The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.

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Inverse Laplace Transform of $\frac{1}{s} - \frac{a}{s^2 + a s^{3/2} \coth\sqrt{s}}$

I got a problem for inverse Laplace transform when solving a PDE, the solution in Laplace space is $$ \widehat{f}(s) = \frac{1}{s} - \frac{a}{s^2 + a s^{3/2}\coth{\sqrt{s}}} $$ where $a$ is a ...
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42 views

Second order differential equation with Heaviside function

I have a differential equation of the form $$y''(x) - a y(x) + b \theta(c - x) = 0, \quad y(0) = 0, \quad \lim_{x \to \infty} y(x) = 0,$$ where $a$, $b$, $c$ are some constants and $\theta(с - x)$ is ...
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On the Laplace transform $\int_0^\infty e^{-sx}d \left( \ \int_2^{e^{1+x}}\frac{dt}{\log t}\right) $

I've read the basics about Laplace transform, and I know that since for $\Re s>1$, $\frac{e^x}{1+x}$ has exponential order, then $$F(s)=\int_0^\infty e^{-sx}\frac{e^{1+x}}{1+x}dx$$ is well defined, ...
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51 views

Solving $xy'+y=x^{k}$

Find a solution to: $$xy'+y=x^{k}$$ Where $k>0$, and on the assumption that the transforms of $f$ and $f'$ exist. I understand that we can take the Laplace of all of the terms and then find ...
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31 views

Calculate the Laplace Transform :

Show that, provided a>0 and f is a real function that : $L\left[ f\left( t-a\right) H\left( t-a\right) \right] =e^{-pa}L\left( f\left( t\right) \right)$ I understand that when we multiply a ...
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24 views

Convolution of $te^{2t}$ and $\delta_1-\delta_2$?

I seek to find $f*g$ where $f=te^{2t}$ and $g=\delta_1-\delta_2$ and $\delta_a(t)= \displaystyle \lim_{\epsilon \to 0^+}d_{a,\epsilon}(t)$; i.e. $\delta$ is the Dirac Delta function. We have learned ...
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37 views

A Function of a Convolution (Laplace)

A paper I am reading makes the following claim: Assume that $a_n$ is a series of of positive, distinct, real numbers. Assume that $\epsilon_n$ are independent random standard exponential variables. ...
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Tauberian theorem when limit is zero

Let $h \geq 0$ be a non-negative increasing function with Laplace transform $H$. Let $\rho \geq 0$ be a constant. A simple Tauberian theorem says that the following two statements are equivalent: I. ...
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Inverse Laplace transform of $\sqrt{H(s)}$

In there any way to find inverse Laplace transform of a function in the following general form \begin{equation} F(s)=\sqrt{\dfrac{a_n s^n+a_{n-1}s^{n-1}+\cdots+a_1 s+a_0}{b_n s^n+b_{n-1}s^{n-1}+\...
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27 views

Solve transport equations by using Laplace transform

I'm trying to solve rather formally one-dimensional transport equation: $$ u_{t}+cu_{x}=0\quad\text{in $(0,\infty)\times(-\infty,\infty)$} $$ with an initial data $u_{0}$, which is bounded and ...
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26 views

Laplace transform of a signal?

Finding the Laplace transform of a signal: How do you setup the step function $f(t)$ (equation of the graph on the image). Even though, I do know know how they setup the equations. I do know how to ...
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17 views

A Shifted Mellin Transform Or Half Moments from a Laplace Transform

I've asked this question in two ways, as I think either is a solution to my problem. Assume that I have some distribution $f(t)$ for which I know the Mellin Transform $M(s)$ and the Laplace transform $...
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15 views

partial differentia equation and Laplace transformate

let the following differential equation: $$ \begin{cases} \dfrac{\partial^2 y}{\partial t^2}= a^2 \dfrac{\partial^2 y}{\partial x^2}, 0<x<l, t>0\\ y(0,t)=f(t)\\ y(x,0)=0\\ \dfrac{\partial y}{\...
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How does multiple integral change into terms multiplying each other in convolution theorem of Laplace?

In the steps of the proofs highlighted below, how does a multiple integral changes in to multiplication of two integral. This is only possible if V is independent of u, but as it turns out V = t - u, ...
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Laplace transform of $\cos(2t-(\frac\pi3))$

Problem I need to find the Laplace transform of $\cos(2t-(\frac\pi3))$ Attempt I've tried to look up some relevant formulae in my book, but I can't find anything that looks useful. I suspect there ...
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109 views

How do you integrate $e^{-st}t\cos(t)$?

I'm doing differential equations and specifically studying Laplace Transformations, where of course the Kernel is: $K(s,t) = e^{-st}$ And the Laplace Transformation $\mathcal{L}$ of a function $f(t)$...
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24 views

Is it possible that a PDE solved by two different analytical methods with same Initial and boundary values give different results?

I have developed two models of same scenario. Both models involve a PDE which is solved with same Initial and Boundary conditions. In one model it is solved with Laplace transform and in other with ...
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inverse laplace of sine function [duplicate]

I use the transformation rule $L(f(t)*t^n) = F^{(n)}(s)(-1)^n$ to find out the inverse Laplace of $\sin(s)$. $F(s) = \sin(s)$ $F''(s)=-F(s)$ $L(f(t)*t^2) = F''(s) = -F(s) = -L(f(t))$ $L(f(t)(1+t^2)...
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21 views

Transfer function unity and output function poles are related?

By solving a few examples, I found the pattern that, given a differential equation in $y(t)$ and $x(t)$, where $y(t)$ can be called the input and $x(t)$ the output, if we make the condition that $y(t) ...
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26 views

Laplace transform and inverse laplace transform

1- Find laplace transform for $4e^2t-3\cos^2(2t)+2\cosh(3t)$ My answer $L(4e^2t-3cos^2(2t)+2cosh(3t))=4L(e^2t)-3L(cos^2(2t))+2L(cosh(3t))$ $=\frac4 {s-2}-3L(\cos^2(2t))+\frac{2s}{s^2-9}$ But how ...
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Finding Laplace Transform of $te^{-t}$

I started with this integral: $$ \int_{0}^{\infty} e^{-st}\cdot te^{-t}dt$$ = $$\int_{0}^{\infty} te^{-(s+1)t}dt$$ let $dv=e^{-(s+1)t}dt, u=t$ and thus $v=-\frac{1}{s+1}e^{-(s+1)t}dt, du=dt$ $\...
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Poissions Equation (Laplace)

$$\begin{align} u''_{xx}&+u_{yy}= x, \quad 0<x<1, \quad 0<y<1,\\ \\ u(x,0)&=u(x,1) = 0, \\ u(0,y)&=u(1,y) = 0,\\ \end{align}$$ Having some problems with Poissons Equation. I'...
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Laplace transform of the square wave to solve PDE

Solve $$y'' + 3y' +2y = r(t)$$ given $y(0)=0$ and $y'(0) = 0$ where $r(t)$ is the square wave, $$r(t) = u(t-1) - u(t-2)$$ I'm just going to type out the answer as I read it and tell you which ...
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Laplace transform of $(3e)^t\sin^2 t$

The existence of Laplace Transform of $(3e)^t\sin^2 t$ exists but can you help me in finding the Laplace transform of this function?
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integral equation and laplace transform

Solve the following integral equation $ u(x)= \cos x - \int_{0}^{x} (x-y)cos(x-y)u(y) dy $ I applied Laplace transforms to the above integral equation and so the initial equation is written as: ...
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Writing a sum of unit step functions as a piecewise function

After taking the inverse Laplace transform of the following $$\mathcal{L}^{-1}\{G(s)\}=\mathcal{L}^{-1}\left\{\frac{e^{-2s}+e^{-3s}}{s^2-3s+2}\right\}$$ I have $g(t)=\mathcal{U}(t-2)[e^{2(t-2)}-e^{t-...
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Compute Heaviside Laplace transform, then use this to solve initial value problem

I've been stuck on this problem for a while, and can't really seem to find where I should go with it, or where I went wrong if I made a mistake. Let $L(x)$ denote the Laplace transform of x. Q: ...
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Dirac Delta Function Problem

In my Differential Equations class, we had the following equation on a test today: $y''+6y'=2\delta(t)$, $y(0)=0$, $y'(0)=1$. I got the following using Laplace Transforms (the only way I know how to ...
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Laplace pairs - proof of summation transform

I am studying this question for my finals revision and I'm lost on how to start it, can anyone suggest something? Probably pretty simple but I've hit a dead end. Here's the question: If $F_i(t)$ ...
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Solving integral equation with Laplace transform.

The equation in question is: $ y(t)+\lambda \int_{0}^{t}y(\tau)d\tau=t $ My work so far is first to Laplace domain: $$ Y(s)+\frac{\lambda Y(S)}{s}=\frac{1}{s^2} \longrightarrow Y(S)=\frac{1}{s^2} \...
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Laplace Transforms Property

Please see image. I'm confused by how the went about to get d[exp(-st)] in the second line Thanks Laplace Transform
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Laplace transforms of powers of cosine (solved!)

During the past several hours, while studying the Laplace transform, I've started wondering what \begin{equation} \mathcal{L} \{ \cos^n(at)\}(s) \end{equation} would look like – since it won't ...
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Find the Laplace transform of the function to show the following result

We have to show that $\mathcal{L}\{t^2e^{-3t}\cos{at}\}=-\frac{s^3-9s^2+(3a^2+9)s+5a^2+33}{(s^2+2s+a^2+9)^3}$. Steps : $\mathcal{L}\{t^2\cos(at)\}=(-1)^2\frac{d^2}{ds^2}\left(\frac{s}{s^2+a^2}\...
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Laplace transform of $\sin(\sqrt t)$

How can I use this differential equation $$4tf''(t) +2 f'(t) + a^2 f(t)=0$$ to show that $$L(\sin(\sqrt{t}))=\frac{1}{2}\sqrt{\pi}\,\frac{1}{s^{\frac{3}{2}}}\,e^{\frac{-1}{4s}}$$
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Difficult Inverse Laplace Transform

I've had this question in my exam, which most of my batch mates couldn't solve it.The question by the way is the Laplace Transform inverse of $$\frac{\ln s}{(s+1)^2}$$ A Hint was also given, which ...
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41 views

Stability and characteristic roots of difference equations

I often hear "For a process to be stable its characteristic roots or poles must be outside the unit circle (for casual process)". All right, consider the recurrence relation: $$y_t-5y_{t-1}+6y_{t-2}...
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24 views

Parseval like theorem for Laplace transform?

I was wondering if there any parseval like theorem for Laplce transform? Some statement such as $\int f(x)g(x) dx=\int F(s)G(s) ds$ where $F(s)=\mathcal{L}\{f(t)\}$ and $G(s)=\mathcal{L}\{g(t)\}$?
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Complicated exponential function inverse laplace transform

The problem is from wave equations for describing flexural vibration of Euler-bernoulli beam. The equations are listed in the following pics. I. equations and 16 elements to be analyzed II.cF:wave ...
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Inverse Laplace transform of exp(-as)/s via an infinite series

It is known that: $$\mathscr L^{-1}[F_1(s)\,+\,F_2(s)\,+\,F_3(s)\, ...] = \mathscr L^{-1}[F_1(s)]\,+\,\mathscr L^{-1}[F_2(s)]\,+\,\mathscr L^{-1}[F_3(s)]\,+\;...$$ where $\mathscr L^{-1}$ is the ...
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Laplace transform of this problem

I'm a bit confused in finding the laplace transform of the following: $f(t) = e^{2t} u(t) $ I know how to solve it if it we're only $e^{2t}$ but the $u(t)$ is confusing.
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What does this “proof” tell us about the Laplace transformation?

An exercise in the book stated "Find $2$ functions $f_1 \not =f_2$ such that $L\{f_1\}=L\{f_2\}$. Now I think I know the answer ($f_1=f_2$ except at any number of discrete points) but previously I was ...
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Why is the Laplace Transformation defined the way it is?

I can see the many uses of Laplace Transformation, but why is it defined the way it is? What is its physical significance?
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Inverse Laplace of a piecewise function

$$ F(s)=\frac{e^{-3s}-e^{-6s}}{s^8} $$ I've tried convolution, partial fractions, and guess and check. I know $t^7$ is involved somehow. Help!
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How do I find the Laplace transform of the function $f(t)=3t$ if $t\le 6$ and $f(t)=18$ if $t>6$?

I found the step function: $u(6-t)\times 3(t-6)+18$ where $u(t)$ is the step function, but I do not know how to find the inverse laplace of this. Do I need to change the form of the step function?
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42 views

Show that $\int_{0}^{\infty} \int_{0}^{\infty} e^{-s(x+y)}f(x)g(y) \ dx \ dy = \int_{0}^{\infty} \int_{0}^{t} e^{-st}f(t-u)g(u) \ du \ dt$.

While learning how to compute the product of two Laplace transform and the inverse transform, I faced with this equality : $$\int_{0}^{\infty} \int_{0}^{\infty} e^{-s(x+y)}f(x)g(y) \ dx \ dy = \int_{0}...
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18 views

Finding the inverse Laplace of this function

Finding the inverse Laplace transform of $L^{-1}\left( \dfrac {1}{\left( x+1\right) ^{2}}\right)$ I understnd that the inverse laplace of $L^{-1}\left(\dfrac {1}{\left( x+1\right) }\right)$ Is ...
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29 views

What is the quickest method of finding the inverse laplace transform of $ 7e^{-6}/(s^2+6)^4 $

Solving for a differential equation, I found this which I need to obtain the inverse laplace of. What method should I use to solve this as quick as possible? Partial fractions and convolution seem to ...
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1answer
27 views

Solve integral equation using Laplace transform and convolution

Solve $$y(x)=e^x(1+\int_{0}^{x}e^{-t}y(t) dt)$$ Here's what I did: take derivative on both sides, $y'(x)=e^x+e^x \int_{0}^{x}e^{-t}y(t)dt +e^xe^{-x}y(x) \Rightarrow y'(x)=e^x+y(x)-e^x+y(x) \...
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30 views

On the relation between Laplace transform of a function and its derivative

Suppose that $s>s_0$. Through integration by parts we can write $$\int_{0}^{\infty}e^{-st} f'(t)~\mathrm dt= \Big[ e^{-st} f(t)\Big]_0^\infty + s \int_{0}^{\infty}e^{-st} f(t)~\mathrm dt. $$ Book ...
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41 views

Convolution of Gaussian and parabolic function.

What is convolution of $\exp(-x^2)$ i.e Gaussian function and $2x^2$? I don't have any idea,as I have found that Laplace transform of Gaussian function involves complementary error function,so inverse ...