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

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Laplace transform problem [closed]

This is the problem. I know ℒ (t^n) = ℒ (t^n-1) x n/s, which should give us -2/sqrt(s). That's as far as I've come and I don't get the hint either. Where does the pi even come from?
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26 views

Cross-correlation, Fourier transform and Laplace transform: measure of how much signal are alike?

I'm studying electrical engineering and use correlation, Fourier transform and Laplace transform a lot. I know how and when to use them, however, the interpretation I've seen in the lectures still ...
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43 views

Solving differential equation using Laplace transform, problem finding inverse

Given $$y'' + 4y' + 5y = H(t-3)e^{-2t}, t>0, y(0) = 1, y'(0)=2 $$ To solve this diff. equation using Laplace transform. Seems very straightforward. On one side, we have $$\mathscr{L}\{y''+4y'+5y\} ...
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26 views

Sine Curve Circular Transform - Parametric Equations

Is there a way to transform a sine curve so that the x-axis of the sine curve would become a circle, with the sine wave oscillating around the now-circular x-axis? What would be the parametric ...
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33 views

What is the domain and codomain of a transfer function? [closed]

Let's say I have the transfer function- $\textbf{H}(j\omega)=\cfrac{1}{1+j\omega RC}$ Where does this function map to and from, and can it be plotted visually?
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41 views

Finding the inverse Laplace transform of this function

Find the inverse Laplace transform of this function (related to my question earlier): $$f(t)=\mathcal{L}_s^{-1}\left[\frac{s}{s+\frac{1}{\tau}}\cdot\frac{A}{s}\left(1-\mathrm e^{-\frac{Ts}{2}}\right)^...
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60 views

Circuit Analysis problem (find the problem)

In this question, I know that $\text{C},\text{R},\text{T},\text{A}\in\mathbb{R}^+$ I've this circuit (the bottom of the resitor is connected to earth ($0$)): When I use Laplace transform I can find ...
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19 views

Expressing the Weierstrass transform in terms of the unilateral Laplace transform

I was looking for a way to express the Weiestrass transform of $f(t)$, $$\mathcal{W}\{f(t)\}(s)=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^{\infty}f(t)\exp\left(-\frac{(s-t)^2}{4}\right)dt$$ in terms of the ...
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23 views

Laplace transform ODE

Use laplace transform to solve the ODE $y''(t)+4y(t)=4u(\pi-t)cos(t)$,,,,,,, $y(0)=y'(0)=0$ u is the unit step function (heaviside function) I use: $u(\pi-t)=1-u(t-pi)$ By inserting this and ...
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37 views

Solve $y'(t)=t+1$ using Laplace transform

I'm studying how to use Laplace transform to solve ODEs. I have thought to use this very simple example: $$y'(t)=t+1 \qquad y(0)=0$$ I can use integration to find $y(t)$: $$y(t)=\int (t+1) \ \ dt=\...
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26 views

Laplace transform of a square wave function

What is the right way to find the Laplace transform of this function: The thing I noticed was: $$f(t)=\text{A}\space\space\space\space\space\space\space\space\space\space 0\le t<\frac{\...
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1answer
43 views

Invere Laplace transform of a function (related to circuit analysis)

I'm studying circuit analysis. I've to solve this inverse Laplace transform to see the response: $$\mathcal{L}_{s}^{-1}\left[\frac{r}{r+\frac{1}{cs}}\cdot\frac{k\tanh\left(\frac{as}{2}\right)}{s}\...
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1answer
28 views

Using laplace trasnform find the convulation of (f*g)(t).

$let \;f(t)=\sin (3t)$ and $g(t)=e^{-2t}$ Using laplace trasnform find the convulation of (f*g)(t). convulation theorem: $h(t)=(f*g)(t)= \int ^t_0 f(u) g(t-u) du$ $L(sin (3t)=\frac{3}{s^2+9}\\ L(e^...
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1answer
25 views

Inverse Laplace of an exponential function $\exp{\{-x\sqrt{(s+h)/k}\}}$

I am having difficulty to figure out to use a Laplace Transform Table formula to verify a particular case. The inverse Laplace transform of $$L^{-1}\bigg[\exp{\bigg\{-x\sqrt{(s+h)/k}\bigg\}}\bigg]$$ ...
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The integral $\int_{o}^{\infty} e^{-3t} cos(4(t-5))u(t-5) dt$ using laplace transform

$$\int_{o}^{\infty} e^{-3t} cos(4(t-5))u(t-5) dt$$ I need to find the laplace transform. Do i consider it as a convolution integral or as f(t)/t and work accordingly. Thank you. Edit: It turns out ...
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First order IVP with unknown continuous function

Solving $y''+4y=g(t)$, $y(0)=3$, $y'(0)=-1$ using Laplace Transforms. I get $Y(s)=\frac{G(s)}{s^{2}+4}+\frac{3s}{s^{2}+4}-\frac{1}{s^2+4}$ Then using Inverse Laplace Transforms, I am not sure of my ...
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1answer
29 views

Stuck relating the input to output (Transfer function)

i want to find the transfer function of a differential equation (given below) $\ddot\theta = a [ ([b\times Xin] - bk\dot\theta) - \ddot\theta] - c\phi $ (where $\phi$ and $\theta$ are time ...
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27 views

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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41 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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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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36 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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124 views

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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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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25 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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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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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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107 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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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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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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1answer
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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1answer
20 views

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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8 views

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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23 views

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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38 views

Laplace transforms of powers of cosine

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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19 views

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}\...