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. D

    MHB Laplace transform of a function II

    $\mathscr{L}\{\sin^{2}4t\}$ $\mathscr{L}\{\sin(3t-\frac{1}{2}\}$for the 2nd prob here's what I have tried $\mathscr{L}\{\sin(3t)\cos(0.5)-\cos(3t)\sin(0.5)}$ $\cos(0.5)\mathscr{L}\{\sin(3t)\}-\sin(0.5) \mathscr{L}\{\cos(3t)\}$ $\frac{3\cos(0.5)-s\sin(0.5)}{s^2+9}$ ---> is this correct? for...
  2. D

    MHB Laplace transform of a function

    please help me solve this problem $\mathscr{L}\{e^{3a-2bt}\}$ here's my attempt $\mathscr{L}\{e^{3a}\cdot e^{-2bt}\}$ from here I couldn't continue I looked up my table of transform but nothing matches the problem above. I'm not sure if the first shift formula would work here. please help...
  3. R

    Power series and Laplace transform

    As it can be read here, http://en.wikipedia.org/wiki/Laplace_transform#Relation_to_power_series the Laplace transform is a continuous analog of a power series in which the discrete parameter n is replaced by the continuous parameter t, and x is replaced by exp(-s). Therefore, computing a...
  4. P

    Laplace transform with Heaviside function

    Hello, I am searching for the Laplace transform of this function u_a(y)\frac{\partial c(t)}{\partial t} where u_a(y) is the Heaviside step function (a>0). Can anyone help me? Thanks in advance! Paolo
  5. A

    Laplace Transform. Is this correct?

    Homework Statement ƒ(t) = 1.15t Homework Equations Standard Laplace Transform Table The Attempt at a Solution L(f(t)) = e(t * ln(1.15) = 1 / (s - ln(1.15))
  6. J

    MHB Laplace Transform, Finding solution: y′′+4y′+4y=f(t)

    y′′+4y′+4y=f(t) where f(t)=cos(ωt) if 0<t<π and f(t)=0 if t>π? The initial conditions are y(0) = 0 , y'(0) = 1 I know that f(t)=cos(ωt)−uπ(t)cos(ωt), the heaviside equation. AND ω is allowed to vary, supposed to find the general solution, i.e. f(t) in terms of ω I think that after...
  7. J

    Laplace Transform to Find Solution

    Use Laplace transfer to find the solution of the following initial value problem: y''+4y'+4y=f(t) where f(t) = cos(ωt) if 0<t<π and f(t)=0 if t>π ? Also, y(0) = 0, y'(0) = 1 Currently, I have gotten to here, but not sure how to perform inverse Laplace: (s+2)² * F(s) − 1 = [s/(s²+w²)]...
  8. M

    Inverse Laplace Transform and Court

    Homework Statement I had a question in my midterm, it was to find inverse laplace tansform of: (4s+5) / (s^2 + 5s + 18.5) Where ^ denotes power. Homework Equations The Attempt at a Solution My answer was to find the complex roots of equation (s^2 + 5s + 18.5) , by them...
  9. A

    Laplace transform for set of differential equations

    I have a set of differential equations with the basic form: dy_n/dt = t*(a_(n-1)*y_(n-1)+b(n+1)*y_(n+1)-2c_n*y_n) So the time depence is a simple factor in front of the coefficient matrix. Does this set of differential equations have closed form solutions?
  10. K

    Laplace Transform solution for 2nd order differential equation

    Homework Statement d^2x/dt^2 - 3 dx/dt + 2x= 2e^3t give that at t=0, x=5, and dx/dt=7 Homework Equations i can't figure out how to derive the values of A, B, and C from the attempted equation solution. please help me out here. thanks The Attempt at a Solution
  11. M

    Solving a Laplace Transform Problem: Where Am I Going Wrong?

    I'm out of college and am brushing up on Laplace Transforms. I have a problem I've solved, but I believe the solution I got is wrong and can't find my error. The problem is 2x''-x'=t*sin(t) x(0)=5,x'(0)=3 My solution... Take the Laplace Transform 2(s^2x-5s-3)-(sx-5)=2s/(s^2+1)^2...
  12. M

    What Is the Intuitive Explanation of the Laplace Transform?

    hello pf! i am wondering if anyone here knows of a geometric, intuitive explanation for the laplace transform? if so, please direct me to the source of if you could, explain to me your understanding? thanks!
  13. P

    MHB Douglas' question via email about Inverse Laplace Transform

    To start with, let's work on the Partial Fraction decomposition. $\displaystyle \begin{align*} \frac{A\,s + B}{s^2 - 16s +128} + \frac{C\,s + D}{s^2 + 16s + 128} &\equiv \frac{3s}{s^4 + 16\,384} \\ \frac{ \left( A\,s + B \right) \left( s^2 + 16s + 128 \right) + \left( C\,s + D \right) \left(...
  14. D

    MHB Solving Laplace Transform: \[(s+1)^2/(s^2-s+1)\]

    \[ \frac{(s+1)^2}{s^2 - s + 1} \] I have simplified it down to \[ \frac{s - \frac{1}{2} + s^2 + s + \frac{3}{2}}{(s - 1/2)^2 + \frac{3}{4}} = e^{1/2t}\cos\Big(t\frac{\sqrt{3}}{2}\Big) + \sqrt{3}e^{1/2t}\sin\Big(t\frac{\sqrt{3}}{2}\Big) + \frac{s^2 + s}{(s - 1/2)^2 + \frac{3}{4}} \] but I can't...
  15. D

    MHB Inverse Laplace transform question

    With a Laplace transform, we can remember common set ups; for example, \[ \mathcal{L}\{e^{-at}\} = \frac{1}{s + a}. \] When it comes to the inverse Laplace transform, I can only find the tables to remember in a book. However, if we go back to the Laplace transform, we can always do \[...
  16. A

    Laplace transform initial value problem-

    Laplace transform initial value problem--need help! Looking at the solutions to these initial value problems, I am very confused as to how the highlighted steps are derived (both use heaviside step functions). I know the goal is to get the fractions in a familiar form so that one can look them...
  17. polygamma

    MHB Evaluating an Integral: Laplace Transform Method

    The typical way to evaluate $ \displaystyle \int_{0}^{\infty} \frac{\cos mx}{a^{2}+x^{2}} \ dx$ is by contour integration. In a recent thread I evaluated that integral using the Laplace transform. http://mathhelpboards.com/analysis-50/advanced-integration-problem-9129.html#post42551My...
  18. T

    How Does the Hankel Transform Solve Problems in Cylindrical Domains?

    Hello, As of recently, I've been working with Laplace transforms and have a question about their relationship to solving differential equations. I know the definition of the laplace transform and I know that a function is essentially being transformed from the time domain to complex...
  19. T

    Arranging this expression into a laplace transform

    Hi Guys, I have an expression that i am struggling to manipulate into a laplace transform. This expression should fit one or a combination of the common transform pairs. I believe the transform the expression should be fitting is either a unit step 1/s a unit ramp 1/s^2 an exponential 1/s+a...
  20. B

    Any suggestions for finding the inverse Laplace transform of 11/(s^2+16)^2?

    Hi, I would like to find the inverse Laplace transform for 11/(s^2+16)^2 I have tried to expand it using the following partial fraction decomp to find the constants and take the inverse Laplace but this did not work C1(s)+ C2/(s^2+16) + C3(s)+C4/(s^2+16)^2 Does anyone have any suggestions?
  21. B

    Laplace transform with 2s(s^2+9) in denominator

    Homework Statement Hi I would like to know how to expand 20/2s(s^2+9) in order to find the inverse Laplace Transform of the function K(s) to gt k(t). The (s^2+9) term in the denominator is throwing my calculations off for the constants because of the s^2 term. Homework Equations...
  22. B

    Laplace Transform for shifted Unit Step Function

    Hello, I have a relatively simple question. after being unable to find it through google I have decided to ask you guys if you know what the Laplace transform of a unit step function that looks like this would look like Us(t-2) From tables, the Laplace transform for a regular units step...
  23. G

    Inverse Laplace Transform Help

    Homework Statement Is there a way to evaluate L^{-1}(\frac{F(s)}{s + a})? I'm sure if it can be evaluate. Homework Equations The Attempt at a Solution
  24. D

    MHB Laplace Transform with two steps

    Given the transfer function \[ H(s) = \frac{Y(s)}{U(s)} = \frac{1}{s + 1} \] and \[ u(t) = 1(t) + 1(t - 1). \] How do I find U(s)? I know I take the Laplace transform of u(t) but with the two step functions how can this be done? The Laplace transform of the step function is \(\frac{1}{s}\)...
  25. D

    MHB Inverse Laplace Transform problem

    I can't seem to part of an inverse Laplace transform correct. \begin{align*} f(t) &= \frac{6}{5}\mathcal{L}^{-1}\bigg\{\frac{1}{s + 2}\bigg\} + \frac{3}{5}\mathcal{L}^{-1}\bigg\{\frac{3s - 1}...
  26. G

    Laplace Transform Within a Domain

    Homework Statement Find the Laplace transform of f(t) = t \forall 0≤t≤T, 0 otherwise Homework Equations The Attempt at a Solution I write the function as tu(t)-t*u(t-T) That is turn on the function t at t=0 and turn the function t off at t=T. It seems to be right to me...
  27. D

    Solve a Laplace transform puzzle

    Hi~ I recently solve a Laplace transform problem as following L[int{t,0}cosh(t'-1)U(t'-1)dt']=? U(t'-1) is the unit step function(=1 for t'>1, =zero otherwise) According the standard Laplace formula : (1)L[cosh(t-1)U(t-1)]=exp(-s)*s/(s^2-1); (2)L(int{t,0}f(t')dt')=F(s)/s...
  28. I

    Trouble with Laplace transform of a function

    I have a probability distribution as follows: \begin{equation}p_j(t)=\frac{1}{N^2}\left|\sum_{k=0}^{N-1}c_{1,k}e^{ij\tilde{k}}\right|^2+\frac{1}{N^2}\left|\sum_{k=0}^{N-1}c_{2,k}(t)e^{ij\tilde{k}}\right|^2\end{equation} where...
  29. alyafey22

    MHB Existence of Laplace transform

    Prove the following Suppose that $f$ is piecewise continuous on [0,\infty) and of exponential order $c$ then \int^\infty_0 e^{-st} f(t)\, dt is analytic in the right half-plane for \mathrm{Re}(s)>c
  30. P

    Solving Integral Using Laplace Transform

    I posted this in the homework section, but I haven't received any help, so hopefully putting it in this section won't be an issue. I'm trying to compute the integral $$ \int_0^ \infty \frac{ \cos xt}{1 + t^2} \, dt $$ using the Laplace transform. The first thing that catches my eye is the 1...
  31. P

    What is the Laplace Transform Integral for the function $\frac{\cos xt}{1+t^2}$?

    Homework Statement $$ \int_0^\infty \frac{\sin xt}{x} \, dt $$ Homework Equations The Attempt at a Solution $$ = \int_0^\infty L(\sin xt) \, dp $$ $$ = \int_0^\infty \frac{x}{p^2 + x^2} \, dp $$ $$ = x \int_0^\infty \frac{dx}{p^2 + x^2} \, dp $$ p = x tan theta: $$ = x \int_0^{\pi/2}...
  32. P

    Question: How do we use the Laplace transform to find the inverse of a function?

    Homework Statement Find $$ L^{-1} \left[ \frac{1}{ (p^2 + a^2)^2} \right] $$ Homework Equations $$ L [ x \cos ax ] = \frac{p^2 - a^2} { (p^2 + a^2) ^2 } $$ The Attempt at a Solution I have no idea. Any thoughts?
  33. J

    Correcting a Laplace Transform Problem

    Homework Statement Why I am getting wrong answer related to this Laplace Transforms problem? According to the book "Higher Engineering Mathematics 6th edition by John O Bird" page no. 583 one should get (e^{-st}/(s^{2} + a^{2}))(a sin at - s cos at)Homework Equations ∫e^{-st}cos at dt The...
  34. I

    Laplace Transform IVP (Easy I think)

    Homework Statement Use Laplace transforms to solve the initial value problems. ##y''+4y=0;## ##y(0)=5;## ##y'(0)=0## Homework Equations The Attempt at a Solution $$y''+4y=0$$ $$L(y'')+L(4y)=L(0)\implies s^{2}Y(s)-sy(0)-y'(0)+4(sY(s)-y(0))=0\implies s^{2}Y(s)+4sY(s)-5s-20=0$$...
  35. T

    Laplace Transform Solution to Second Order ODE IVP

    Homework Statement y''+6y=f(t), y(0)=0, y'(0)=-2 f(t)= t for 0≤t<1 and 0 for t≥1 Homework Equations The Attempt at a Solution L{y''}+6L{y}=L{t}-L{tμ(t-1)} where μ(t-1) is Unit Step Y(s)=L{y} sY(s)-y(0)=L{y'} and y(0)=0 s2Y(s)-sy(0)-y'(0)+6Y(s) where y(0)=0 and...
  36. 1

    Inverse Laplace Transform of s/(s^2+1)^2)

    Homework Statement ##\mathcal{L}^{-1}\Big\{\frac{s}{(s^2+1)^2}\Big\}## I'm trying to figure out how to find the inverse Laplace transform of this expression. Is this something you just look up in a table or is there a way to find it directly, maybe by Convolution?
  37. L

    Laplace transform interpretation

    Hello, We were introduced to Laplace transforms in my ODE class a few days ago, so naturally I went online to try to figure out what this transform actually is, rather than being satisfied with being able to compute simple Laplace transforms. The Wikipedia page for the transform says it...
  38. I

    Laplace Transform of t: Using Integration by Parts

    Homework Statement Apply the definition in (1) to find directly the Laplace transforms of the functions described (by formula or graph). 1) f(t)=t Homework Equations The Attempt at a Solution Seems pretty easy... Question is, I don't understand the directions exactly.. Am I...
  39. G

    Laplace transform of Heaviside function

    Homework Statement What is the laplace transform of H(-t-17) Homework Equations Shifting theorem: L(H(t-a)) = (e^-as)/s The Attempt at a Solution This is the only part of the problem that I can not get (this part is from a larger differential equation I'm trying to solve). I'm can't seem...
  40. M

    Laplace Transform with Integral Convolution

    Homework Statement Determine the Laplace Transform of ∫(from 0 to t) (t-τ)cos(2(t-τ))e-4τ dτ using Laplace Transform tables. Homework Equations I know the basic convolution theorem is (f*g)(t) = ∫f(τ)g(t-τ)dτ The Attempt at a Solution I'm not sure if this is double convolution...
  41. D

    Calculate the range of S using Laplace Transform

    Homework Statement find the range of values for s: f(t)=e^(-t/2)u(t)Homework Equations The Attempt at a Solution what I did initially was to take the integral from 0 to infinity of e^(-st)e^(-t/2)u(t) dt this gave me 2/(2s+1) which when I sub in -.5 to find a divide by 0 or discontinuity then...
  42. N

    MHB How Do You Compute the Inverse Laplace Transform of a Power Series?

    Please refer to the attachment. For part a) so far I have: $e^x = 1 + \frac{x}{1!} + ...+ \frac{x^n}{n!}$ So $S^\frac{-1}{2}e^\frac{-1}{S} = S^\frac{-1}{2}(1 -\frac{1}{S} + \frac{1}{2!S^2} - \frac{1}{3!S^3} + \frac{1}{4!S^4} + ... - ...)$ I don't think my $S^\frac{-1}{2}$ on the outside is...
  43. N

    MHB Find Laplace Transform of $ t^2+1)^2 $ without Expansion

    How do you find the laplace transform of this without expanding it? $ L(t^2+1)^2 $
  44. T

    Solve Laplace Transform of e^-te^tcost

    Homework Statement $$L\{ { e }^{ -t }*{ e }^{ t }cost\}$$ Homework Equations The Attempt at a Solution $$L\{ { e }^{ -t }*{ e }^{ t }cost\} \\ =L\{ \int _{ 0 }^{ t }{ { e }^{ -\tau }{ e }^{ t-\tau }cos(t-\tau )d\tau } \} \\ =\frac { L\{ { e }^{ t }cost\} }{ s } \\ =\frac {...
  45. A

    Inverse Laplace transform (Initial Value Problem)

    Homework Statement I'm stuck trying to find out the inverse Laplace of F(s) to get y(t) (the solution for the differential equation): Y(s) = 1 / [ (s-1)^2 + 1 ]^2 The Attempt at a Solution I tried using a translation theorem and then apply the sine formula, but the denominator...
  46. Y

    Can the Laplace Transform Be Determined for Specific Functions?

    Homework Statement Can people help me on these two questions, please. Q1) Does f(t) have a Laplace transform F(s) for sufficiently large real value of s, where f(t) = et/(t4-1). Q2) Either find a function f(t) for which F(s) = L{f(t);t→s} = es, or explain why no such function f(t)...
  47. S

    Laplace Transform Question

    In my differential equations book (Edward and Penny) there are many examples of Laplace transforms being applied to linear differential equations with constant coefficients and no examples of them being applied to linear differential equations with variable coefficients. My question is, can this...
  48. N

    Derive Laplace Transform of the Third Derivative

    Homework Statement Derive he Laplace Transform of the third derivative of f(t). Homework Equations The Attempt at a Solution So, I'm not at all sure how to do this. I think I can start with: L{f'''(t)} = But I'm honestly not sure how this works. Any guidance would be...
  49. M

    Inverse Laplace Transform of a product of exponential functions

    I am reviewing some material on Laplace Transforms, specifically in the context of solving PDEs, and have a question. Suppose I have an Inverse Laplace Transform of the form u(s,t)=e^((as^2+bs)t) where a,b<0. How can I invert this with respect to s, giving a function u(x,t)? Would the inverse...
  50. I

    Laplace transform of unit step function

    Homework Statement The Attempt at a Solution I know that u(t) is a unit step function and holds a value of either 0 or 1. In laplace transform, when we integrate f(t) from 0 to infinity, we take u(t) to be 1. In this case, since u(t) is u(-t), does this mean it holds a value of 0? Does not...
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