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. B

    Can Laplace Transform Solve Differential Equations with Arbitrary Constants?

    hi guys i cannot solve this. say that y'' + y = g(t) derive the formula y(t) = c1*cost + c2*sint + convolution of sint and g(t)
  2. P

    Code for Laplace Transform

    Hi all How hard is it to write a symbolic laplace solver ? (I am aware that there are many available out there, but I wanted to write one in MATLAB which doesn't use symbolic toolbox). Where can I start to get a good reference for the code ? Thank you Prabhakar
  3. H

    Laplace Transform of Noncausal systems

    This is some homework from my Analog Signal Processing class, it's a cool class and I've not had much trouble, but this is giving me a bit of a problem and I was hoping someone could help me see what I'm doing... Here's the problem: Sketch, Get the Laplace Transform of, and Sketch the complex...
  4. G

    How to Solve a Differential Equation Using Laplace Transformations?

    Homework Statement Solve df/dx (x)+ f(x)= 3 under the condition f(0)= -2 using the Laplace transformations Homework Equations The Attempt at a Solution Not even sure where to start on this one. Dont have any examples that point me in the right direction. Any ideas anyone??
  5. L

    Fourier transform of laplace operator

    Hello Everybody. I gave a quick look onto the internet but i couldn't get anything interesting. Heres my problem. Im solving the differential equation given by: (-\Delta+k^2)^2u=\delta Where \delta is the dirac delta distribuiton (and u is thought as a distribution as well) The...
  6. P

    Initial Value Problem with Laplace Transforms

    Solve the following given y(0) = 0 & y'(0)=1: y′′+3y′+2y = u2(t), such that u2(t) is a heaviside step function Here's what I've got so far, =>s2Y(s)−sy(0)−y′(0) + 3sY(s)−3y(0) + 2Y(s)= exp(−2s)/s Y(s) = (exp(−2s) + s) / (s(s2+3s+2)) Y(s) = exp(−2s)/(s(s2+3s+2))* + 1/(s2+3s+2)** The...
  7. M

    Solving Integral Equation with Laplace Transform

    Homework Statement y(x) = e^{3x} + \int_0^{x}\hspace{1mm}y(t)\hspace{1mm}e^{x-t}\hspace{1mm}dt Homework Equations (f*g)(x) = \int_0^{x}\hspace{1mm}y(t)\hspace{1mm}g(x-t)\hspace{1mm}dt L(f*g;s) = L(f;s)L(g;s) I will use Y(s) to denote L(y;s) The Attempt at a Solution I tried...
  8. C

    Hom. heat equation in cylindrical coordinates using Fourier & Laplace transforms

    I'm trying to solve the homogeneous heat equation of a semi-infinite cylinder in cylindrical coordinates for a semi-infinite cable (no theta dependence): \frac{\partial U}{\partial t}=D\left(\frac{\partial^{2} U}{\partial r^{2}}+\frac{1}{r}\frac{\partial U}{\partial r}+\frac{\partial^{2}...
  9. G

    Where can I find practice problems for mastering Laplace transforms?

    I read through that 800 something page Dover book on differential equations, but I feel like I haven't had enough practice. I need several problems and solutions, but mostly Laplace transform problems. In fact even just Laplace transform problems would be greatly appreciated.
  10. S

    Engineering LTspice iv Laplace circuit design help

    Homework Statement Find Io(t) for t>0 using Ltspice 4 Homework Equations n/a The Attempt at a Solution Here is what i have so far. I am unsure if i have the voltage source on the left correct or not. I am also stuck on how to use the current dependent current source from...
  11. G

    Engineering RLC Circuit Second Order Differential and Laplace

    Homework Statement Derive the second order differential equation relating x(t) and y(t). Using the Laplace transform, find the total response as a function of the zero input response and the zero state response in the following form.Homework Equations Y(s)=Yzs(s) + Yzi(s) The Attempt at a...
  12. M

    Prerequisites for Laplace Transforms

    Hi, I was wondering what the prerequisites are for doing Laplace Transforms. I'm just a little confused with one of the examples : \int_0^{\infty} = e^{-sx} dx = [e^{-sx}/s]_0^{\infty} = 1/s I understand that e^{-s(0)} is 1. But where does the e^{-s(\infty)} part go? Is there something I...
  13. S

    Laplace Transforms: Homework Help

    Homework Statement Okay, I'm pretty lost in my differential equations class and I'm not sure how to transform the functions Homework Equations f(t)= 1 / (s+2)^2 f(t)= 1 / (s+3)^3 f(t)= 1 / (s^2-3^2) They seem pretty simple, but i have no clue if whether I'm suppose to use 2 formulas or...
  14. I_am_learning

    What is the significance of Laplace Transform in determining system stability?

    After solving a simple circuit with a inductor L in series with resistor R Driven by Voltage source V0 by using Laplace Transform we get I(s) = \frac{V}{s (R + Ls} Why do we call this the frequency Domain response? What does 's' represents?
  15. H

    Inverse Laplace of a Mass-Spring System

    Homework Statement Given a transfer function in the Laplace Domain Detemine an expression for x(t), given f(t) is a sinusodial input with frequency omega = root(k2/m2) and amplitude of 1 N (initial conditions equal 0)Homework Equations...
  16. M

    Efficient Laplace Transform Solution for Sin Square Root of T

    Laplace Transform of... Few months back my friend called me to help him (and his whole class!) finding the Laplace Transform of \sin \sqrt{t}. Since then I found myself wasting hours and days (even before non-mathematics exams!) trying to solve the problem. I also checked the provided answer...
  17. G

    Laplace Transform of sin(t)/t help

    Homework Statement \mathcal{L}\left[\frac{\sin(t)}{t}\right]=? Homework Equations \mathcal{L}\left[f(t)\right]=\int_0^\infty e^{-zt}f(t)dt The Attempt at a Solution \int_0^\infty e^{-zt}\frac{\sin(t)}{t}dt=\int_0^\infty \frac{e^{-zt}}{t}\left(\frac{e^{it}-e^{-it}}{2i}\right )dt...
  18. C

    What is the inverse laplace transform of this?

    F(s) = \frac{1}{K^s} where K is a positive real.
  19. R

    Differential equation, Laplace transform

    Homework Statement solve the differential equation by using Laplace transform x^{''}(t)+7x^{'}(t)+10x(t)=sin3t x(0)=0, x^{'}(0)=0 Homework Equations X(s)s^2+X(s)7s+X(s)10=\frac{3}{s^2+9} then i get X(s) = \frac{3}{s^2+9}\cdot\frac{1}{(s+5)(s+2)}...
  20. M

    Laplace Transfrom to Solve ODE Help

    Hi, so we just started learning about Laplace transforms yesterday, and I have a problem which I am not sure what to do: My question is about the second term, if it was a constant coefficient I could do this fine, but none of the 2 examples we did in class for solving ODEs with Laplace...
  21. T

    Multiplication in a LaPlace Transform

    Homework Statement Find the Laplace transform of the given function, a and b are real constants. Hint: cos(bt)=(e^ibt+e^-ibt)/2 and sin(bt)=(e^it-e^ibt)/2i. f(t)=e^(at)*sin(bt) The attempt at a solution The LaPlace transform of e^at is 1/(s-a). The LaPlace transform of sin(bt) is...
  22. T

    Laplace transform of function with independent variables OTHER than time

    Hi all Conventionally we used to seeing the Laplace transform applied to problems that use time as the independent variable, can anybody point me at some examples that do not use time as the independent variable? Thanks Tim
  23. A

    Laplace transform, with integral

    Homework Statement I'm trying to solve a definite Laplace transform. The function is defined as f(t) = sin(t) over the interval [0, \pi] Homework Equations The integrate to evaluate is: \int_0^\pi e^{-st} sin(t) dt The Attempt at a Solution To evaluate, first use integration by...
  24. T

    What is the laplace transform of t(sinh(3t))?

    Homework Statement So I finally had my final today, and I was so worried because I needed a 90% at least to pass the class with an A. :\ So I'm really worried with a few of my answers. One of those (that I can remember) was the laplace transform of: t(sinh(3t). Homework Equations laplace t...
  25. J

    How can I use the Dirac delta function in a Laplace transform?

    Homework Statement 2y'' + y' + 4y = ∂(t-π/6)sin(t); y(0)=0, y'(0)=1/2. Homework Equations Dunno The Attempt at a Solution The left side of the equation is what's tripping me up. There's nothing on my Elementary Laplace Transforms table that has the Dirac delta function...
  26. J

    Deriving some Laplace transforms

    Homework Statement Find the Laplace transform of each of the following functions: ... 2. f(t)=eatcos(bt) 3. f(t)=tn, were n is a positive integer Homework Equations As you well know, taking the Laplace of f(t) means ∫f(t)e-stdt from 0 to ∞ The Attempt at a Solution...
  27. F

    What is the proof for the Laplace transform of f(t)/t?

    Hello, I am trying to find the laplace transform of \frac{f(t)}{t} http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/mathematics-2/node92.html Lemma 10.3.9 I don't understand why we are taking the limit as s goes to infinity Can anyone help me? thank you
  28. Char. Limit

    What is the Laplace Transform of the Sinc Function?

    Just out of curiosity, I've been looking into the Laplace transform and noticed that... \mathcal{L}(sin(t))=\frac{1}{s^2+1}=\frac{d}{ds}(tan^{-1}(s)) So, I was wondering if the Laplace transform of the sinc function was the inverse tangent function. In LaTEX...
  29. G

    Differential Eq: Inverse Laplace Transforms

    Homework Statement I have a couple of problems that I am stuck on. The following: y'' - 6y' + 9y = t y(0) = 0 y'(0) = 1 and y'' - 6y' + 13y = 0 y(0) = 0 y'(0) = -3 Homework Equations y'' = s2Y(s) - sf(0) - f'(0) y' = sY(s) - f(0) y = Y(s) The Attempt at a Solution...
  30. S

    How do I find B using the boundary condition ux(0,t) = -Q in Laplace transforms?

    I'm deriving the heat equation using Laplace transforms. I'm up to u(bar) = Aesqrt(s)x + Be-sqrt(s)x. My boundary conditions are lim(x tends to 0) = 0 & this makes A = 0, so u(bar) = Be-sqrt(s)x My other BC is ux(0,t) = -Q My question is how do I get B using this BC? I tried using the def...
  31. E

    Inverse laplace transofrm of natural logarithm

    Homework Statement the inverse laplace transform of ln\frac{s+2}{s-5} using the inverse Laplace transform of the derivative Homework EquationsL^{-1}{\frac{d^{n}}{ds^{n}}F(S)} = (-1)^{n}t^{n}f(t) The Attempt at a Solution the integral of ln\frac{s+2}{s-5} I worked to be (s+2)ln(s+2)-(s+2)...
  32. J

    Laplace Transform to Solve a Diff. Eq. System spent hours on this one =*(

    Homework Statement http://img710.imageshack.us/img710/3381/38442384.jpg The Attempt at a Solution [PLAIN][PLAIN]http://img695.imageshack.us/img695/6295/56987528.jpg Thanks in advance, JJ
  33. C

    Find Laplace Transform of 1+t - No Table Needed

    Find the laplace transform {\cal L} (1 + t) Without using the table. im really having a hard time working these out please help!
  34. I

    What is the Laplace Transform of (t-3)u2(t) - (t-2)u3(t)?

    Compute L[(t-3)u2(t) - (t-2)u3(t)] I come up with using unit step function but I don't know how to solve it
  35. G

    Solve Laplace Transform: L{tcos(t)}

    Homework Statement L{tcos(t)} Homework Equations Using the laplace transform find the equation. The Attempt at a Solution I already have a really long answer. . .I was just wondering if someone can explain this: Another method (appart from using integration by parts...
  36. B

    A laplace transform question

    one of the questions my diff eq teacher gave us is L{(t^3)(sin^3(t))} (laplace of t cubed sine cubed t) i'll just cut to the chase and ask, would you be of the opinion that a question like this is in the scope of an intro to ODE class?
  37. W

    Laplace Transforms to solve First Order ODE's.

    Homework Statement Solve the system of first order differential equations using Laplace Transforms: dx/dt = x - 4y dy/dt = x + y, subject to the initial conditions x(0)=3 and y(0)=-4. Homework Equations So far I've used the limited knowledge of Laplace Transforms for first order...
  38. R

    Laplace Transform, what am I doing wrong?

    Homework Statement Solve using the Laplace Transforms (can not use partial fractions) f '(t) + \int2f(u) du = 2 + 3f(t) Homework Equations Using Laplace f '(t) gets replaced with sF(s) -f(0) \int2f(u) du gets replaced with \frac{2F(s)}{s} Please correct me if I'm wrong on...
  39. R

    Proof of an inverse Laplace Transform needed please

    Homework Statement \frac{s}{(s-a)(s-b)} Homework Equations Now I know that it results in: [tex]\frac{ae^(at)-be^(bt)}{a-b}[\tex] The Attempt at a Solution OK, I don't have the slightest clue where to begin. Could someone point me in the right direction? I've looked at all...
  40. L

    Laplace Transforms with IVP and linear first ODE

    Homework Statement y'' + ty' - y = 0 plugging in the known Laplace formuals i get this... [s^2Y(s) - sy(0) - y'(0)] + [-sY'(s) - Y(s)] - Y(s) = 0 Homework Equations y(0) = 0 y'(0) = 3 The Attempt at a Solution simplying to a linear first order DE -sY'(s) + (s^2-2)Y(s) = 3...
  41. T

    Why Does L+ {f(−at)} Equal Zero for a Causal Signal?

    Homework Statement Let f(t) denote a causal signal. Show that L+ {f(−at)} = 0 for any a > 0. Homework Equations The Attempt at a Solution I know that the laplace{f(at)} = 1/a*F(s/a) but by simply placing -a whenever determining the Laplace, my signs gets confused within the...
  42. M

    Transforming Cosine with Unit Step?

    Homework Statement Find the Laplace transform of cos(t)h(t) if the LT of h(t) (unit step) is 1/s Homework Equations --- The Attempt at a Solution Since the unit step isn't shifting anything anywhere, can I just treat it as 1? Seems too simple, haha. Any ideas? thanks.
  43. N

    Solving the Laplace Equation on a Circular Wedge

    Homework Statement I need to solve the Laplace equation, u_rr + (1/r)u_r + (1/r^2)u_{theta}{theta} = 0, on a circular wedge with radius R, angle {alpha}, where u(r,0) = 0, u(R,{theta}) = 0, and u(r,{alpha}) = 50. Homework Equations The Attempt at a Solution Separate variables -...
  44. P

    Laplace Eq with Dirichlet boundary conditions in 2D (solution check)

    Homework Statement The steady state temperature distribution, T(x,y), in a flat metal sheet obeys the partial differential equation: \frac{\partial^2{T}}{\partial{x}^2}+{\frac{\partial^2{T}}{\partial{y}^2}}=0 Separate the variables and find T everywhere on a square flat plate of sides S with...
  45. M

    Solving Laplace Inversion with Erfc

    When trying to solve a pde using Laplace transform, I need to invert an expression of the form \frac{\exp{(-as-b\sqrt{s})}}{s^2} A friend told me that Mathematica cannot invert such expression. I try using convolution but a bit loss when trying to evaluate the integral of Erfc(.)...
  46. P

    Inverse Laplace Transform Help

    Homework Statement Find the unit impulse response for the operator D^4 + I, using the Laplace transform. Homework Equations The Attempt at a Solution w^(4) + w = δ(t) s^4W + W = 1 W = 1 / (s^4 +1) Now, I need to find the inverse laplace transform of W. But I don't know how to simplify W...
  47. P

    Engineering Laplace Transform for a circuit

    Homework Statement It's been a while since I've done laplace transforms, but I need some help with one! I am supposed to find the laplace tranform for the following signal. This is for my circuits class btw. xa(t) = 2u(t-1) Homework Equations Laplace tables...
  48. L

    Finding Laplace Transforms Sin(bt)

    Homework Statement Simply find L{Sin(bt)}Homework Equations My Professor gave us a hint "Use Integration by Parts TWICE" The Attempt at a Solution so i start with the usual limit as N -> infinity the integral of e^(-st)sin(bt)dt Integrating by parts i get ( u = sin(bt) du = bcos(bt) dv =...
  49. M

    Laplace Transform for Solving a Second-Order ODE with Piecewise Functions

    Homework Statement Using laplace transform, solve: y'' - y = \frac{x^2-3x+2}{|x^2-3x+2|} y(0)=y'(0)=0 Homework Equations The Attempt at a Solution Just to know if I'm on the right path - Since the quadratic has 2 roots, at 1 and at 2, the entire thing can be equal to...
  50. B

    Combining Laplace Transforms to Obtain Time Domain Behavior

    Homework Statement Not a homework problem exactly, but in an EE textbook I saw something to the following effect: To obtain the time domain behavior for \frac{s}{s^2+\frac{\omega_0}{Q} + {\omega_0}^2} the following Laplace transforms are combined to cancel the term in the numerator...
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