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

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How to compute the inverse laplace transform of this term? ${-{{3}\over{10}}s-{{1}\over{5}}\over{(s+1)^2+1}}$

So, I have been asked to solve $y'-2y=e^{-t} *cos(t)$ where $y(0)=-2.$ I applied the Laplace transform, getting $$\mathcal{L}(y)={{-2s^2-3s-3}\over{(s-2)((s+1)^2+1)}}$$ I set up a partial fraction ...
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Why Fourier transform is not sufficient and we have to use Laplace transform?

Is there an easy way to explain the motivation behind the use of Laplace transform instead of Fourier transform? Isn't that any periodic function can be represented by sines and cosines? Why to ...
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27 views

Why is the Laplace tranform of the pdf of a random variable called the Laplace transform of that variable?

We know that moment generating function of a random variable $X$ is $$M(t)=E[e^{tX}]$$ and if we replace $t$ with $-s$ then we get the Laplace transform as follows: $$\int_{-\infty}^{\infty}e^{-sx}f(x)...
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50 views

Is going from $V_{\text{L}} = L \frac{di_{\text{L}}}{dt}$ to $\frac{ V_{\text{L}} } {i_L} = L \frac{d}{dt}$ allowed?

The Laplace transform of $\frac{d}{dt} f(t)$ would be sF(s), when f(0)=0, which is something you can find in a Laplace transform table. If there is a rule that prohibits mathematical operations from $...
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14 views

What is the most general context to study Laplace transform?

In undergraduate course, one learns the fourier transform of continuous absolutely integable functions using Riemann integrals. Then, one learns the fourier transform in the context of measure theory ...
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44 views

Inverting Laplace Transform with Geometric Series: $\mathcal{L}C(t) = \frac{\mu e^{-\beta \tau}}{1-\mu e^{-\beta \tau}}$

Question Am I correctly resuming the series to invert this Laplace Transform? Specific points of interest are bullet pointed. The Laplace Transform of a function, $C(t)$, is given by, $$ F(\beta) = \...
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1answer
37 views

Can you use Laplace transform to show a function is convex?

First, I should say I just know about Laplace transform through Wikipedia. My question is, Can you use Laplace transform to show a function is convex? If so, is there any link to an example? I have ...
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31 views

N-order differential equations

Suppose that we have n-order differential equation like $$h(x)=?$$ Is it possible to find a general solution for all n? $$(x^n+1).|h'(x)|^n=const.$$.
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How to solve $y''+y=x^2$?

I need to solve: $$y''+y=x^2$$ Taking the Laplace transform (and using the fact that it is a linear operator) on both sides I get: $$\mathscr{L}(y)=\frac{2}{s^3(s^2+1)}+y(0)\frac{s}{s^2+1}+y'(0)\...
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1answer
28 views

Is it possible to use the convolution theorem on a finite interval integral ? (Laplace)

Say I have the following equation : $$\int_{0}^{1}\cos(t-\tau)x(\tau) d\tau = t\cos(t)$$ if we replace 1 in the integral for t it is easily solvable using the convolution of Laplace and the answer ...
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29 views

Definition of Laplace transform

I need to use the definition of laplace transform to determine $L(s)$ where $f(t)=e^{-t}$ on $0\leq t\leq 3$, and $2$ on $t>3$. My solution $$\begin{aligned} \int_0^3 e^{-st} e^{-t} dt + \int_3^{...
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1answer
45 views

Clarify and justify how get the derivative of the Laplace transform of the Buchstab function

I would like to justify that the derivative with respect to $s$ of the Laplace transform of the Buchstab function is $$\int_1^\infty u\omega(u)e^{-su}du=\frac{e^{-s}}{s}\exp\left(\int_0^\infty \frac{e^...
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0answers
42 views

Methods for solving nth order semilinear elliptic PDEs

I am looking for names of methods, and examples of their use that can be used to find solutions for semilinear elliptic PDE equations of the below types: $$\frac{\partial^ny}{\partial x^n}+\frac{\...
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2answers
56 views

Calculate an inverse Laplace transform

I need to calculate the inverse Laplace transform of $$\frac{s-2}{(s+1)^4}$$ Not quite sure how to do this one. I see that you should break the numerator up into $$\frac{s}{(s+1)^4}-\frac{2}{(s+1)^4}$...
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1answer
42 views

A question on the Laplace Transform of $f(t)=t e^{at}\sin (bt)$ [closed]

I would like to solve the Laplace transform of the following function: $$t \mapsto t e^{at}\sin (bt).$$ I know that $\mathscr{L}\left(\sin(bt)\right)=\dfrac{b}{s^2+b^2}$ and that you have to ...
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1answer
36 views

Laplace transform in ODE

Use any method to find the laplace transform of coshbt Looking to get help with this example for my exam review
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23 views

Analysing the modes of a signal with Laplace transform

If I have a linear dynamical system (assume continuous time for the time being) I can create the transfer function, let's say: $$\frac{1}{(s+a_1)(s+a_2)}$$ and the pole-zero map (this one is for e.g. ...
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1answer
35 views

Laplace transform of bell-shaped functions

A real smooth function $\varphi$ is said bell shaped iff as the Gaussian : $\varphi''$ is positive on $(-\infty,a) \cup (b,+\infty)$ and negative on $(a,b)$. I'm interested in the bilateral Laplace ...
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1answer
41 views

Solve equation $y^{(iv)}+y=1$ with Laplace

help me with this exercises, Laplace transform $$y^{(iv)}+y=1$$ with $\ \ \ \ \ y(0)=y'(0)=y''(0)=y'''(0)=0$ I got $$Y(s)=\frac{1}{s(s^4+1)}$$ But, I don't know how to continue: $s^4+1=(s^2+\...
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2answers
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laplace differential equation with conditions

I have to solve this differential equation with laplace $y'' + 6y' + 9y = \begin{cases}5t & 0 < t \le 3 \\ 0 & t>= 3\end{cases}$ and $y(0)=1, y'(0)=1$ I know what to do with the left ...
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1answer
61 views

Method for solving 2nd order linear PDE of three variables

For the 2nd order linear PDE below, please give method(s) to solve it, working, a solution, and what conditions the solution can exist? $$\sin(t)\frac{\partial^2y}{\partial t^2}+\cos(t)\frac{\partial ...
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On a second set of calculations for the Buchstab function

In this post I've added simple calculations deduced for the Buchstab function as claims. Question. Can you choose the worst of these claims and show where were my mistakes or inaccurancies? Also ...
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Laplace transform of $\,\sin\left(t\right) \,\frac{d^2y}{{dt}^2}$

I would like to know what is, and how to work out the Laplace transform with respect to $t$ of: $$\sin\left(t\right)\,\dfrac{d^2y}{{dt}^2}$$ I know that the transform of $\sin\left(t\right)$ is $\,\...
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On a first set of calculations for the Buchstab function

In this post I've added simple calculations deduced for the Buchstab function as claims. Question. Can you choose the worst of this claims and show where were my mistakes or inaccurancies? Also ...
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1answer
41 views

Are there expressions which have no inverse Laplace?

For a function $f(t)$ to have a Laplace transform, it must be piece-wise continuous of exponential order. But what about the inverse Laplace ? Is there expression $F(s)$ which has no inverse Laplace ?...
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23 views

Laplace Transform of unknown functions

I don't know how to go about solving these questions the second one seems to want integration by parts but I don't know how that will work out, an the first one seems to need the definition of Laplace ...
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1answer
40 views

How to compute this Laplace Transform! [closed]

How to solve this using the Laplace transform? $$ y''+4y = u_{2\pi}(t)\sin(t-2\pi), \qquad y(0)=0,\,y'(0)=0 $$ And how to compute $y\left(\frac{\pi}{2}\right)$ and $y\left(\frac{5\pi}{2}\right)$ ?
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How to prove this equation for moment generating function?

Let $\mathcal{L}_I(s)$ be the Laplace transform of $I$ which is given by $\mathcal{L}_I(s)=\left(\frac{2}{r_d^2-r^2}\int_r^{r_d}\mathbb{E}_H\left[r\exp\left(-sHr^{-\eta}\right)\right]dr\right)^k$ $\...
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56 views

Solving coupled second order ODEs via Laplace transforms & Function theory.

I have used Laplace transforms to transform a system of 2 coupled second order ODEs into 2 simultaneous equations. 1st ode: $$\frac{3d^2y}{dt^2}+\frac{dy}{dx}=0$$ 2nd ode: $$\frac{5d^2y}{dx^2}-\...
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Evaluate $\int _{ 0 }^{ \infty }{ \frac { { x }^{ n }-1 }{ \ln { x }}} dx$ using residue theorem.

$$\int _{ 0 }^{ \infty }{ \frac { { x }^{ n }-1 }{ \ln { x }}} dx$$ I couldn't solve this problem using the residue theorem. Can anyone help me get the answer? I know the steps like taking the ...
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1answer
35 views

Finding the inverse laplace transform using complex analysis.

I've been able to prove simple laplace transforms like $\dfrac {1}{(s+a)} $ quite easily but what about $\dfrac {1}{(s+a)^3+b^2} $ this does not seem easy to do since you cannot easily compute the ...
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46 views

Laplace Transform for Solving Differential Equation

I solved the following task, but since I am new in this field I need to check if it is correct or if there is anything I am missing or doing wrong. Task : Solve differential equation using Laplace ...
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37 views

Asymptotics of Inverse Laplace transform of a function with a branch point and singularities

consider the inverse Laplace transform $f(x)=L^{-1}[\tilde{f}]$ of a function $\tilde{f}(s)$. I would really like to find the large-$x$ asymptotics of $f(x)$ for the following case: $$\tilde{f}(s)=\...
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81 views

Laplace Transforms and Inverse Laplace

Please can you check my answers for the below Laplace questions. thank you. Question 1 Find the Laplace transform forms of the following piecewise function: $$g(t)=\left\{ \begin{align} & (...
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1answer
21 views

Second order system - find -3dB frequencies and magnitude response analytically

Let's take some simple second-order system like $H(s) = \frac{j\omega T}{(1+ j \omega T)^2} $. I know that the magnitude response is simply the absolute of the function and the -3dB frequencies can be ...
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Laplace transform of $\int_{0}^\infty\frac{e^{-t}\sin^2t}{t}dt$

Laplace transform of $\int_{0}^\infty\frac{e^{-t}\sin^2t}{t}dt$. So far I've calculated that $\frac{e^{-t}\sin^2t}{t}$ transformed equals $\frac{1}{8}(\ln((s+1)^2+4)-2\ln(s+1))$. My question is what ...
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1answer
40 views

Rewriting $8H(t-\pi)(sint)$ without use of the heaviside function

I was given a differential equation to solve using Laplace transformation. and I got a term that had : $-8H(t-\pi)(sint)$ The question asks to rewrite the solution without the use of the heaviside ...
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Simple way to prove that $e^{-x^2}$ doen't admit Laplace inverse trasform

The question is already contained in the title. Is there any criterion that one can use to show this or is it necessary to apply Mellin's inverse formula and verify that the integral doesn't converge?
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22 views

Null Laplace Transform

As the title says, if I had a real signed measure $\nu$ defined on Borel sets of $\mathbb{R}^m$ with Laplace Transform vanishing on every $m$-tuple, can I say that $\nu=0$?
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34 views

Another Laplace transform of a function with square roots.

This question is very much related to this (one). Let $F\colon \mathbb{C}\to \mathbb{C}$ be defined as $$F(s) = \frac{1}{4+3s+\sqrt{s(4+s)}}.$$ My question is what is the inverse Laplace transform ...
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Laplace transform of a product of K simple functions

Is there a closed form expression for the Laplace Transform of a function which is a product of K simple functions, where the i-th function is of the form (1 - exp(-k_i *t))?
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Complicated Laplace Transform

I have found the following Laplace Transform in a list $$\int\limits_0^{\infty}e^{-st}\frac{e^{-u^2/4t}}{\sqrt{\pi t}}dt = \frac{e^{-u\sqrt{s}}}{\sqrt{s}}.$$ I am wondering how to prove this? I ...
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51 views

Does the following have a Laplace transform?

I've looked at several resources and used Wolfram alpha but have been unable to find a Laplace transform for the following function: $$f(s) = {s\over \sqrt{a^2-\left({s\over 2}\right)^2}}$$ For a = ...
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1answer
30 views

Inverse laplace transform of $\dfrac{\alpha s}{s+\beta}$

I want to know the inverse laplace transform of $$\dfrac{\alpha s}{s+\beta}$$ where $\alpha, \beta$ are non-zero constants I already know the result for $$\dfrac{\alpha }{s+\beta}$$ Which is $\...
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1answer
44 views

Inverse Laplace transform of function with square roots.

Let $F\colon \mathbb{C}\to \mathbb{C}$ be defined as $$F(s) = \frac{1}{2+s+\sqrt{4s+s^2}}.$$ My question is, what is the inverse Laplace transform of $F$? From solving similar problems I have a ...
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How to derive through a convolution?

Let $f(t) = \alpha e^{-\beta t}$, where $\alpha, \beta$ are constants Let $g(t) = y(t)$ Then the resulting convolution $f\ast g$ is: $$f \ast g = \int_0^t \alpha e^{-\beta (t-\tau)} y(\tau) d\tau$$...
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How to calculate the Laplace transform

$$ h: t \in[0,+\infty[ \to \int_{t}^\infty \frac{1}{e^s\sqrt{s}}ds$$ I have to calculate the Laplace transform of $h$ in $0$ I know that $L[\int_{0}^\infty f(t)dt](p)= \frac{1}{p}L[f](p)$ but i ...
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46 views

Laplace Transform Derivation Help

Please see image. These are screenshots of a lecture slide from a Control Engineering module, regarding determining the transfer functions of mechanical systems. However I can't seem to understand how ...
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29 views

Derivative property of Bilateral Laplace Transform

According to the definition of the bilateral Laplace Transform: $$ X(s)=\int_{-\infty}^{+\infty}x(t)e^{-st}dt$$ where $s=\sigma+i\omega$. So to get the derivative property, $y(t)=\frac{dx(t)}{dt}$: $$...
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Solving ode with Laplace transform

What i want to ask is both question 20 and 21 with Laplace transforms Actually i can solve question 20 and 21 with method of undetermined coefficients but . In laplace transform, i don't know how to ...