What is Laplace transform: Definition and 776 Discussions

In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable



t


{\displaystyle t}
(often time) to a function of a complex variable



s


{\displaystyle s}
(complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication.For suitable functions f, the Laplace transform is the integral






L


{
f
}
(
s
)
=

∫

0


∞


f
(
t
)

e

−
s
t



d
t
.


{\displaystyle {\mathcal {L}}\{f\}(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}

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  1. T

    Laplace Transform - Step Function

    Homework Statement What is the Laplace transform of f(t) = t^2 - 18 for 0 < t < 3 and f(t) = (t-3)^2 for t>3?Homework Equations Laplace Transforms 3. Work Using Heaviside/step function, made equation into: f(t) = t^2 - 18 + u(t-3)( (t-3)^2 - (t^2-18) ) Then, using Laplace transforms...
  2. C

    Using fourier and laplace transform to solve PDE

    PDE is type of heat equation. Many book only gives an example of solving heat equation using Fourier transform. An exercise asks me to solve it for using Fourier and laplace transform: In the heat equation, we'd take the Fourier transform with respect to x for each term in the equation...
  3. V

    Finding an inverse Laplace Transform for a function - solving IVPs with Laplace

    Homework Statement Use Laplace Transforms to solve the following initial value problems a. t\,y''\, - \,t\,y'\, + \,y\; = \;2\;\;\;y(0)\;=\;2\;\;\;y'(0)\;=\;-1 b. y''\,+\,2\,y'\,-3\,y\;=\;\delta(t\,-\,1)\,-\,\delta(t\,-\,2)\;\;\;y(0)\;=\;2\;\;\;y'(0)\;=\;-2 Homework Equations Laplace...
  4. I

    Inverse Laplace transform Help

    I couldn't figure the sol'n for this problem Could somebody help? Thanks a lot Find the following Inverse Laplace transform (L^(-1))*[1/(4s+1)]:frown:
  5. G01

    Laplace Transform of a periodic function

    Homework Statement Suppose f(t) is a periodic function with period T. Show that the Laplace transform of f is: L(f) = \frac{1}{1-e^{-sT}}\int_0^T f(t)e^{-st} dtThe Attempt at a Solution I started with the definition of a Laplace Transform for f: L(f) = \int_0^\infty f(t)e^{-st}dt Using the...
  6. G01

    Solving Diff EQ using a Laplace Transform

    Homework Statement Solve the initial value problem: \frac{dy}{dt} + 2y = u_2(t)e^{-t} y(0) = 3 Where u_2(t) is a Heaviside Function with the discontinuity at t=2. Homework Equations The Laplace transform of a Heaviside function multiplied by another function: L( u_a(t)f((t-a) ) =...
  7. N

    Solving Differential Equation w/ Laplace Transform

    Hi all, Just a small question: I came across a problem to solve a differential equation using Laplace transform. I solved the major part but only still a part where I had to inverse transform the following experssion: (exp(-s))/(s^2) I looked in Laplace Transform Tables but did not...
  8. N

    How Accurate Are These Inverse Laplace Transforms?

    Gents, I have this problem: find the inverse laplace transfor for Y1(s) = exp(-s)/s^2 Y2(s) = {1/[4*(s+1)]}*exp(-2*s) my solution is: using the 2nd shifting theroem y1(t) = (t-1) H(t-1) y2(t) = (1/4)*exp(2-t)*H(t-2) Is my solution correct?
  9. R

    Need help on laplace transform and PDE

    Need urgent help on laplace transform and PDE ! I'm stuck with this 2 questions ... q1) Using laplace transforms, solve: y" + 4y = r(t), where r(t) = {3sint, 0<t<pi, -3sint, t>pi y(0)=0, y'(0)=3. this is what i get after rewriting for the step function: 3sint [1-u(t-pi)] + (-3sint)u(t-pi)...
  10. O

    Trouble Finding an Inverse LaPlace Transform

    I am trying to find the inverse LaPlace transform of F(s)=\frac{2s+12}{s^2+6s+2\sqrt{2}} Using partial fraction decomposition on the above rational expression seems tedious. Is there any other method? Thanks
  11. S

    Using convolution for Laplace transform

    Homework Statement Using convolution theorem for Laplace theorem,, show that Homework Equations inverse Laplace transform (1/(S^3/2*(s-1)) = (2*e^t)/Pi^1/2 intregral (from 0 to t) e^-x*x^1/2dx. The Attempt at a Solution The inverse Laplace above is a product of 1/s^3/2 and...
  12. S

    Proving that the Laplace Transform of J0(t) is 1/(s + 1)^1/2

    [b]1. The zero order bessel function J0(0+) = 1, J'0(0+) = 0. and J0(t) satisfies teh differential equation [b]2. ty''(t) + y'(t) + ty(t) = 0, t>0 Prove thata the Laplace transform of J0(t) is 1/(s + 1)^1/2 The Attempt at a Solution Could anyone help on this. How to I go about with...
  13. C

    Solve Laplace Transform: t^m = m!/s^m+1

    I am trying to show that L[t^m] = m!/s^m+1, unfortunately I can not understand why integral from 0 to inf of (t^m)(e^-st)dt = (-d/ds)^m. integral 0 to inf of e^-stdt... ?
  14. C

    Laplace Transform: Integral of x from 0 to ∞

    \int_{0}^{\infty} x dx = \left[ \frac{1}{2}x^2 \right]_{0}^{1} = \frac{1}{2}
  15. S

    How does the inverse Laplace transform work in signal processing?

    My book on signal processing says that: f(t) = \frac{1}{2\pi j} \int_{c-j\infty}^{c+j\infty} F(s) e^{st} ds = \lim_{\Delta s \to 0} \sum_{n = -\infty}^{\infty} \Big[ \frac{F(n\Delta s)\Delta s}{2\pi j} \Big] e^{n\Delta s t} I don't get this. How/Why can you write a integration over a...
  16. P

    Physical meaning of Laplace transform

    Hi every body, I am not so sure this kind of question is suitable here. In styding maths, I have some difficulties with the Laplace transform. The formulas are OK, I can apply them, but I still do not understand the physics meaning behind that. Can anyone can explain to me a little bit more...
  17. C

    How Do I Prove the Laplace Transform Formula for Functions with Finite Jumps?

    how do i go about solving the laplace transform of sint * cost ? i know the answer becomes 1/(s^2 + 4) but what is the method? thanks.
  18. S

    Laplace transform w/ partial fraction

    Homework Statement solve the following initial value problem using Laplace transforms y"+2y'+y=(8/3)cos(2t)-2sin(2t) y(0)=1 y'(0)=7/3 Homework Equations L[d^2y/dt^2]=s^2Y-sy(0)-y'(0) L[dy/dt]=sY-y(0) L[coswt]=s/(s^2+w^2) L[sinwt]=w/(s^2+w^2) The Attempt at a Solution so...
  19. E

    Inverse Laplace Transform homework

    Homework Statement 1/s^5 Homework Equations I was thinking I need to use n!/s^n+1 The Attempt at a Solution But n would then be 4, so n! will be 24. Not sure what to do?
  20. A

    Laplace transform for solving ODE with variable coefficients

    Can we use laplace transform to solve an ODE with variable coefficients? Like this one: 4x y" + 2 y' + y = exp (-x)
  21. A

    Laplace transform of exp (-t^2)

    Hi everyone, I need to solve this ODE using the laplace transform: y" + y - exp (-t^2) = 0 , y(0) = y'(0) = 0 My question is on how to find the laplace transform of exp (-t^2). I used different properties of laplace transform to solve it but I was not succesfull. I was thinking that I...
  22. H

    Verifying Laplace Transform Solution for x' + 2y' + x = 0, x' - y' + y = 1

    Can someone tell me if I did this right because my solution seems wrong, but I've done it a couple times and get the same answer. I'm given the following: x' + 2y' + x = 0 x' - y' + y = 1 and the initial values of x(0) = 0 and y(0) = 1 The idea is to solve this initial value problem. Can...
  23. Reshma

    Proof for this Laplace transform

    Can someone help me out with the proof of the Laplace transform of the function tn? I did have a go at this one. L[t^n] = \int_0^{\infty} t^n e^{-st}dt = t^n\frac{-e^{-st}}{s}\vert_0^{\infty} + {1\over s}\int_0^{\infty} nt^{n-1}e^{-st}dt ={n\over s}\int_0^{\infty} t^{n-1}e^{-st}dt ={n\over...
  24. C

    Inverse laplace transform question

    How do i find the inverse laplace transform of s/(s-2)^2
  25. C

    Laplace transform please check my answer

    I have to find the laplace transform of (t+2)sinh2t my answer is (4s/(s^2 + 4)^2 ) + (4/(s^2 - 4)) is this simplified as much as possible...?
  26. C

    Help with Laplace Transform of (t+2)sinh2t

    My understanding of the laplace trasnform isn't so great so i would appreciate some help with this question please: find the laplace transform of (t+2)sinh2t now i know the laplace transform of sinh2t is 2/(s^2 -4) as this is a standard rule... looking through textbooks they show the...
  27. R

    Using Laplace transform to solve ODE

    Here's the problem: Solve ty'' + 2y' + ty = 0 with lim_{t\rightarrow0+} y = 0 and y(\pi) = 0 So here's my working: -\frac{d}{ds}[s^2Y - sy(0) - y'(0)] + 2[sY - y(0)] - Y' = 0 -[2sY + s^2Y' - s] + 2[sY - y(0)] - Y' = 0 Arranging terms, I get: Y' = \frac{s-2y(0)}{s^2+1} and -Y' =...
  28. B

    How Can You Find the Inverse Laplace Transform of \(\frac{s}{{(s + 4)^4}}\)?

    Hi I'm having problems find the inverse Laplace transform of \frac{s}{{\left( {s + 4} \right)^4 }} via a table/look up method. In another question part I found the inverse Laplace transform of (s+4)^-4 by considering \frac{{d^3 }}{{db^3 }}\left[ {\left( {s + b} \right)^{ - 1} } \right] so...
  29. G

    [help] A laplace transform definition problem

    http://xs104.xs.to/xs104/06322/ssss.gif Hello all ^^ What is the difference between the two definitions thanks a lot
  30. N

    Laplace Transform (correct, or not?)

    I wish someone would help me to verify my answer and working. I don't have the answer for the question. :frown: http://www.mrnerdy.com/forum_img/laplace.jpg Is there a better working method?
  31. D

    Laplace Transform With Initial Values

    Okay, I know this is alot... but I am stuck, so here goes... Use the method of Laplace transform to solve the initial value problem y''+3ty'-6y=0, y(0) = 1, y'(0) = 0 L\{y'' + 3ty' - 6y\} = L\{0\} s^{2}Y(s) - sy(0) - y'(0) + 3L\{ty'\} - 6Y(s) = 0 s^{2}Y(s) - s(1) - 0 -...
  32. B

    Piece wise function and Laplace Transform

    Hi, I need to find the piece wise function whose the the Laplace Transform is: 2/s+e-3x(1/s2)+4e-3x(1/s) I found for f as function of the unit step function u: f(x)=2+x*u(x-3)+u(x-3) Now I have difficulty to put the function in piece wise form like: f(x)= 2 for 0<x<2 and f(x)=...
  33. S

    Integration by parts (Laplace transform)

    First off, I hope these images show up - I don't have time to figure out this latex stuff atm, so it's easier just to throw the formulae together in openoffice. I'm working on the Laplace Transform for http://home.directus.net/jrc748/f.gif Which is obviously...
  34. S

    Laplace Transform of cos(at) * cos(bt)

    I have to find the laplace transform of cos(at) * cos(bt) and express it as a ratio of two polynomials. I converted both of the cosines into exponentials, and took the laplace transform of those. I think I'm getting confused on the complex side of things. I get a few things like...
  35. S

    How Can You Prove the Laplace Transform of t^n?

    Our professor asked us to derive an expression for the laplace transfrom of t^n. I did a few examples in MatLab and gathered that the Laplace Transform of t^n = n!/s^(n+1). I'm pretty sure this is correct, but I don't think my professor will be happy with it. I don't really know how I should...
  36. J

    How can I find the Laplace transform of erf without using tables?

    how do i find the laplace transform of the following error function without using tables? f(t)=erf(t^(1/2)) i've been trying really long but i seem to be stuck in a loop of erf
  37. G

    Understanding Laplace Transform of f(t)

    i have f(t) defined piece-wise and continous.. f(t) = 0, t < 2pi t-pi , pi <=t<2pi 0 , t >=2pi i have so far g(t)=U_pi*f(t-pi)-U_2pi*f(t-pi) if i do the laplace, i get e^-pis/s^2-e^-2pis/s^2 in the book, they have e^-pi*s/s^2 -e^-2pi*s/s^2 (1+pi*s) I am not...
  38. A

    Can Laplace Transform Converge for Periodic Functions on [0,infty)?

    I need to show that for f(t)=f(t+T) on [0,infty), that the Laplace Transform is: \mathcal{L}\left[f(t)\right]=\frac{\int_0^Te^{-st}f(t)\,dt}{1-e^{-sT}}. The first thing I did was to write the transform as...
  39. S

    Laplace Transform and complex contours

    In the HW section, the following equation was proposed: ty(t)=\int_0^t \tau^{1-\alpha}y(t-\tau)d\tau;\quad \int_0^\infty y(x)dx=1 Link to HW thread: https://www.physicsforums.com/showthread.php?t=97562 Using Laplace Transforms: Y(p)=\mathcal{L}\left\{y(t)\right\} and noting the...
  40. H

    Restrictions on Alpha for Validity of Laplace Transform Solution

    I have the problem, ty(t)= \int_{0}^{t}\tau^{\alpha-1}y(t-\tau)d\tau subject to the constraint that \int_{0}^{\infty}y(t)dt=1. In need to get the answer in the form of, Y(p)=something (where Y(p) is the Laplace transform of y(t)). I can see that the right hand side is...
  41. L

    Question about the maclaurin serie and laplace transform

    Question about the maclaurin serie and the laplace transform. For maclaurin serie i wonder, the function used for teh maclaurin development must be derivativable to infinity? What is the difference between the fouri transform and the laplace transform? As i understood it, it's just the...
  42. M

    Find the laplace transform of log[x]

    how do i find the laplace transform of log[x]
  43. L

    Neeed help with a laplace transform

    What is the laplace transform of http://img150.imageshack.us/img150/8145/laplacetransform6wk.jpg
  44. S

    Laplace transform, fraction expansion

    I have y"+y=t , y(0)=1, y'=0 After Laplace transformation a got: (S^3+1)/(S^2(S^2+1)) After I made a partial fraction expansion (S^3+1)/(S^2(S^2+1))=a1/S^2+a2/S+a3/(S^2+1) (1) It comes to a system where a2=1 a1+a3=0 (2) a2=0 a1=1 Here I am getting confused...
  45. F

    Laplace Transform - (stinking partial fractions)

    Ok... I'm working on this laplace transform, and I'm getting stuck on the partial fractions part on this one problem. If someone could help me out with setting it up, I would be very appreciative. \frac{s}{(s^2+4)(s^2+\omega^2 ) } After trying to set it up, I get something like...
  46. M

    How would I find the inverse laplace transform

    Hi How would I find the inverse laplace transform of this? I(s) = \left( \frac{1}{s(1+e^{-s})}\right) \left( \frac{1}{Ls+R}\right) i(t)=? L, R are constants. I recognize the first term to be a geometric progression (square-wave function). With an infinite number of terms in that...
  47. F

    Need help simplifying equation to find the laplace transform of it

    there are two loops in an electrical circuit. I've got two equations and two unknowns. here are the equations v(t) = I1*R1 + 1/c*int(I1*dt) - 1/c*int(I2*dt) - first loop 0 = LdI2/dt + R2*I2 - 1/c*int(I1*dt) + 1/c*int(I2*dt) - second loop the capacitor is in the branch between the two...
  48. M

    Where did I go wrong? [Laplace Transform]

    L is Laplace Transform operator. Question is: Let f(t) = t^2. Derive L[f] from L[1] So I know f(1) =1 and L[1] = \frac{1}{s} Carrying out the Transform... L[f] = \int_{0}^{\infty} e^{-st}t^2 dt Integration by parts u = t^2, dv = e^{-st} dt I do the integration, the uv...
  49. E

    Perhaps delta function or inverse Laplace transform?

    Hello everyone, i have this question and not even sure how to approach it: \frac {di}{dt}+4i+3\int_{0^-}^t{i(z)dz = 12(t-1)u(t-1) and i(0^-) = 0 find i(t) last topics we covered were laplace transforms (and inverse) and dirac delta function. At least some hint to get me started...
  50. E

    Laplace transform with abs value in exponential

    Hi everyone, I have this problem and just need someone to check or correct: f(t) = -4e^{-3|t|}(u(t + 3) - u(t-1)) where u(t) is step function: u(t) = 1 for t >= 1 and 0 elsewhere; so, i guess I need to break abs value into 2 cases and have 2 different equations? anyway, here's what I have...
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