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Questions tagged [laplace-transform]

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 of Functions with Infinite Discontinuities

I know it's possible (generally speaking) to take the Laplace transform of step functions with a finite amount of discontinuities, such as $f(t) = u_0(t)$, $f(t) = u_3(t)\sin(t)$, etc. where $u_x(t)$ ...
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What algebraic step is taken here?

We have as part of a Laplace transform: $s^2F(s) - s - 1 + \frac{s^2F(s) - s - 1}{s^{\frac{1}{2}}} + F(s) = \frac{1}{s} + \frac{1}{s^2}\\ \implies F(s)(s^2+s^{\frac{3}{2}} + 1) = (\frac{1}{s} + \frac{...
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Can the antiderivative of a function $f(x)$ be represented as $f^{(-1)}(x)$?

I was going over some common practices for solving ODE's today, and the following question came to mind: Can we solve a "Differential Equation" of the form $\int {af(x)dx} + bf(x) = c$? I know it ...
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What is the inverse Laplace transform of $\frac{s^3}{(s+1)^3}$?

I tried to expand the given function into partial fractions, but since the degree of numerator and denominator are same, I can't use it.
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Laplace Transform Simultaneous Equations

If I have the matrix down below how can I get rid of I1 so I can have everything in terms of I2? $ \begin{pmatrix} (10+s) & (-s-6) \\ (-s-6) & (s+\dfrac{4}{s}+6) \\ \end{...
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Visualizing the Laplace Transform (for beginners)

I'm taking a differential equations course and have just been introduced to the concept of the Laplace Transform. I know its usefulness as a tool for solving IVPs, but I'm having trouble visualizing ...
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Laplace transform of $sin(\sqrt 3t)$

This is an interesting question for me: Find the Laplace transform of $\mathcal{L} \{sin(\sqrt 3t)\}$. I know the Laplace transform of $sin(\sqrt t) = \frac{\sqrt\pi}{2s^{3/2}}e^{-1/4s}$, but ...
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A second order differential equations with initial conditions solved using Laplace Transforms

Below is a problem I did from the book Differential Equations by K.A. Stroud and Dexter Booth. I got the right answer but I am not sure I did it right especially when I took the inverse Laplace ...
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Finding an unknown function out of an DE (involving an integral and an inverse Laplacetransform)

Well, I've the following problem for an unkown function $x(t)$: $$x'(t)\cdot\text{a}+\text{b}\cdot\frac{x'(t)}{x(t)+\text{c}}+\frac{\partial}{\partial t}\left\{\int_0^tx(\tau)\cdot\mathcal{L}_\text{s}...
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I'm stuck with these Laplace problems

I'm stuck on solving (4)(ii) and 4(b), I have tried to solve them but got stuck in the middle, any hints? This is where I have gotten in (4)(a)(ii) and (4)(b)(i), I have no idea how to start solving ...
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Laplace functional for a point process of exceedances

Let $$\mathit N_n(\cdot)=\sum_{i=1}^n \varepsilon_{\frac in}(\cdot)\mathit I_{(\mathit X_i>u_n)}$$ be a point process of exceedances where $(\mathit X_n)$ is a sequence of random variables and $(...
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Give the Laplace transform of the expression!

Given the equation $u_π(t)sin(at)$, where $u_τ(t)$ is the Heaviside step function. Give the Laplace transform $Y(s)$. In order to use the formula $L\{u_c(t)\cdot f(t-c)\}=e^{-cs}\cdot F(s)$ I have ...
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What is the Laplace transform of $\frac{\Gamma(t,z)}{\Gamma(t)}$

What is the Laplace transform of : $$\frac{\Gamma(t,z)}{\Gamma(t)}$$ w.r.t. the real parameter $t$, Where $\Gamma(\cdot,\cdot)$ is the incomplete gamma function, and $z\in \mathbb{C}$ EDIT From ...
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Prove the inverse transform of unilateral Laplace transform

I'm reading this article and having a question. Consider a function $f$ and its Laplace transform $\hspace{3.0cm} F(s) = \int_0^\infty f(t) e^{-st} dt$, with $\{s|\text{Re}(s) = 0\} \in \text{ROC}[...
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Finding the Laplace transform from a graph

Find $\mathcal{L}(F(t))$ where $F(t)$ is the perioidic function shown graphically below: I know a few basic things, but don't know where to get started. For example, I know that the period, T = 2, ...
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Solving the Airy Equation using Laplace Transform

I have been trying to read the book Airy Functions and Applications to Physics, Olivier Vallee & Manuel Soares, for research on the Airy Functions, but I am stuck on using the Laplace transform to ...
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Help with explanation to the notations in a paper about notations and use of Fourier and LaPlace Transformations

I'm looking at a paper 'Recent applications of fractional calculus to science and engineering' (https://www.hindawi.com/journals/ijmms/2003/753601/abs/) but some of the notations in it baffled me and ...
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Laplace transform and frames vs Bases

The Laplace transform $$F(s) = \int^{∞}_{0}f(t)e^{-st} dt$$ can be understood much like the fourier transform, as a change of basis of an $L^2$ function to the eigen functions of the differential ...
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How to deduce Fourier / Z transform from Laplace transform through a CAS-Calculator.

I have a calculator with CAS ( https://en.wikipedia.org/wiki/Computer_algebra_system ) and in particular I have a Casio Algebra FX 2.0 Plus ( https://en.wikipedia.org/wiki/...
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Laplace transform on a Boundary value problem (ODE)with . Am i attempting correctly?

From a system of PDEs where i used the following ansatz: $$\theta_w(x,y) = e^{-\beta_h x} f(x) e^{-\beta_c y} g(y)$$. $F(x) := \int f(x) \, \mathrm{d}x$ and $G(y) := \int g(y) \, \mathrm{d}y$ So, $$\...
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Find the inverse laplace (if there is one)

I need to find if F(s) can be a laplace transform of a continuous exponential function: F(s)=s^3/(s^3+2s^2+s+1) any ideas?
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Inverse Laplace transform of rational function

Is it true that the inverse Laplace transform of a proper rational function (where all the coefficients are real and positive), \begin{equation} s \mapsto \frac{a_1 \, s^{m-1} + a_2 \, s^{m-2} + \...
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Relationship Between Laplace and $z$-Transforms.

I've recently come across the relation $s= \frac{2(z-1)}{T(z+1)}$ between the Laplace and $z$-Transforms with inverse $z= \frac{2+sT}{2-sT}$ in some lecture slides, however there was no elaboration ...
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Complete sufficient statistic for shifted exponential distribution

How can I find CSS for $\mu$ of $\text{Exp}(\mu,1),\;\mu\in\Bbb{R}$? I just derived $X_{(1)}$ is a SS for $\mu\in\Bbb{R}$ where $X_{(i)}$ is the i-th order statistic Now I'm struggling to show $\...
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Properties of laplace type transform of $t^{\alpha - 1}$

Let $p>2, \frac{1}{p} < \alpha < 1- \frac{1}{p}$ and define $g_\alpha(t) := t^{\alpha - 1} \chi_{[1, \infty]}$. Then $g_\alpha \in L^p(\mathbb R)$. Define $$f(z) := \int_1^\infty g_\alpha(t) \...
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Integral of Laplace transform

Consider the following theorem: Let $u: \mathbb{R}\rightarrow \mathbb{C}$, $u\in T$, the set of Laplace-transformable functions, and $v \in T$ as well, $v(t)=u(t)/t$. Then $\int^{\infty}_{s}\...
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Laplace transform for solving differential equations [closed]

I am confused about Laplace transform. the only thing that is important in Laplace theorem is the value of $f(t)$ in $t \geq 0$. laplace transform: $\mathcal{L}\{f(t)\}$ $=$ $\int_0^\infty \! e^{-st}...
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How to resolve this differential equation with Laplace transform?

$$\mathcal{L}\left\{ f'(t)\right\} =s\mathcal{L}\left\{ f(t)\right\} -f(0)$$ $$\begin{cases} x' +x - y' = -t \\ x' + y' + y = 1 \end{cases}$$, $$X=\mathcal{L}\left\{ x\right\}, \ Y=\mathcal{L}\left\{...
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Extension of the initial value theorem

I'm trying to prove this extension of the initial value theorem for Laplace transforms. If $$\lim_{s\to \infty}s^{v+1}L\{f(t)\}=C$$ , with $L$ the laplace transform then $$D^{v}f(0)=C$$ I know the ...
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Solving the Laplace transform $\mathcal{L}_t\left\{sin\left(at^n\right)\right\}\left(s\right)$

I was solving an integral and i stepped in a Laplace transormation of the form $\mathcal{L}_t\left\{sin\left(at^n\right)\right\}\left(s\right)$ and I was curious on a generalized solution. After some ...
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Laplace Transform of $y^n$

When the Laplace transform of $y$ is denoted $Y(s)$, we have formulas for the derivatives of $y$ without actually knowing what $y$ is. Is there an explicit formula for $y^2$? More generally, $y^n$?
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Jordan lemma and Laplace inverse transform, textbook incongruence

I have a doubt about Laplace antitransform and Jordan lemma. Let's say I have a certain Laplace transform $U(s)$ and want to compute the antitransform by means of the formula $$u(t)\theta(t)=\frac{1}...
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How to find the transfer function for advection equation?

The Problem: I have to find the transfer functions of the two equations in my system and of the whole system but I am not sure about something and as I haven't had mathematics lessons for my three ...
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Laplace inverse of functions involved with square roots

Let $ F(s) = \sqrt(s+ia)g(s) $, where $s$ is a complex number with $\Re(s) >0 $. I want Laplace inverse of $F(s) $. I have tried contour integration and convolution theorem but couldn't come up ...
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Some sort of Laplace transform

Let $K$ be some compact set and $\mu$ be a probability measure on $K$. Let $f:K\to\mathbb S^{d-1}$ be a continuous function, where $\mathbb S^{d-1}$ is the unit sphere in $\mathbb R^d$ And $g:K\to\...
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convergence- Laplace Transform

The bilateral Laplace transform is defined as, $$X(s) = \int_{-\infty}^{\infty}x(t)e^{-st}dt~~,s = \sigma +j\omega$$ where both $\sigma$ and $\omega$ are real. Then, $$X(s) = X(\sigma+j\omega) = \...
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Laplace transform of “shifted” modified Bessel function

Dear all: i'm trying to derive a closed form the following integral, $$ X_n(R)=\int_0^\infty \exp(-p\, r)I_n(\omega (r+R))\, dr, $$ where $I_n$ is the standard modified Bessel function of the first ...
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Find the periodicity with the help of Laplace transform

I have a function $$x(t) = \pi\cos(21\omega_0t)+0.1\cos(39\omega_0t)$$ that I want to solve T from the periodicity identity, $x(t)=x(t+T)$. What I have tried now is basically just solving $x(t)=x(...
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Laplace transforms - second shift theorem

Hi, i'm struggling with understanding how this expansion came about for determining the LT of f(t). Is anyone familiar with this? thank you
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Confusion with this inverse Laplace Transform

I have earlier posted this question in Physics StackExchange but I feel that it is more relevant here. The question is about a contour integral and I have written most of the equations needed. Please ...
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Inverse Laplace Transform of $F(s)=\frac{1}{\sqrt{s} \coth(\sqrt{s})-1}$.

Dear All (best wishes for this new year 2019), I try to find the inverse laplace transform of the function $\displaystyle F(s)=\frac{1}{\sqrt{s} \coth(\sqrt{s})-1}$. I check numerically that this ...
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Finding conditions to an equality in an integral

First of all, a happy new year to everybody. Well, I'm working on a problem. That problem stated that I've to find the conditions for which this equality holds: $$\mathcal{I}_\text{n}\left(\epsilon\...
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Laplace transform of square root of a trigonometric function

Need help with this question from my university paper. My question : Find Laplace Transform of the following: $\sqrt{1 + \sin(4t)}$ I do know how to solve $\sqrt{1 + \sin(t)}$ By taking $1 = \...
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Integrate $\int_{0}^{\infty}e^{-pt}\sin\left(\sqrt{t}\right)\mathrm dt$

I need the following Laplace transform to solve the Differential Equation $$\int_{0}^{\infty}e^{-pt}\sin\sqrt{t}\, dt, \quad \text{where} \ \ \ p>0$$ I tried Integration by parts after ...
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Problem with 'simple' step function

I have a function in terms of a Heaviside function which I have worked out is: t-t.H(t-1). I need to get it into a form for Laplce transform . The answer to that is apparently t-(t-1)H(t-1)-H(t-1) I ...
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Laplace Transformation of sin(t+1)*DiracDelta(t)

why is Laplace transformation of sin(t+1)*DiracDelta(t) = sin(1)? I thought: L[f(t)-u(t-a)] = e^(-as)*L[f(t+a)] and according ...
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How can I solve this integral equation with the inverse Laplace Transform?

This question is related to Solving an integral equation with inverse Laplace transform. Let $\alpha,\beta,\mu>0$ with $\alpha/\beta>\mu$ and $X\sim\operatorname{Gamma}(\alpha,\beta)$. I am ...
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Laplace transform in probability question

A component has a failure rate of $\lambda$ (i.e. the probability of failure in period ∆t is λ∆t). Show that the probability that the component will have failed by time t is $F(t) = 1 − e^{−\lambda t}...
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how resolve this difference equation used Laplace transform?

Solve the equation or system of equations using the Laplace transform. I need a solution to this task. Anyone could do this for me? \begin{cases} x' +y' -x = 1 \\ x' + 2y' = 0 \end{cases} $$x(0) = ...
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Am I using the Laplace Transform pair correctly?

I am given a question of Laplace Transform which is as follows: $$ e^{t}sin2t $$ for $$t\leq0$$ Now I know that by using the transform pair we get: $$ [e^{-at}sin\omega_0 t]u(t) \rightarrow ...