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. T

    Engineering RLC Circuit problem using laplace transforms

    Homework Statement The switch at S is closed at time t=0 and a constant ... va−vh=v0 is applied. Use Kirchoff's laws to write 3 equations for i,i1,i2, and q(t). What is the relation between q(t) and i,i1,i2? Given that the charge on C and all currents are initially zero, find an ODE for i1(t)...
  2. R

    Getting to Grips with Control Theory: Exploring the Laplace Transform

    Good day to you people, I have just started learning control theory at Uni, as part of my course, and I have to admit it is quite difficult to grasp. I am starting from the basics, and I am having difficulty understanding what 's' is supposed to represent as regards the Laplace transform...
  3. M

    MHB Solve Laplace Transform ODE: $\sin t, 0<t<\infty$

    Solve $\begin{eqnarray*} {{u}_{t}}&=&{{u}_{xx}},\text{ }0<x<\infty ,\text{ }0<t<\infty \\ u(0,t)&=&\sin t,\text{ }0<t<\infty \\ u(x,0)&=&0,\text{ }0\le x<\infty . \end{eqnarray*}$ I need to apply the Laplace transform to solve this, so by applying it I get...
  4. M

    Laplace Transform Help: Solving Equation (1)

    Hi guys, Would you help me in finding how Laplace transform is done for equation (1) in the attached image? Equation (2) is the correct solution but I don't know how we get it! I know that we can split the integration in (1) into two parts; one is from t0 to 0 and the other part is from 0...
  5. J

    IVP Laplace Transform Problem - Tricky Inverse Laplace Transform

    IVP Laplace Transform Problem -- Tricky Inverse Laplace Transform Homework Statement Solve x"+x'+x=1, given x(0)=x'(0)=0 Homework Equations The Attempt at a Solution Plugged in transforms: s2*Y(s)-s*y(0)-y'(0)+s*Y(s)-y(0)+Y(s)=1/s Plugged in initial value points, simplified...
  6. S

    Strategies for Solving Inverse Laplace Transform Problems with Complex Roots

    Homework Statement Hello all, Having difficulty with this one question that involves complex roots. Here it is: F(s)=\frac{s+3}{s^3+3s^2+6s+4} I tried two different ways to tackle it. First method I divided it right away: F(s)=\frac{s+3}{s^3+3s^2+6s+4}\rightarrow{s^2+6-\frac{14}{s+3}} Is there...
  7. M

    Laplace Transform (Integration Theorem)

    Hi guys, would you please help me in my question which is in the attached image? Thanks
  8. U

    Laplace equation derivation, where does the potential go

    Homework Statement Since the potential field is only a function of position, not velocity, Lagrange's equations are as follows: (Wikipedia, image 1) Homework Equations (Wikipedia, image 2) The Attempt at a Solution Now, -\frac{\partial{V}}{\partial{q_{j}}} How is speed...
  9. A

    Initial value problem and laplace transform

    Homework Statement I understand how to do initial value problems but I'm slightly stuck when the initial values are y(0) = y'(0)=0 The question is Solve: y''+3y''+2y=f(t), y(0)=y'(0)=0 where f(t) is a square wave. Homework Equations \Im{y'} =s\Im{y}-y(0)...
  10. M

    Tension on a capacitor from Laplace domain to time domain.

    The problem ask me to find the tension on a capacitor after a switch has been opened. I have everything in terms of equations in s-domain and I'm sure they aren't wrong because I checked on the book. My unique problem is to understand a certain passage necessary to find the voltage knowing...
  11. S

    Laplace Transform of this Differential Equation

    I tried doing this problem soooo many times for several days and keep failing and then I'm so worn out I can't keep thinking straight and I can't move on because of this! I'm attaching my work and it would be great if I can get back and forth feedback if necessary to find my mistake(s). The...
  12. A

    Laplace transform of a matrix exponential

    Homework Statement show that the Laplace transform of e^(At) = (sI - A)^(-1) \mathcal{L}\left\{ e^{At} \right\}(s) = \left(sI - A \right)^{-1} The Attempt at a Solution I find \left( e^{At} \right)_{ij} = \sum_{k=0}^{\infty} \frac{(A^k)_{ij}t^k}{k!} and since \mathcal{L}\left\{...
  13. A

    Substitution in Laplace Transform: Simplifying the Process of Finding Transforms

    Hi, I'm learning about Laplace and I was wondering what substitution is when using laplace. Just say you have to find the laplace transform of {x^4 e^4x} I know what the individual transforms are = 4!//S^5 and 1/S-3 but how do you kinda smush them together, apparently that's...
  14. P

    Laplace Transforms and Phasors in Circuit Analysis

    So last semester, I had a Circuit Analysis course where I learned about phasors. Basically, when dealing with AC circuits, I should convert everything to the frequency domain where X = j\omega L and X = \frac{1}{j\omega C}. I feel like I understood this part really well. However, in Circuits...
  15. fluidistic

    Solving Laplace equation in a hollow sphere

    Homework Statement Determine the stationary temperature distribution in the hollow sphere a<r<b where r=a is kept at T1 and r=b is kept at T2. Homework Equations \triangle u =0. But with the Laplacian in spherical coordinates. The Attempt at a Solution I think I must solve the given equation...
  16. A

    What's the inverse Laplace transform of this?

    Homework Statement L^{-1}\{\frac{1}{(s^2+4)^2}\} Homework Equations The Attempt at a Solution I have no idea how to solve this. Any idea to being solving the problem would be appreciated.
  17. A

    Does the dirac delta function have a Laplace transform?

    If yes, how can we find it?
  18. fluidistic

    Laplace equation in 2d problem, Dirichlet problem

    Homework Statement Consider a circle of radius a whose center is in (0,0). Let (r, \phi) be the polar coordinates and (x,y) the corresponding rectangular coordinates of the plane. Calculate the solution to Dirichlet problem (interior) for Laplace equation \nabla ^2 u =0 with the following...
  19. P

    Laplace Transform Homework: Solving Equations & Unit Step

    Homework Statement Homework Equations The Attempt at a Solution I don't know if it is possible to use Derivative of a transform while it is on unit step function procedure. How can these questions be solved? Thank you.
  20. G

    RLC inverse Laplace transform question

    The link below shows a circuit and the s-domain equation for I1(s) that I have derived; My question is on the portion of the equation containing the "(s+a)" terms. I am trying to arrive at the time-domain equation for I1(s) but instead of arriving at the Laplace transform pair shown in the...
  21. O

    Solve Laplace Transform: y''+4y'+5y=3u^4(t)+7(t*u(t)*δ(t-1)

    Homework Statement could u help to find result? I don't know laplace of u^4(t)?? Homework Equations y''+4y'+5y=3u^4(t)+7(t*u(t)*δ(t-1) The Attempt at a Solution The only one i couldn t found is 3u^4(t),,
  22. P

    Laplace transform using properties

    Homework Statement Find the laplace transform of f(t)=(t-5)e^{-17(t-5)}u(t-5) The Attempt at a Solution The answer I got was \frac{e^{-5(s+17)}}{(s+17)^{2}}. I thought I finally understood the process, but when I plugged it into wolfram alpha, the answer I got was different...
  23. H

    Finding the Inverse Laplace Transform for Solving Fractional Equations

    Homework Statement Compute the inverse Laplace transform of Homework Equations http://img267.imageshack.us/img267/696/66451772.png The Attempt at a Solution I tried using partial fraction but no luck.
  24. A

    Laplace Transform: Solving for f(x) and Finding Area Under a Curve

    So the question is: A curve rise from the origin of the xy plane into the 1st quadrant. The area under the curve from (0,0) to (x,y) is 1/5 the area of the rectangle with these points as opposite vertices. So I'm solving for f(x): So far what i have is: Area(D)=1/5 xy=integral 0 to x...
  25. P

    Laplace Transform of Systems of ODEs with variable coefficients

    Homework Statement Say you have: EQ1: y1''*t+y1'*t+y2=0 and EQ2: y2''*t+y2'*t+y1=0 y1(0)=0,y1'(0)=0,y2(0)=0,y2'(0)=0 Homework Equations The Attempt at a Solution I can get it so far, but having both y1 and y2 really gives me fits: Eq1: Y1(-2s-1)+dY1/ds(-s2-s)=-Y2...
  26. S

    Differential Equations, Laplace Transform

    Hello all, Next semester I will be taking a Network Analysis course in my EE degree. Moreover, we will be utilizing numerous mathematical concepts I have not yet seen. If anyone has (preferably free) access to any of the concepts to follow that they would be willing to share, I would be...
  27. T

    Laplace Operator in Polar Coordinates: Steps & Solutions

    Homework Statement The Laplace operator Δ is defined by: Δ= Show in polar coordinates r and Θ, that the Laplace operator takes the following form: http://upload.wikimedia.org/wikipedia/en/math/0/7/a/07a878276cffd0c680f3f827204aba24.png Homework Equations x=rcos(Θ), y=rsin(Θ), r ≥ 0, Θ ∈...
  28. M

    Why Are All Singularities of the Laplace Transform Left of the Line Re(s)=c?

    f(t)=\int^{c+i\infty}_{c-i\infty}F(s)e^{st}ds Why we suppose that all singularities are left from line Re(s)=c?
  29. T

    Laplace Transform of Product of Two Functions

    Homework Statement Laplace Transform of u(t-∏/2)et (u is unit step function) Homework Equations Laplace Transform Table (any) The Attempt at a Solution I tried using the Laplace transform for the unit step function and the exponential function. L{u(t-∏/2)} = e-(∏s)/2...
  30. T

    Existence of Laplace Transform of Piecewise Functions

    Homework Statement Let f(t) = t if 0<t<3 et if t>3 a. Is f(t) piece-wise continuous? b. Is f(t) of exponential order α? Either prove it by producing an M, T and α that satisfies the definition, or show that no such constants exist. c. Does the Laplace transform of f(t) exist...
  31. R

    Understanding the Inverse Laplace Transform of Fractional Expressions

    Homework Statement I'm attempting to find the inverse laplace transform of (5/((2x+3)(4+x^2))) The Attempt at a Solution Here's the solution: http://www.wolframalpha.com/input/?i=inverse+laplace+transform+%285%2F%28%282x%2B3%29%284%2Bx%5E2%29%29%29 There's 3 terms in the solution and 2 are...
  32. E

    Why Does My Laplace Transform Calculation Differ from the Textbook's Result?

    Homework Statement f(t)=e^(t+7)Homework Equations £{f(t)}=∫e^(-st)f(t)dtThe Attempt at a Solution so i insert my f(t) into the formula, came up with ∫e^(-st+t+7)dt using u substitution, u=t(-s+1)+7, du=(-s+1)dt so it follows that 1/(-s+1)∫e^(u)du=e^(u)/(-s+1) so I plug u back in, and should be...
  33. A

    Laplace Equation in Cartesian Coor.

    Solving the Laplace equation in Cartesian Coordinates leads to the 2nd order ODEs: \frac{X''}{X}=k_1, \qquad \frac{Y''}{Y}=k_2 \qquad \frac{Z''}{Z}=k_3 In each case the sign of k_i will determine if the solution (to the particular ODE) is harmonic or not. Hence, if two people solve the...
  34. M

    Laplace Transform: Explaining Theory & Solving ODE

    Hello, What does laplace transformation exactly 'do'? If I have PDE of second order and use LT on it, what do i get to solve? ODE? or if I have ODE of second order, what do I need to solve afet transformation? How does this work? is there any rule?!
  35. L

    Dealing with Derivatives in Laplace Transform Problems

    I'm working on a LaPlace transform problem. Part of it was this: -ty' I elected to do this first: (-1)\frac{d}{ds}L(y') Which I then expanded to: -\frac{d}{ds}(-y(0)+sL(y)) By the given initial conditions y(0)=0 -\frac{d}{ds}(sL(y)) So next I need to expand this out: -(L(y)+sL(y)') Now I'm...
  36. J

    Laplace equation in rectangular geometry

    [/itex][/itex]Homework Statement A battery consists of a cube of side L filled with fluid of conductivity s. The electrodes in the battery consist of two plates on the base at y = 0, one grounded and one at potential V = 12 Volts. The other sides of the battery casing are not conductive...
  37. R

    Inverse Laplace Transform of 2/(s + 4)^4: Explained

    Hello I'm struggling to understand some basics here with the laplace transform.. I'm given the laplace transform of 2/(s + 4)^4 and I need to take the inverse of this to get back to y(t) Looking at my tables the only transform similar to this is 1/(s + a)^2 I understand I can pull...
  38. K

    How Do You Apply Shifting in Inverse Laplace Transforms?

    Hello everyone, I'm currently enrolled in Control Theory at my University, and part of the coursework requires differential equations; which wouldn't be a problem, if not for the fact it's been 2 years since I've taken D.E. Anyway, over the course of the problem I ran into this little function...
  39. K

    How Do You Find the Inverse Laplace Transform of (1/6)/((s+2)^2)?

    Hello everyone, I'm currently enrolled in Control Theory at my University, and part of the coursework requires differential equations; which wouldn't be a problem, if not for the fact it's been 2 years since I've taken D.E. Anyway, over the course of the problem I ran into this little function...
  40. M

    Find the inverse laplace transform of this function

    Homework Statement F(s) = 3/(s(s^2 +2s + 5)) Homework Equations The Attempt at a Solution I have used partial fraction using coefficients. F(s) = (3/5)s - (3/5) ((s+2)/(s^2 +2s + 5)) and reduce s^2 +2s +5 by completing the square F(s) = (3/5)s - (3/5) ((s+2)/((s+1)^2 + 4))...
  41. A

    How to take the Laplace transform of a function involving the step function?

    so I am working on a problem and i have a question about the step function. let's say that a decaying force is f(t). with u (t-11)= {0 if t< 11, 1 if t>=11}. the function of the force is then g(t)=[1-u(t-11)]* f(t) or g(t)=f(t)-[u(t-11)*f(t)] as i understand it. in order to do the...
  42. K

    Simplifying Inverse Laplace Transform of s/(s+4)^4

    Homework Statement the questions asks to determine inverse laplace transform of s/(s+4)^4 Homework Equations The Attempt at a Solution this can supposedly be done just using laplace transform tables so I am guessing i need to simplify that to something that's workable but i...
  43. C

    Laplace Transform of a signal?

    Can someone explain the steps of this solution? In my linear systems class, we are doing Laplace transforms using transform tables and the properties. I can usually do the problems that closely resemble the table, but when they involve heavy algebraic manipulations, I don't know what to do...
  44. D

    Green's Function using Laplace Transformation

    I was wondering if someone could help me go through a simple example in using Green's Function. Lets say: x' + x = f(t) with an initial condition of x(t=0,t')=0; Step 1 would be to re-write this as: G(t,t') + G(t,t') = \delta(t-t') then do you multiply by f(t')\ointdt' ? which I...
  45. J

    Finding the laplace of a Heaviside system

    I need to use Laplace transforms to find the solution to this system of Heaviside functions but I'm not sure where to start because the two different x's in the system are confusing me. Should I start by taking the laplace transforms of both sides where the laplace of H(t-1) = e-s/s These...
  46. C

    Laplace Transform: Solving 2D Wave Equation Now

    How do I take the laplace transform of a two dimensional wave equation? I need an answer now!
  47. D

    How to solve this partial differential equation which is a Laplace equation

    I was trying to solve this partial differential equation which arose because I wanted to find a general solution to the Laplace equation in the case f=f(x,y). \frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{\partial y}^{2}}=0 Thanks in advance.
  48. W

    How can I use Laplace transforms to solve for cos^3 t?

    I need to do a laplace transform on cos^3 t. I understand laplace but the trig is tripping me up. cos^3 t = Cos^2 t * Cos t = cos t * (cos 2t + 1)/2 (double angle formula) so i have (cos t)*(cos 2t)/2 + (cos t)/2. my book's solution says (cos t)*(cos 2t)/2 = (1/2)(cos (2t+1) + cost...
  49. romsofia

    Laplace equation in spherical coordinates

    Homework Statement Verify by direct substitution in Laplace's equation that the functions (2.19) are harmonic in in appropriate domains in ℝ2 Homework Equations (2.19)= {u_n(r, \theta)= \lbrace{1,r^{n}cos(n \theta), r^{n}sin(n \theta), n= 1, 2...; log(r), r^{-n}cos(n \theta), r^{-n}sin(n...
  50. W

    Laplace Transform of Heat Equation

    Homework Statement T(x,t) is the temperature distribution for t > 0 in a semi-infinite slab occupying x > 0 T(x, 0) = T_0 e^{-ax} for x > 0 (with a positive constant) T(0, t) = T_1 for t > 0 \tau(x, s) is the Laplace transform of T(x, t) show that \tau(x, s) = \frac{T_0}{s - Ka^2}e^{-ax} +...
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