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. Sirsh

    Calculus Resources for learning Fourier's and Laplace transforms/series in 2 weeks

    Does anyone know of any good sources, websites, books etc. That would be best for trying to become proficient in these topics in as short of a time period possible? I have a good grasp on calculus concepts as is, but I'm undertaking a unit that involves these concepts in the application to...
  2. D

    A question on Laplace transform

    x(t) and y(t) are related by y(t)=1/(x(t) -k), how should I derive Y(s)/X(s)?
  3. H

    Understanding the Laplace Transform for Right-Handed Notes

    for the right part of the notes, why the integral of (e^-su)f(u) from 0 to T will become integral of (e^-st)f(t) from 0 to T suddenly ? why not integral of (e^-s (t-nT) )f(t-nT) from 0 to T ? as we can see, u = t +nTd given/known dataHomework EquationsThe Attempt at a Solution
  4. J

    Laplace Transform unit step function

    Homework Statement g(t) = (t+1)us(t) - (t-1)us(t-1) - 2us(t-1) - (t - 2)us(t-2) + (t-3)us)(t - 3) + us(t-3) Homework Equations unit step function us(t-3) is same as u3 (t) Shift in time: L[f(t - T)us(t-T)] = e-TsF(s) us(t) ↔ 1/s t ⇔ 1/s2 The Attempt at a Solution 1/s2 + 1/s - e-s/s2 + e-s/s -...
  5. A

    How Do You Apply Laplace Transforms to Vehicle Suspension Analysis?

    Homework Statement Diagram for a vehicle suspension is given. Displacement of wheel is given by 'x' and and displacement of body is 'y'. Spring constant, k = (7*10^4) Nm Damping coefficient, c = (3*10^3) N/m/s mass,m = 250kg a) Make a Laplace Transform of system and utilize it to predict 'y'...
  6. j3dwards

    Explicit check for Laplace transform?

    Homework Statement Solve the following initial value problem using Laplace transforms: y' + 4y = 3t3 e−4t ; y(0) = 0 . Useful information: Recall that the Laplace transform of y 0 is pY − y(0), where Y is the Laplace Transform of y. The Laplace transform of tk e−at is k!/(p + a)k+1 . Confirm...
  7. R

    Solving ODE by Laplace Transform: Where Did I Go Wrong?

    Homework Statement Use Laplace transform to solve the following ODE Homework Equations xy'' + y' + 4xy = 0, y(0) = 3, y'(0) = 0 The Attempt at a Solution L(xy'') = -\frac{dL(y'')}{ds} L(4xy) = -\frac{4dL(y)}{ds} L(y'') = s²L(y) - sy(0) - y'(0) = s²L(y) -3s L(y') = sL(y) - sy(0) - y(0) =...
  8. ELiT.Maxwell

    Point charge inside a electrically neutral cavity in conduct

    bear with me, i know that this question has been asked many time , but i would like a definite answer, now, starting off the external charge density on the outer surface of sphere WILL be uniform by unique solution of Laplace equation and letting the sphere be huge, so, electric field due to...
  9. L

    MHB Rieman inversion formula in Laplace transform

    Hello everybody! I'm sorry if it's not the right section to post in. I'm trying to solve this exercise: \frac{1}{2i\pi}*\int_{8-i\infty}^{8+i\infty}\frac{e^{s(t-5)}}{(s+4)^2}ds The request is to find the result in function of t I know i must use the Riemann inversion formula, and so the request...
  10. J

    How Does Laplace Transform Solve the Harmonic Motion Differential Equation?

    (2.) Let the differential equation ¨x + 2 ˙x + 2x = 6 sin(t)U(t − 3π/2) , x(0) = 2, x˙(0) = 2 Solve for the position function x(t) using the Laplace transform:
  11. I

    MHB Laplace Transform: Basic Concepts & Applications

    Did I do this right? View image: 20151201 143310 1
  12. I

    MHB How do I find the Laplace Transform for U3(t)(t-3)^5/2?

    I'm working on a few problems to find Laplace transforms and I got stuck on this one. ${U}_{3}(t){(t-3)}^{5/2}$ It looks different from the other I've been doing so I don't really know how to get started
  13. Crush1986

    Can Laplace Transforms Reveal Bessel Functions as Solutions?

    Homework Statement xy''+y'+xy=0, y'(0)=0, y(0)=0 Using the method of Laplace transforms, show that the solution is the Bessel function of order zero. Homework Equations -(d/ds)L{f(x)} The Attempt at a Solution The only thing I got out of this when trying to solve it was y=0. Obviously not...
  14. Julio1

    MHB Laplace equation and Median Value Property

    Suppose that $u$ is the solution of the Laplace equation $u_{xx}+u_{yy}=0$ in $\{(x,y)\in \mathbb{R}^2: x^2+y^2<1\}$ $u(x,y)=x$ for all $(x,y)\in \mathbb{R}^2$ such that $x^2+y^2=1.$ Find the value of $u$ in $(0,0).$ Use the property of median value.
  15. Mark Brewer

    How can partial fractions be used to solve inverse Laplace transforms?

    Homework Statement L-1{(2s2+3)/(s2+3s-4)2} The Attempt at a Solution I factored the denominator f(t)=(2s2+3)/((s-1)(s+4))2 now I've tried partial fractions to get (2s2+3)/((s-1)(s+4))2 = A/(s-1)2 + B(s+4)2 (2s2+3)=A(s+4)2 + B(s-1)2 by substitution, s=1 and s=-4 5=A(25) A=1/5 35=B(25)...
  16. jdawg

    Understanding the Inverse Laplace Transform: Solving for 3/s√(π)

    Homework Statement L-1{3/s1/2} Homework EquationsThe Attempt at a Solution 3L-1{1/s1/2} 3L-1{(1/sqrt(π))(sqrt(π)/(sqrt(s))} 3/(sqrt(π))L-1{(sqrt(π))/(sqrt(s))} 3/(sqrt(π))(1/(sqrt(t)) This is what I got from the solution for this problem. What tipped them off to multiply by sqrt(π)? And...
  17. Julio1

    MHB Proving Laplace Equation in $\Omega_{(a,b)}$

    Let $\Omega$ an open domain in $\mathbb{R}^2.$ Suppose that $u$ is an application $C^2$ that satisfy the Laplace equation $\Delta u=0$ in $\Omega.$ Let $\Omega_{(a,b)}=\{(x+a,y+b): (x,y)\in \Omega\}$ and define $v(X,Y)=u(X-a,Y-b)$ for all $(X,Y)\in \Omega_{(a,b)}.$ Show that $v$ is an...
  18. H

    How to solve this problem using laplace transform?

    Homework Statement The differential equation given: y''-y'-2y=4t2 Homework EquationsThe Attempt at a Solution I used the laplace transform table to construct this equation,and then I did partial fraction for finding the inverse laplace transform.But I'm now stuck at finding the inverse laplace...
  19. M

    Inverse Laplace : with Convolution

    Homework Statement Use the convolution property to obtain the inverse Laplace transform of F(s)= e-3s * ((3s+15)/s2+25) Homework EquationsThe Attempt at a Solution = (3*(s/s2+52) + 15*(1/s2+52)) *e-3s Using table of Laplace: 3*(s/s2+52) = 3*cos(5*t) = T7 15*(1/s2+52) = 15/5*sin(5*t) =T18 e-3s...
  20. J

    What would the laplace inverse of a laplace be?

    For example: If F(s) = L{t3e-16tcos(3t)sin2(t)} What would L-1{F(s)} be?
  21. J

    How would I take the laplace transform of f(t)= te^tsin^2(t)?

    How would I take the laplace transform of f(t)= te^tsin^2(t)?
  22. D

    Laplace tranform of unitstep (heaviside) function

    Homework Statement [/B] I know for t[u(t)-u(t-2)], we can simplify that to tu(t)-((t-2)+2)u(t-2) which then gives us tu(t)-(t-2)u(t-2)-u(t-2). Now, the laplace transform seems trivial but I am having problems with this equation: sin(t)[u(t)-u(t-pi)] Homework...
  23. M

    A Laplace equation on a trapezoid

    Hello everybody! I know how to solve Laplace equation on a square or a rectangle. Is there any easy way to find an analytical solution of Laplace equation on a trapezoid (see picture). Thank you.
  24. B

    Identifying Reverse Laplace Transforms

    Homework Statement Hello, I have just started studying Laplace transformations and I am struggling to identify reverse Laplace transforms. I understand how to perform the transform, but going the other way is really confusing me. i.e, given ##F(P)## find ##f(t)##. If I have that ##F(P) =...
  25. F

    Laplace equation in spherical coordinates

    Homework Statement Solve the Laplace equation inside a sphere, with the boundary condition: \begin{equation} u(3,\theta,\phi) = \sin(\theta) \cos(\theta)^2 \sin(\phi) \end{equation} Homework Equations \begin{equation} \sum^{\infty}_{l=0} \sum^{m}_{m=0} (A_lr^l + B_lr^{-l -1})P_l^m(\cos...
  26. Linder88

    How Do You Apply Laplace Transform to an ODE with a Derivative of Time?

    The ordinary differential equation, with initial values,shall be solved using Laplace transform. The ODE looks like this \begin{equation} y''(t')+2y''(t)-2y(t)=0 \end{equation} And the initial conditions are \begin{equation} y(0)=y'(0)=0, y''(0)=0 \end{equation} The problem is with the first...
  27. Aristotle

    Laplace Transform Method for Solving Initial Value Problems

    Homework Statement Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teacher is against using it..) y'' + 2y' + 2y = 2 ; y(0)= y'(0) = 0 Homework Equations Lf'' = ((s^2)*F) - s*f(0) - f'(0) Lf' = sF - f(0) Lf = F(s) The...
  28. Aristotle

    Initial Value Problem using Laplace Transform help?

    Homework Statement Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teach is against using it..) y'' - 4y' + 3y = 0 ; y(0)=2 y'(0) = 8 Homework Equations Lf'' = ((s^2)*F) - s*f(0) - f'(0) Lf' = sF -...
  29. J

    Is there a mistake in my calculation or in my reasoning?

    Homework Statement y'' + 3y' + 2y = r(t), r(t) = u(t - 1) - u(t - 2), y(0) = y'(0) = 0. I need to solve this by convolution, which I know is commutative. The problem is that my calculation gives (f * g) =/= (g * f). Could someone please tell me where my mistake is? Homework Equations...
  30. O

    Quick laplace transform question

    Homework Statement So I know 1/(s-a)=e^(a1), but why is say, 2/((s+4)^2) equal to 2xe^-4x? Do I just simply add an X if the numeration is a constant other than 1?
  31. I

    Solving ODE with Laplace transform

    Homework Statement Let ##f(t)= \begin{cases} \sin t , \; \; 0 \le t < \pi \\ 0 , \; \; \; \; \; \text{else.} \end{cases}## Use Laplace transform to solve the initial value problem ##x'(t)+x(t)=f(t), \; \; \; x(0)=0.## Homework Equations Some useful Laplace transforms...
  32. SarahAlbert

    Laplace in Spherical and Cylindrical Coordinates

    Homework Statement I'm suppose to verify the given Laplace in (a) Cartesian (b) Sperical and (c) Cylindrical coordinates. (a) was easy enough but I need to know if I'm doing (b) and (c) correctly. I don't need a solution, I simply need to know if the my Spherical formula is correct, my...
  33. Italo Campoli

    Inverse Laplace (stuck @ Partial Fraction)

    Homework Statement Find the Inverse laplace transform of: http://www4c.wolframalpha.com/Calculate/MSP/MSP14541hg721e74730d4fb00004644i96f59549h1d?MSPStoreType=image/gif&s=30&w=201.&h=40. Result...
  34. C

    Laplace Transforms: Transfer Functions and Impulse

    Homework Statement I uploaded the question as a picture and attached it. Homework Equations Unit step function - u_c (t) = \begin{cases} 1 & \text{if } t \geq c \\ 0 & \text{if } t < c \end{cases} Impulse function - δ(t) = \displaystyle\lim_{Δ\rightarrow 0} δ_Δ (t) Multiplication Property...
  35. J

    Keep getting the wrong answer in this lengthy Laplace problem

    Homework Statement y'' + 4y = 8t^2 if 0 < t < 5, and 0 if t > 5; y(1) = 1 + cos(2), y'(1) = 4 - 2sin(2). Use the Laplace transform to find y. Homework Equations t-shift, s-shift, unit step function. The Attempt at a Solution I have been trying to solve it for hours, but keep getting the wrong...
  36. C

    Laplace Transforms: Transfer Functions, and IVT/FVT Problems

    Homework Statement I uploaded the problem statements as a picture as well. I have completed these and was wondering if someone could check my work, and let me know if it is correct. Problem 1.3: Find the expression for the transfer function of this linear time-invariant causal system with...
  37. C

    Laplace Transforms Involving: Unit-Step, and Ramp Functions

    Homework Statement Here is an imgur link to my assignment: http://imgur.com/N0l2Buk I also uploaded it as a picture and attached it to this post. Homework Equations u_c (t) = \begin{cases} 1 & \text{if } t \geq c \\ 0 & \text{if } t < c \end{cases} The Attempt at a Solution Question 1.1 -...
  38. Italo Campoli

    Inverse Laplace Transform of a fractional F(s)

    Homework Statement [/B] Having a little trouble solving this fractional inverse Laplace were the den. is a irreducible repeated factor 2. The attempt at a solution tryed at first with partial fractions but that didnt got me anywhere, i know i could use tables at the 2nd fraction i got as...
  39. Italo Campoli

    Prove a Laplace Transform Equality

    Homework Statement Prove that http://www4f.wolframalpha.com/Calculate/MSP/MSP26931c1531g07285beh7000062h7f6g1ggd95eea?MSPStoreType=image/gif&s=5&w=98.&h=38. =http://www4f.wolframalpha.com/Calculate/MSP/MSP6901c153574d0bdbh20000048829f0g4d1fi1d0?MSPStoreType=image/gif&s=5&w=69.&h=35. The...
  40. khurram usman

    A discussion about Fourier and Laplace transforms and calculus

    i have read many of the answers and explanations about the similarities and differences between laplace and Fourier transform. Laplace can be used to analyze unstable systems. Fourier is a subset of laplace. Some signals have Fourier but laplace is not defined , for instance cosine or sine...
  41. C

    Can You Help With Finite Element Analysis in Cylindrical Coordinates?

    I am trying to numerically calculate the electric potential inside a truncated cone using the finite element method (FEM). The cone is embedded in cylindrical coordinates (r,phi,z). I am assuming phi-independence on the potential, therefore the problem is essentially 2D; I am working only with...
  42. S

    Second shifting theorem of Laplace transform

    Can anybody direct me towards a proof of the second shifting theorem for Laplace transforms? I'm understanding how to use it but I can't figure out where it comes from. I've been learning from Boas, which doesn't offer much in way of proof for this theorem. Are there any good resources online...
  43. Destroxia

    Step Function Laplace w/ piecewise

    Homework Statement ##y'' + 2y' + 2y = h(t); y(0)=0, y'(0)=1, h(t) = \begin{cases} t &, \pi \leq t < 2\pi \\ 0 &, 3\pi \leq t < \infty \\ \end{cases}## 2. Homework Equations The Attempt at a Solution Take the Laplace of both sides:[/B] ##\mathcal{L}(y'' + 2y' + 2y) = \mathcal{L}(h(t))...
  44. Destroxia

    Step Function IVP Differential Equation w/ Laplace Transform

    Homework Statement (didn't know how to make piecewise function so I took screenshot) Homework EquationsThe Attempt at a Solution My issue here with this problem is that I have absolutely no idea where to start... I have read through the textbook numerous times, and searched all over the...
  45. H

    Need explanation on Laplace Transform and Fourier Transform

    Hello guys. I need an easy explanation regarding Laplace Transform and Fourier Transform. I know it is quite a mathematics question but I need an explanation in which it has something to do with engineering. I already search a bit about them but still cannot find and explanation that easy enough...
  46. kinbeat

    Comp Sci How to Implement Laplace Expansion in Fortran for Matrix Determinants?

    Hi, I'm dealing with an exam problem (which i have 8 days to solve, with no outside help limitations), which is to write a fortran95 program that solves linear systems. the first part asks to find the determinant of a NxN matrix with the laplace expansion, implementing it as a recursive...
  47. A

    Can the Laplace equation be used to solve for y⁴?

    hi , i was asked to solved this but i have no idea how to laplace y^4, can anyone please help? its the question in the middle
  48. J

    Simple inverse Laplace using PFE not so simple?

    Hello, When evaluating the step response of a circuit, the resulting Laplace representation is: $\frac{I_{pd}}{s^2 C1 R1}$ If I look this up on a table of Laplace Transforms, this results in $\frac{I_{pd}*t}{C1 R1}$. However, I'm struggling to solve this via partial fraction expansion--is...
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