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

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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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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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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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36 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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27 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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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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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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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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Solving 2nd Order ODE w/Laplace Transforms + Heaviside

This is a similar problem to the one I posted earlier with some differences. Attempt at solution: Write $g(t)$ as a heaviside function. Take Laplace transform of LHS and RHS. Solve for Y. Take ...
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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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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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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
41 views

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

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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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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30 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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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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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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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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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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Laplace transform of $f(t)/t$ [closed]

For a function $f(t)$ Laplace transform is defined as $F(s)=\int_0^{\infty} f(t)e^{-st}dt$. I have to show the property that the Laplace transform of $f(t)\over t$ is $\int _s^\infty F(s')ds'$. ...
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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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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 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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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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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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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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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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130 views

(Laplace Method) $y'' - 4y' = 6e^{3t} - 3e^{-t}$

For this problem $y(0) = 1$ and $y'(0) = -1$ I need to solve this problem using this: \begin{align*} y(t) &\longrightarrow Y(s)\\ y'(t) &\longrightarrow sY(s) - y(0)\\ y''(t) &\...
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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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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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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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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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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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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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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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Criteria for $L^1$ convergence looking at Laplace transforms

Let $(X_n)_{n \geq 0}$ be a sequence of integrable ($\mathbb{E} |X_n| < \infty$) random variables and denote by $l_n(t)$ the Laplace transforms of $X_n$. Similarly, let $X$ be a r.v. and $l(t)$ ...
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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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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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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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Laplace transform of the square root of a generic function

Let $f(t)$ be a function (for example of time $t$). Is there a general expression of the laplace transform of $\sqrt{f(t)}$ ? Same question for the inverse Laplace transform : Let $f(s)$ be the ...
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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 $\cos(at)$

I need to find the Laplace transform of $\cos(at)$ I know that $L\{\cos(at)\}= \int_{0}^{\infty} e^{-st} \cos (at) dt$ but I am having trouble finding the integral Thank you
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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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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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Laplace Transform of $tf(t)$

Q. prove that $\mathfrak{L}\{tf(x)\}=-\frac{d\mathfrak{L}\{f(x)\}}{ds}$ where the notation used is standard one. Attempt I tried what would seem obvious way to start: $$\mathfrak{L}\{tf(x)\}=\int_0^\...
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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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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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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 ...