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. C

    Inverse Laplace with given limits

    Homework Statement Problem reads: find inverse Laplace transform of f(t) of F(s)=(2s+3)/(s(s2+7s+10) What is the value of the function f(t) at t=0 and t=∞?Homework Equations Inverse laplace transformThe Attempt at a Solution I solved F(t) down to F(t)= .3/s+0.166/s+2-0.465/s+5 thus lead me to...
  2. M

    Solving mx'' + cx' + kx = 0 Using Laplace

    Hello , can anyone help me solve this using laplace ? mx'' + cx' + kx = 0 , x(0)=0 , x'(0)=0 , m > 0 , c > 0 , k > 0 and c^2-4km > 0 thanks in advance for your answer.(first time using this forum)
  3. P

    Solve Laplace Transform: t{e^{ - t}}u(t - 1)

    Homework Statement Is there an easier way of solving this rather than doing the integral? Find the laplace transform of: t{e^{ - t}}u(t - 1) Homework Equations The Attempt at a Solution \int_1^\infty {t{e^{ - t}}} {e^{ - st}}dt = \int_1^\infty {t{e^{ - t(1 + s)}}} dt =...
  4. C

    Laplace transform and fourier transform

    Homework Statement F{f(t)} is the Fourier transform of f(t) and L{f(t)} is the Laplace transform of f(t) why F{f(t)} = L{f(t)} where s = jw in L{f(t)} The Attempt at a Solution I suppose the definition of F{f(t)} is ∫[f(t)e^-jwt]dt where the lower integral limit is -∞...
  5. F

    Laplace transform with heavyside step function

    Homework Statement I need to solve the ODE y''-y = t - tH(t-1); y(0)=y'(0)=0 Homework Equations - The Attempt at a Solution I'm fine with the process of solving the ODE, but I need a little help regarding the first t. From what I understand from lectures, all of the 't' terms need to be in...
  6. R

    How Do You Complete the Square for Inverse Laplace Transform Denominators?

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  7. R

    Laplace transform and tabular integration

    Homework Statement The Attempt at a Solution I can't get the jump to [e^-st(-s cost bt + b sin bt)]/(s^2+b^2) They say they have to use integration by parts but when e and a trig ratio are involved that requires tabular integration. They're obviously not using tabular...
  8. R

    Understanding Laplace Transform: Step 1 to 2 Difficulty

    Homework Statement The Attempt at a Solution I can understand step 2 to 3, but I can't get step 1 to 2. For simplicity sake we'll just call e^(3-s)t = N since it will = N anyway ultimately. I think the answer should be [N/3-s - 1/(3-s)^2)N - (0 - 1/(3-s)^2)0 The second term...
  9. K

    What are the benefits of studying Fourier series in physics?

    Use of Laplace Transforms! What is the use of Laplace transforms in differential equations? I mean, why are they used? And can you please explain the "s" domain in the Laplace Transform! I studied that it is a variable in the complex plane! But, i want to grow upon that! Please help!
  10. E

    Need Help with Laplace Homework Equations: Solving for Capacitor Voltage

    Homework Statement Homework Equations For a capacitor: i=C(de/dt) The Attempt at a Solution Using Kirchhoff's current law at the note above the resistor R2 I get the following equation, which I believe is right: C(e0' - ei') + (e0 - ei)/R1 + e0/R2 = 0 Then, since the source has a...
  11. M

    Taking the inverse laplace of this?

    How do I take the inverse laplace transform of something that looks like this? It's part of a larger piecewise-defined second order differential equation, but this is the part I'm stuck on. (-4s-1)/(4s^2 + s + 4) Completing the square doesn't work for the bottom, so I figure I need to...
  12. H

    Laplace DE Problem: Proving Laplace Delta(t-2)

    Homework Statement Homework Equations Laplace's transforms. The Attempt at a Solution Okay, so applying the Laplace on both sides yields: s^2Y(s) - sy(0) - y'(0) + 2sY(s) - 2y(0) + 10Y(s) = ? + 13 / s - 1 Is e^{-2s} / s the Laplace \delta(t - 2)? Our instructor gave us the result of...
  13. T

    I have 3 questions which are bessel, half range expression and laplace transform

    the questions are together with this file and my solutions are also attached. hope someone can comment on my solutions. thanks a lot and i hope i won't get any warning any more.
  14. E

    Finding a Transfer Function using Laplace

    I'm trying to find the Transfer function of the circuit below so that I can draw the Bode plot for it. I chose to use Laplace because I think it made it a little simpler. My equations so far: KVL of Left Loop: EQUATION1 Vs=I1(900+1800)-900I2 KVL of Right Loop: EQUATION2...
  15. H

    Why can't I substitute a Laplace transform into an integration?

    Hi, I was given the attached question and have given my wrong attempt at the answer. I know how to work the answer out (also shown) but I would like to know why my first attempt is wrong. Many Thanks
  16. H

    Laplace Transform: Solving Hi Everyone - Help Needed

    Hi everybody, can somewone tell me where am i wrong in this solution?(attached here) thank's
  17. S

    Laplace Transformation Help: Simplifying Denominators and Finding Solutions

    1. Homework Statement [/b] [b] I'm required to find the current i0(t), i wrote KVL equations for each loop, I0 can be expressed interms of i1 and i3, i3 being the current source on the right. i managed to get the solution down to the following : i0 = (s + 4)/(3*s^2 + 4*s + 1) I...
  18. G

    Solving the 2D laplace equation. Quick question about the final stage

    Homework Statement ## \frac{\partial^2 V}{\partial x^2} +\frac{\partial^2 V}{\partial y^2} = 0 ## it's defined for ##V=0 ## for ## x=0, 0<y<b##, ## x=a, 0<y<b##, ## y=0, 0<x<a## ##V=V_0 x/a ## for ## y=b, 0<x<a## Homework Equations None. The Attempt at a Solution I've got ##V(x,b) =...
  19. Twinflower

    Easy laplace conversion from s to t-domain gone wrong

    Homework Statement I(s)=\frac{2}{s^2 + s} Convert this to the time domain The Attempt at a Solution Step 1 - Divide by s I(s)=\frac{\frac{2}{s}}{s + 1} Step 2 - Substract and add 1 to create new fractions I(s)=\frac{\frac{2}{s}-1+1}{s + 1} Step 3 - Split into new fractions...
  20. A

    Solving a Simple Laplace Transform Equation: Tips for Beginners

    I'm new to Laplace and having slight difficulty with what looks like an obvious equation. I can do basic first and second order equations. Why is it that the seemingly easy equations are always the ones to stump you? Homework Statement \frac{dy}{dx}=-y y=2, x=0, y_{0}=2 Homework...
  21. G

    Laplace Inverse of a solution

    Homework Statement laplace inverse of (40/(s^2+4s+5)^2)? Homework Equations I completed the square in the denominator to get 40/((s+2)^2+1)^2 I know that I will get cosines and sines from the shape of it in the laplace inverse; however I'm stuck. The Attempt at a Solution
  22. G

    Laplace and transfer function ( input output voltage)

    Homework Statement Given transfer function : H(s)=Vo/V1=(10/s)/(4+0.4s+(10/s)) and the input voltage is given as V1=10(u(t)-u(t-0.1))V. Use the convolution integral to find Vo for t>=0.1 Homework Equations convolution integral..its hard to write and laplace table The Attempt at a...
  23. E

    Solving the Differential Equation Using Laplace Transform

    Homework Statement Using the laplace transform technique, solve the ordinary differential equation: d2y/dt2+4.2dy/dt+4.5y=0 initial conditions: y(0)=1 and y'(0)=1 Homework Equations From laplace tables: d2y/dt2=s2Y-sy(0)-y'(0) dy/dt=sY-y(0) [b]3. The Attempt at a Solution...
  24. S

    MHB Using definition of Laplace transform in determining Laplace of a step function

    I have a question that has stumped me a bit, i am not sure how to use the definition to calculate it, i can use the tables, but i don't think that's what is needed. Using the definition of the Laplace transform, determine the Laplace transform of I can do it with the table but i am not sure...
  25. P

    Integral (related to Laplace transform)

    Hi. Homework Statement \int\limits_0^{ + \infty } {\frac{{dt}}{{{e^{st}} \cdot (1 + {e^t})}}} Homework Equations The same as... Laplace\left[ {\frac{1}{{1 + {e^t}}}} \right] The Attempt at a Solution Found no elegant properties related to Laplace transform here. So figured my...
  26. W

    Radial solutions to laplace equation

    Hi, When can I assume that the solution the laplace equation (or poisson equation) is radial? That is, when can I look only at (in polar coordinates) \frac{\partial u^2} {\partial^2 r} + \frac{1}{r} \frac{\partial u} {\partial r} = f(r,\theta) instead of \frac{\partial u^2} {\partial^2 r}...
  27. Femme_physics

    Engineering Laplace plane basic electronic circuit (topic from control engineering)

    Homework Statement C = 4 μF L = 0.2 Hy E is DC current. No internal resistance. Write an expression the the laplace plane transfer function, that relates current (that flows through resistor R) to the voltage E. (There is another question. I will just address the first one for...
  28. Twinflower

    Partial fractions before Inverse Laplace

    Homework Statement I have this lowpass circuit which I have transformed to the S-domain. The circuit is to be exposed to a unit step, and then I shall convert the transient response to the time domain. Here's the transfer function of the lowpass circuit: H(s) = \frac{\frac{1}{LC}}{s^2 +...
  29. B

    Laplace Inversion: Why Contour Must Exceed Singularities

    Maybe I am just being stupid, but I don't understand why in the Laplace inversion formula (\mathcal{L}^{-1} F)(t) = \frac{1}{2\pi i} \int_{\sigma-i\infty}^{\sigma+i\infty} e^{st} F(s) ds the contour of integration must be chosen so that \sigma is greater than the real part of all...
  30. H

    Need HELP with Laplace transform

    So I've been searching on the internet and in my textbook and I can't find a simple enough explanation of the Laplace transform of a product of functions. Is it absolutely necessary to evaluate the integral using different methods or is there an easier way?
  31. B

    Laplace Inverse Problems: 1/(s^2-6s+10) and s/(s+1)^2 Solutions

    Homework Statement I have two problems i can't figure out , laplace inverse of 1/(s^2-6s+10) and laplace inverse of s/(s+1)^2 Homework Equations I use the table of laplace transforms in my book The Attempt at a Solution For the first one i did the quadratic formula to solve...
  32. A

    Domain in Laplace - Physical Explanation

    I need a physical explanation of s domain..Is s-domain a higher dimensional plane..?
  33. T

    Finding Laplace transform of a Differential Equation

    Homework Statement Consider the following initial value problem: y''-6y'-7y=sin(9t) y(0)=-4, y'(0)=-3 I need to solve for L[y(t)]Homework Equations The Attempt at a Solution Here are the steps I've taken to solve it: L(y'')-6L(y')-7L(y)=L(sin(9t))...
  34. G

    Inverse laplace transform

    Homework Statement Hi all, I'm struggling to find the Inverse Laplace transform of the following function: F(s) = (1+ 4e(-s) - 5e(-3s)) / s(s2 + 11s + 55), where F(s) is a Laplace transform Solution should be in terms of complex exponentials and unit step functions. Homework EquationsThe...
  35. G

    What is the inverse Laplace transform of F(s)?

    Hi all, I'm struggling to find the inverse Laplace transform of the following function: F(s) = (1+ 4exp(-s) - 5exp(-3s)) / s(s^2 + 11s + 55)Any help is appreciated. Thanks
  36. R

    Physical Meaning of T's in Laplace equations

    Dear All, When I have a equation 1/sT its an integrator with time constant i.e Assuming a T of 5s, if I input a unit step function, the output of the integrator reaches unity in 5 seconds atleast I undestand it that way. Please correct me if I am wrong in the same way can someone help...
  37. A

    Computing the Inverse Laplace Transform: A Step-by-Step Guide

    Hi, I recently posted another question about a Laplace transform and now have a question about taking the inverse Laplace transform. Again, my professor did not cover this topic as well as I could have hoped for, and so I am stuck on this problem as well. Again, I understand the idea of Laplace...
  38. A

    Master the Laplace Transform with Expert Help | Find Solutions Now!

    Hi, I've been asked to find the Laplace transform of a function and I have not the slightest clue where to begin. My professor derived the basic Laplace transforms in class(sin, cos, delta function, step function, etc), all of which I understood perfectly. However, he never really gave us an...
  39. Femme_physics

    Laplace plane meaning in control engineering ?

    We use this plane called the "Laplace plane" to solve problem in control-engineering, second order systems. Can anyone help explain the Laplace plane in simple words? I find the wiki article too fancy for me... it says that it "is a mean or reference plane about whose axis the instantaneous...
  40. G

    Solve Laplace Transform for y'' + w^2y = cos(t), y(0) = 1, y'(0) = 0

    Homework Statement y'' + w^2y = cos(t) y(0) = 1 y'(0) = 0 w^2 not equal to 4 Homework Equations Laplace integral, transform via table/memory... Y(s) = F(s) or whatever you like to use The Attempt at a Solution s^2Y(s) - sy(0) - y'(0) + w^2Y(s) = s/(s^2 + 1). Right side is...
  41. M

    Can you help me solve this differential equation using laplace transforms?

    Homework Statement Solve the differential equation using laplace transforms. Assume zero intial conditions and that the forcing functions are zero prior to t=0 x''(t) + 6x'(t) +8x(t)=sin3t Homework Equations The Attempt at a Solution I took the laplace transform which I found to be after...
  42. H

    Laplace transform formule

    Hello, I'am looking for de correct transformation formule:\mathcal{L}[f(t).g'(t)] (and proof). I'am not looking for method to solve it by means of integrating g'(t), offcourse this a possible way. But assume that g(t) is much work to calculate. So is there a good one to one formule...
  43. A

    Relation of laplace transform with power series

    Hi, just wonder if anyone can help Homework Statement Apparently there is a relation between laplace transform and power series. http://www.jstor.org/stable/pdfplus/2305640.pdf?acceptTC=true states that if the discrete variable n of a power series is replaced by a continuous variable lambda...
  44. T

    Laplace Convolution proof - can someone please explain the diagram

    Homework Statement http://gyazo.com/22f5c655071e49adf3f4652e8042873b I derived the proof myself, but by arbitrary changing the t limit to infinity. I'm hoping that by understanding the diagram in the link above I will have a better idea of why I changed the limit. I currently look at it...
  45. E

    Finding the particular/complementary solution from a laplace transform

    Say you find the laplace transform V(s) and want to switch it back to the time domain, once you've done this, how do you determine which parts of the total solution correspond to the complementary solution and particular solution respectively? Do you just find which parts approach zero as time...
  46. K

    Engineering Laplace Circuit Problem: Solving for Capacitor Voltage in the Laplace Domain

    Homework Statement Determine the capacitor voltage in the laplace domain. Homework Equations http://picturepush.com/public/8001507 http://picturepush.com/public/8001507 The Attempt at a Solution http://picturepush.com/public/8001546 http://picturepush.com/public/8001546 Using nodal...
  47. P

    MHB Laplace equation

    A metallic spherical shell occupying the region given in terms of spherical polar coordinates \( (r, \theta, \phi)\) by \(r \le a \) has its surface \(r = a\) maintained at temperature \(u(a,\theta)=1 + \cos(\theta)+2 \cos^2(\theta) \) Using the general solution for the Laplace equation \(...
  48. M

    Extended tables of Laplace transforms

    I'm looking for "extended" tables of Laplace transforms i.e. ones which have examples beyond the basics commonly shown in tables. I have already linked to this one in another thread: http://www.me.unm.edu/~starr/teaching/me380/Laplace.pdf which is a good indicator of what I am looking for...
  49. R

    Laplace Transforms of Kirchhoff's Laws

    Back again with another problem. These transforms are difficult... Homework Statement Had to take a screenshot: http://i.imgur.com/LM38A.png The Attempt at a Solution His instructions are to take the Laplace Transform of each of Kirchhoff's laws. However, when we do, nothing useful comes...
  50. M

    Laplace analysis of RLC and RLCC circuits

    (I hope I have posted this in the appropriate forum. I have consulted the posting guidelines and this seems to be OK. The subject of my post certainly isn’t homework or academic coursework, rather it is self-study for its own sake. If this is the wrong place – apologies, and maybe an admin could...
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