) = 2*e^{-4t}u(t-1) The Attempt at a Solution Essentially we're told for a time shift we multiply the Laplace transform pair of the function (without the delay) by e^{-as} So here a = 1 (for the delay) The Laplace transform for e^{-4t}...
  5. R

    Finding the inverse laplace transform

    I'm attempting to find the inverse laplace transform of \frac{25}{(1-s)^2*(4+s^2)} I get to this point but can't get the values of A B and C when equating coefficients. 25/(s-1)=A(4+s^2)+(Bs+C)(s-1) Also for a separate question: (s+1)/(5-4s+s^2) How do you find the inverse...
  6. W

    Laplace tranforms with boundary conditions

    Homework Statement Here's the question: Use laplace transforms to find X(t), Y(t) and Z(t) given that: X'+Y'=Y+Z Y'+Z'=X+Z X'+Z'=X+Y subject to the boundary conditions X(0)=2, Y(0)=-3,Z(0)=1. Now I have learned the basics of laplace transforms, but have not seen a question in...
  7. D

    RC circuit and Laplace transforms

    Hi everyone, I know this should be obvious, but there's something I am just NOT getting. Imagine a simple series RC circuit with a DC source as shown in the attachment. As can be seen from the picture, I have solved the differential equation in capacitor current in the time domain. In...
  8. T

    What is the solution to the inverse Laplace transform of a given integral?

    Homework Statement I've attached the multiple choice question Homework Equations The Attempt at a Solution I inversed laplaced the problem and i get an answer of 3 H(t-3) sin(9(t-3)) which isn't any of the options so I chose F but that was wrong. So far I've confidentally...
  9. T

    Another laplace differential equation.

    Homework Statement question is attached Homework Equations L(1) = 1 /s L(t^n) = n!/s^(n+1) L{f '(t)} = s F(s) - f(0) L{f ''(t)} = s^2 F(s) - s f(0) - f '(0) The Attempt at a Solution okay so i laplace transformed both sides and i got: (s^2) Y(s) + 5s -5 -5s Y(s) -25 - 6Y(s) = -84exp(-5s) /...
  10. T

    Laplace transforms of differential equations

    Homework Statement I've attached the problem.Homework Equations L(1) = 1 /s L(t^n) = n!/s^(n+1)The Attempt at a Solution because the question only asks for X(s) I only considered the x' + x + 4y=3 equation. I applied laplace tranforms and got: s X(s) - x(0) + 1/s^2 + 4/s^2 = 3/s. since x(0) =...
  11. S

    Using Laplace to solve a differential equation

    Use Laplace to solve x''+3x'+5x=sin(8t) Initial conditions x(0)= 2 and x'= -3 I have worked it down to (s2+3s+5)X(s)= 2s+3+ [8/(s2+64)] but stuck. Please help
  12. S

    Struggling to Find X(t) Using Inverse Laplace Transform

    Homework Statement Having found the laplace transform of a differential equation. I must now find X(t). All of my work is attached. The problem I am having is fitting my function of s to my table of transforms. I tried using partial fractions but it took me in a loop.The Attempt at a Solution...
  13. T

    Laplace transform integral problem

    Homework Statement Hey guys, I've attached the question that's troubling me. I've also attached the table and formulas of Laplace transform for you convenience. Homework Equations attached The Attempt at a Solution Right now, I am thinking the best way to do this problem is...
  14. M

    2D Fourier of 3D (vector) function - laplace of it?

    Hi, I have a vector function (momentum \vec{p}(x,y,z)=\vec{p}(\vec{r},z) for \vec{r}=(x,y)) and need to transform it, also I use the Fourier transform F(\vec{p})=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \vec{p}(\vec{r},z) exp^{i \vec{\chi}\cdot \vec{r}}d\vec{r} and then when I have...
  15. T

    Basic question about Laplace and signals properties

    Hey guys, I have a butterworth high-pass filter, and I was asked to find it's temporal response equation to the u(t) function. That part was easy, using basic Laplace tables I was able to find the following equation: y(t)=√2 e^(-31100t) *cos⁡(31000t+π/4)u(t) However, I'm supposed to be able to...
  16. S

    Solving DE with Laplace Transforms and g(t) Function

    Homework Statement Solve the differential equation y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms. The g(t) function is a piecewise function and is attached as g.jpg. Homework Equations Laplace transforms of regular and unit step/Heaviside functions. The Attempt at...
  17. S

    Laplace Transform for ∫0tf(t)dtA Guide to Solving Laplace Transforms

    Homework Statement We have been given a table of laplace transforms and have been asked to show them using the definition. ∫0∞e-stf(t)dt. But this one I have no clue where to begin ∫0tf(t)dt the laplace transform of this is F(s)/s. Can anyone tell me what to do with this one? Thank you...
  18. F

    Using laplace to solve 2 equationsplease check workthanks

    Homework Statement use laplace transforms to solve system of two equations x' = x -4y + e^4t y' = -x + y -4e^4t x(0) = 0 y(0) =1 Homework Equations i am uncertain about the correct order to solve these problems.. am I taking inverse L.T. at right time and place in order for the prob...
  19. F

    Confused about unit step function wrt laplace transforms

    Homework Statement y'' + 4y' + 2y = u_pi(t) + u_2pi(t) y(0) = 0 and y'(0) = 0.Homework Equations the step function equation: u_c(t) = u(t-c) --> (laplace) --> e^-cs/sThe Attempt at a Solution i am having major probs with getting my head around step function probs wrt laplace transforms. What...
  20. F

    Using laplace transforms to solve system of (two) equations

    Homework Statement x' = 3x - 4y y' = 2x + 3y x(0) = 1 y(0) = 0Homework Equations y' = sY(s) - y(0) The Attempt at a Solution i am confused as to how to take laplace of either equation when i am left with a term i can't take the laplace of: specifically, x' = 3x - 4y sX(s) - 1 -3X(s) = -4Y(s)...
  21. N

    Laplace transform of unit step function

    Homework Statement Could someone please explain this to me? I have read several notes on it, but do not really follow the reasoning: The Attempt at a Solution When t = 0, -1/s*e^-st = -1/s because e^0 = 1. When t goes to infinity is the part I do not fully understand. Why...
  22. D

    Understanding the Intuitive Interpretation of Laplace and Fourier Transforms

    Thought it's pretty related to this forum. I'm familiar with applications of both Laplace and Fourier in physics and differential equations. However I still struggling trying to figure out the intuitive interpretation of both transforms or at least a mathematical illustration that shows their...
  23. M

    MHB Solving PDE by using Laplace Transform

    Given $\begin{aligned} & {{u}_{t}}={{u}_{xx}},\text{ }x>0,\text{ }t>0 \\ & u(x,0)={{u}_{0}}, \\ & {{u}_{x}}(0,t)=u(0,t). \end{aligned} $ I need to apply the Laplace transform to solve it. I'll denote $u(x,s)=\mathcal L(u(x,\cdot))(s),$ so for the first line I have $s\cdot...
  24. MathWarrior

    Laplace Transforms - Just looking up tables?

    I recently have started learning Laplace transforms, it seems like its just a bunch of looking up tables. Along with having a bunch of standard Laplace transforms memorized. Is this how it usually is when dealing with transforms for the first time? I feel like I am not really even doing math...
  25. QuarkCharmer

    Laplace Transform, what does it mean?

    Homework Statement Homework Equations The Attempt at a Solution We went over the Laplace transformation today in my DE course. We only covered essentially, "how to do it", so: L(f(t)) = \int_{0}^{\infty}f(t)e^{-st}dt (also, how do you make the fancy f and curly brackets in Latex?)...
  26. F

    Using laplace transforms to solve IVP problems with cramers rule

    Homework Statement y'' + 4y = sin(2t) y(0) = 1 y'(0) = 1 Homework Equations cramers ruleThe Attempt at a Solution after isolating Y(s) i end up with: Y(s) = 2 / (s^2+4)^2 + s/(s^2+4) after i do partial fraction decomposition, i get 4 equations with 4 unknowns, A, B, C and D i inserted B...
  27. B

    Laplace transform of multiplication

    Hi everyone, I have a problem with finding the transfer function of a linear system. It happens that in some terms the input and the output are multiplying and I have no idea how to do the Laplace transform of this. The system is a car and I am only studying its speed. The input is the...
  28. F

    Using laplace transforms to solve IVPplease check work thanks

    Homework Statement y' - 3y = 13cos(2t) y(0)=1Homework Equations y' = sY(s) - y(0)The Attempt at a Solution heres all my work.. i am confused as to why its not matching book solution.. i think (geussing) that I probably messing up the decomposition step..thanks for any help with this...
  29. T

    Inverse Laplace Transform: How to Manipulate Fractions for Completing the Square

    Homework Statement Find the inverse laplace transform of \frac{3s + 7}{s^{2} - 2s + 10}Homework Equations completing the square. e^{at}sin(bt) = \frac{b}{(s-a)^{2} + b^{2}} e^{at}cos(bt) = \frac{s-a}{(s-a)^{2} + b^{2}} The Attempt at a Solution F(s)= \frac{3s + 7}{s^{2} - 2s + 10} F(s) =...
  30. W

    Laplace transform and bessels equation

    Homework Statement I'm trying to find the Laplace transform of tJ''0(t), it's from bessels equation, but that doesn't matter too much at the moment, I just need to integrate (e^-st)*t*J''0(t) but am unsure how to go about this with the J''0(t) in there.
  31. W

    Laplace Transform of a Bessel Equation

    Hi guys, I have this question on Laplace transforms, but am not sure how to start it. The zero order Bessel function Jo(t) satisfies the ordinary differential equation: tJ''o(t) + J'o(t) + tJo(t) = 0 Take the Laplace transform of this equation and use the properties of the transform to find...
  32. M

    Laplace transform as rotation. In what space?

    Sorry for thumb terminology, I just would like to grasp the main idea, as a physicist, without unnecessary complications, associated with system of axioms and definitions. Fourier transform can be seen as rotation of basis in space of all complex-valued functions from basis of delta-functions...
  33. A

    How Do You Perform an Inverse Laplace Transform on 1/(s^2+s+1)?

    Using the laplace transform, find the solution to the differential equation: y'' + y' + y = 0 , y(0)=0, y'(0)=1 Using the laplace transform and its properties I end up with: f(s) = 1/(s2+s+1) How can I find the inverse of this/ does anyone know the inverse of it? Setting y=eax I got a...
  34. alexmahone

    MATLAB  Laplace transform of 1 (MATLAB)

    How do I find the Laplace transform of 1 using MATLAB? >> laplace(1) gives the error ? Undefined function or method 'laplace' for input arguments of type 'double'.
  35. alexmahone

    MHB What is the Inverse Laplace Transform of 2s+7-e^-2s/(s+1)^2?

    Find the inverse Laplace transform of $\displaystyle \frac{2s+7-e^{-2s}}{(s+1)^2}$.
  36. alexmahone

    MHB What is the Laplace transform of a step function?

    Find the Laplace transform of $\displaystyle f(t)=1$ if $\displaystyle 1\le t\le 4$; $\displaystyle f(t)=0$ if $\displaystyle t<1$ or if $\displaystyle t>4$.
  37. D

    How Do You Apply and Invert the Unit Step Laplace Transform?

    Someone give me a crash course on Unit Step Laplace Transform and Inverse Laplace Transform. Need help on the formula: U(t-a)f(t-a) to (e^-as)F(s) from what i can deduce, the way to use it is (Laplace) - Find a from U(t-a) - group (t-a) together - replace (t-a) with ____ (?) What...
  38. S

    System of Laplace Transforms (Question and Solution Included).

    I attached the question along with its solution. Upon trying to find X(s), I get (s - 2)/(s^2 + 3s - 1) which is correct but after that I have to take the inverse Laplace transform and I don't know how to get the (-9/2 + sqrt(13)/2) and (-3/2 + sqrt(13)/2) parts and if someone could show me...
  39. J

    Solve Laplace Eq. in 1D: Head & Darcy Vel.

    Homework Statement Solve the Laplace equation in one dimension (x, i.e. (∂^2h)/(∂x^2)= 0) Boundary conditions are as follows: h= 1m @ x=0m h= 13m @ x=10m For 0≤x≤5 K1= 6ms^-1 For 5≤x≤10 K2 = 3ms^-1 What is the head at x = 3, x = 5, and x = 8? What is the Darcy velocity...
  40. fluidistic

    Using Laplace Transform to Solve Differential Equations

    Homework Statement I'd like to solve a DE using Laplace transform. \ddot y (t)+\omega ^2 y(t)=f(t) for all t>0. Initial conditions: y(0)=\dot y (0)=0. The dot denotes the derivative with respect to t. Homework Equations \mathbb{L}( \dot y )=s\mathbb{L}( y )-y(0). Convolution: if h=f*g...
  41. G

    Inverse laplace transform for unique diffusion type problem

    Hi all I have been working on some unique solutions to advection-diffusion type problems. One inverse Laplace transform that I seem to continue to encounter is the following: Inverse Laplace[F(s)] where F(s)=[(1/(((s-α)^2)+β)*exp(-x*sqrt(s/D))] In their classic 1959 text, Carslaw...
  42. T

    PDE, 2D Laplace Equation, Sep. of Variables, Finding Potential

    Homework Statement A square rectangular pipe (sides of length a) runs parallel to the z-axis (from -\infty\rightarrow\infty). The 4 sides are maintained with boundary conditions (i) V=0 at y=0 (bottom) (ii) V=0 at y=a (top) (iii) V=constant at x=a (right side) (iv) \frac{\partial...
  43. P

    Laplace transform IVP 2nd order

    Homework Statement y''+6'+10y=0 y(0)=2 y'(0)=1 Homework Equations The Attempt at a Solution Laplace everything and I get s^2*Y(s)-2s-1+6s*Y(s)-12+10Y(s)=0 isolate Y(s) Y(s)=(2s+13)/(s^2+6s+10) split into 2 terms, bottom can be rearranged by completing the square...
  44. H

    Simultaneous Laplace transforms

    I have to try and solve the following simultaneous Laplace transform problem and don't really know which path to take can someone give me a nudge in the right direction please. dx/dt=4x-2y & dy/dt=5x+2y given that x(0)=2, y(0)=-2 this is what i have so far for dx/dt=4x-2y sx-x(0)=4x-2y...
  45. S

    Laplace Transform, transfer function

    Homework Statement Given transfer function H(s)=s^2+4 and input x(t)=sin(2t), find the ouput y(t) in time domain, and show whether bounded or unbounded. Okay, so I know L^{-1}[sin(2t)]=2/(s^2+4)=2/[(s+j2)(s-j2)] and that Y(s)=H(s)X(s)=2 Therefore, y(t)=2\delta{(t)} However, I am a...
  46. P

    Laplace transform - step function

    Homework Statement f(t)= 1 if 0≤t≤1 ; 0 is t>1 find the laplace transform Homework Equations The Attempt at a Solution I know u(t)= 0 for t<0 and 1 for t≥0 I know I have to shift it and get u_a(t)=u(t-a)= 1 if 0≤t≤a, 0 if a>1 am I even going the right way? then I think...
  47. R

    Laplace Transform Homework | Rob K

    Homework Statement f(t) = (1/(a^2))(1-e^-at - ate^-at) Homework Equations f(t) = (1/a)(1-e^(-at)) F(s) = 1/s(s+a) f(t) = t e^(-at) F(s) = 1/(s+a)^2 The Attempt at a Solution F(s) = (1/s(s+a^2)) - (1/(s+a)^2)Totally lost now, but I think this is wrong anyway, can anyone help me please. I...
  48. M

    Trying to find integral using laplace method

    ∫∞x exp(N(lns-s)) ds how do integrate this when x>1 x<1 and x=1 using laplace method? the maximum point is at x=1 i have the answer for x<1 and x=1 but I am struggling for x>1 as the stationary point is no longer inside the interval.
  49. P

    How do I solve the Laplace transform of te^t without using a table?

    Homework Statement f(t)=te^t, find laplace Homework Equations The Attempt at a Solution I started doing integration by parts and after doing it three times I wasn't sure if I was going in the right direction/making any progress. I'm not supposed to use a table to solve this (I...
  50. N

    Laplace transform with Convolution

    Homework Statement Use convolution theorem to solve: \mathfrak{L} \left \{t\int_{0}^{t} \sin \tau d\tau \right \} Do not solve the integral. Homework Equations "Convolution Theorem" in textbook is stated as: \mathfrak{L}\left \{ f*g \right \}=F(s)G(s)f*g=\int_{0}^{t} \ f(\tau )g(t-\tau...
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