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

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Determine the domain of convergence of Laplace inverse trasform

I'm trying to solve a problem that gives these three functions $$ f_1 (p) = \cos(p) e^{-p}\\ f_2 (p) = e^{-p^2}\\ f_3 (p) = (p+2)^k/(p-3)^{k+1} \text{, k>1} $$ (where $p$ is a complex variable) and ...
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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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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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29 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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40 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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44 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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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 ...
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26 views

Inverse Laplace Transform with squared irreducible quadratic in denominator using convolution theorem

Please help me find the inverse Laplace transform of $$F(s) = \dfrac{4}{(s^2+2s+5)^2}.$$ The answer I got is $\frac 1 5 (e^t - e^{-t}) \cos 2t - \frac 1 2 (e^t + e^{-t}) \sin 2t$. I first applied ...
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42 views

Laplace Transform with initial value

Use the Laplace transform to solve the following initial value problem: $$y'' + y = 2t$$ with $y(\pi/4) = \pi / 2 $ and $y'(\pi/4) = 2 - \sqrt{2}$. I understand this type of problems ...
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Solve a Laplace in Poolar Coordinate under two minutes?

I trouble with calculating the following example from previous exam with short solution on this Link. OP says there is a Laplace ٍPoolar Coordinate: $\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\...
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Does the scaling property of Laplace transforms also hold for complex scaling?

Consider the following Laplace transform (it arose in the context of Borel resummation) $$ \int_0^{\infty}e^{-\zeta}\phi(z\zeta)d\zeta $$ my textbook says that $$ \int_0^{\infty}e^{-\zeta}\phi(z\zeta)...
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Initial Value Problem using Laplace transformation: What is the ${\cal L}$ transform of $u(t-5)$?

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses a single difficulty. If it's any indication of difficulty, the exercise ...
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Asymptotic behavior of inverse laplace transform [duplicate]

My question may be quite rough. Let $F(\lambda)$ be the Laplace transform of some function $f(t)$, $$ F(\lambda)= \int_0^\infty e^{-\lambda t}f(t) dt. $$ If I have knowledge about $F(\lambda)=O(\...
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Statistics - Laplacian Characteristic Function

I was asked to find the characteristic function of the Laplacian random variable. And, from that, to find the mean of it. Im having trouble solving this. That's what I have done so far, but it doesn'...
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How to calculate the inverse Laplace transform of $F(s)=\frac{1}{1-e^{-s}}$?

How to calculate the inverse Laplace transform of this function? $$F(s)=\frac{1}{1-e^{-s}}$$
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Inverse Laplace transform of $(s^2-1)^{-1/2}$

please help with this. Not derived from any differential equation. Also found the answer $\mathcal{L}^{-1}(\dfrac{1}{\sqrt{s^2-1}})$
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Trigonometric functions and complex numbers

I solving the inverse Laplace transform using the method of Heaviside. This is part of the problem: I understand the division between complex numbers and that $e^{it} = Cos(t) + iSin(t)$, but I ...
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How to solve a differential equation with a distributional free term?

I tried to solve this type of differential equation $$y'' + y = \delta + \delta' .$$ I tried using the Laplace Transform, but I'm stuck at that $\delta$ (Dirac function). The only thing I know is ...
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What exactly is the variable of Laplace transform

I try to find the solution for a hard differential equation. I could not solve it with any orthodox method. However, if I use Laplace transform and then replace its term with its Maclaurin series, it ...
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Laplace and Transfer Function Problem

I have to show that all initial condition problem where $t = 0$ with constant coefficients: $ a_{n} y^{n}(t) + a_{n-1}y^{n-1}(t)+...+a_{2}y''(t) + a_{1}y'(t) + a_{0}y(t) = x(t) $ with $ y^{k} = y_{...
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extracting a function out of an equation

I encountered the following problem in my thesis. We have an equation as follows: $\phi(s)=\sum^\infty_{n=1}P(n)\int^{\infty}_{0}e^{-st}f(t|n)dt=\sum^{\infty}_{n=1}[(1-q)M_1(s)]^{n-1}qM_2(s)$ in ...
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Solve non-linear pde

so i wonder the next thing : if i consider a pde like heat equation Fourier transform works very well. Now, if i consider this equation : $\frac{\partial u(t,x)}{\partial t}-k(u(t,x))\frac{\partial^2 ...
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55 views

What is the inverse Laplace transform of $\lfloor s \rfloor$?

How can we find the inverse Laplace transform of: $[x]$ (floor function) ? My question isn't LLaplace transform of floor function i asked the "inverse" laplace transform of floor function $$\mathcal{...
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How do I proove that function is solution of the Laplace equation?

How do I proove that for $\vec{r}=(x,y,z)\in \mathbb{R}^3,\vec{r}\neq 0$, function is $u(x,y,z):=1/(-ln\left \| \vec{r} \right \|)$ a solution of the Laplace equation $\Delta u=\frac{\partial^2 u}{\...
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Laplace Transform of $t^2$ , for $t\ge1$ .

$$f(t) = t^2 , t>=1$$ $$f(t) = 0, 0<t<1$$ what is the laplace transform of $f(t)$, It is solved In my sheet as $$t^2 = (t-1)^2 + 2t -1 $$ $$ L(f(t)) = e^{-s} L(t^2 +2t +1) = e^{-s}(2/s^3 + ...
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Laplace transform to solve a pde

i have to find the laplace transform of this : $$f(u(x,t))\frac{\partial u(x,t)}{\partial t}$$ I have no idea how to solve this... Thank you for your help.
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Laplace transform using change of scale property [closed]

If Laplace transform of $f(t)=\phi(s)$, then Laplace transform of $e^{bt}f(at)$ is
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prove the result of a Laplace transformation

I have to prove the next problem $$\mathcal{L} \left(\int_{0}^{t}\frac{1-e^{-u}}{u}du,s\right) = \frac{1}{s}\log\left(1+\frac{1}{s}\right)$$ I'm quite new in the subject and I have troubles with ...
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Laplace Transform on this Poisson Process?

Two independent Poisson processes with parameters $\lambda_1$ and $\lambda_2$. The waiting is exponentially distributed with mean of $\frac{1}{\mu_1}$ and $\frac{1}{\mu_2}$. Knowing this, how to ...
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Why is Laplace Transform of $\delta(t)$ $F(s)=1$, not $0$ or $\frac{1}{2}$?

Let $\varepsilon \in \mathbb{R}$, the following integral $$ \int_{\varepsilon}^\infty \delta(t)e^{-st}dt. $$ converges to $1$ if $\varepsilon \to -0$ and $0$ if $\varepsilon \to +0$. This shows ...
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laplace transform probability distribution not concentrated on 0

This seems intuitively obvious but how to prove that $\hat{\mu} < 1,$ when $\theta >0$ and $\mu$ is a probability measure not concentrated at $0,$ where $\hat{\mu}$ is defined as below $$\hat{\...
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13 views

Get transfer function of a nonlinear diff. equation

I have this equation: $$\frac{\partial v}{\partial t} = -g + c\left(u(t) - v(t)\right)^2$$ g and c are constants. u(t) is my input and v(t) is my output. I need to reach the transfer function $\frac{...
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Non-trivial inverse Laplace transform

I'm trying to compute the inverse Laplace transform of $f(s) = s^c/(N + s^{ir} )$ where $c,N \in \mathbb{C}$ and $r \in \mathbb{R}^+$ using the Bromwich integral $$ F(t) = \frac{1}{2 \pi i} \int_{- ...
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Laplace transform of two exponential distribution

1) Type 1 clients take exponential time with average of $\frac{1}{\mu_1}$ to be served. 2) Type 2 clients take exponential time with average of $\frac{1}{\mu_2}$ to be served. 3) Type 1 and 2 ...
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Alternative integration limits in a Laplace transform

The unilateral Laplace transform of $f(t)$ is $\int_0^\infty e^{st} f(t) \mathrm{d}t$. If we define the transform as $\int_{a}^\infty e^{st} f(t) \mathrm{d}t$, would it conserve all the nice ...
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Workbook recommendation in preparation for Electrical Engineering

I'm currently preparing myself for starting my graduate degree in Electrical Engineering. The mathematics courses given are outlined as follows: Mathematics 1 Real functions Continuity, limits, ...
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23 views

Laplace Transform - Partial Fractions [closed]

Please see attached image. I keep coming up with a irrational/complex coeffecient which is correct. Can you please help me put it partial fractions please Thanks
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Inverse laplace transform using equating coeffecients method [closed]

Please see attached image. Could you please help me do the inverse of this laplace transfrom. I'm using the method of trying to equate the coeffecients. Thanks
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Coupled differential equations: Laplace transform

By using Laplace transforms find the steady state values of w, u and v:$$\frac{du}{dt}=-\frac{\Gamma}{2}u+\Delta v,$$$$\frac{dv}{dt}=-\frac{\Gamma}{2}v - \Delta u +w \Omega,$$$$\frac{dw}{dt}=-\Gamma-\...
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Solving ODE using Laplace transformation without table of transform

Solve $$y'' + 4y' -5y = 2xe^{-3x}$$ using the Laplace transform method. For the the Laplace transform of the ODE you must do the integration (do not use table). Table can be only used when computing ...
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Laplace of a function raised to a power

For example: $y' = y + y^2$ The Laplace of the first two terms is $s(F(s)-f(0))$ and $F(s)$. But what is the Laplace of $y^2$?
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Differential equations using Laplace transforms

By using Laplace transforms find $x(t)$ from the coupled differential equations$$\frac{dx}{dt} = -kx+gy+E,$$$$\frac{dy}{dt} = -ky-gx,$$for some functions $x(t),$ and $y(t)$, where $E, k, g$ are real. ...
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Laplace transform of a convolution-like function

Is there a way to calculate the Laplace transform of the following function? $$ \sum_{k=1}^{+\infty}f(t-(g(t)-\theta_k))h(g(t)-\theta_k), \qquad t>0. $$ Thanks in advance.
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Consider the equation $y'' + 4y = f(t), y(0) = 1, y'(0) = 0$. Use the Laplace Transform to compute the Green’s function for this equation.

Consider the equation $y'' + 4y = f(t), y(0) = 1, y'(0) = 0$. Use the Laplace Transform to compute the Green’s function for this equation. $y'' + 4y = f(t) \rightarrow L\{y'' + 4y = f(t)\}=L\{f(t)\}$...
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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?