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

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Partial fraction decomposition of $\frac{21}{s^{2}+4}$ for inverse-Laplace transform

So I have this number which I want to do inverse-Laplace transformation on, which is kind of complicated. So it would be easier to do some partial fraction decomposition first. I am trying to do the ...
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Solving for $x$ in a Laplace equation

So I have this Laplace equation: $$s^{2}x+4sx+5=\frac{s}{s-1}$$ And I want to solve for $x$. My result is the following: $$x = \frac{5-4s}{s^{3}+3s^{2}-4s}$$ Which is also the same answer that for ...
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Solving Bessel's equation by Laplace transform

I am learning Bessel function the solution of Bessel equation by book 'Advanced Engineering Mathematics' by Peter V.O'Neil and here i found its derivation by Laplace transform. In this derivation of ...
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Laplace transform identity $F(s) = \mathcal{L}(t^{-3/2} \mathrm{e}^{-1/t})$

I'm asked to prove the following result: If $F(s)$ is the Laplace transform of $f(t) = t^{-3/2} \mathrm{e}^{-1/t}$, show that $F'(s)=-s^{-1/2}F(s)$. I'm having a lot of troubles to prove this ...
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Laplace Transform of Dirac Delta function

I've seen everywhere that that the Laplace Transform of Dirac Delta function is: $$L[\delta(t-a)] = e^{-sa} \text{ when } a > 0$$ But they never explain what happens when $a < 0$. Can I assume ...
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Laplace Question $f(t) = e^{-t} \sin(t)$

I need help with this Laplace question. \begin{equation} f(t) = e^{-t} \sin(t) \end{equation} Answer should be $\dfrac{1}{s^2 + 2s + 2}$ What I'm currently doing is as follows: $u = \sin(t)\qquad$ ...
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Laplace transform problem with heaviside functions

Find the Laplace transform of (a) $[u(2pi/3)(t)]e^{-3t}cos(4t)$ (b) $[u(2pi/3)(t)]e^{-3t}(t)cos(4t)$ [Hint: Use the result from (a)] u is the heaviside function. For part a I got an answer of ...
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Tauberian theorems in queing theory

I'm trying to use Tauber's theorem below (Feller 1971, chapter XIII.5) "Let U be a measure with a Laplace transform $\omega(\lambda)$ defined $\forall \lambda >0$ and $t,\tau>0$ s.t. $t\tau=1$, ...
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42 views

Laplace Transform of Square Wave Function

I am given a problem in my textbook and I am left to determine the Laplace transform of a function given its graph (see the attached photo) - a square wave - using the theorem that $$F(s) = ...
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26 views

How to write a transfer function (in Laplace domain) from a set of linear differential equations?

Provided I have a system of linear differential equations (in time domain) such as: $$\begin{cases} \dot{x}(t)=Ax(t)+By(t)+Cz(t)\\ \dot{y}(t)=A'x(t)+B'y(t)+C'z(t)\\ \dot{r}(t)=B''y(t)\\ \end{cases}$$ ...
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23 views

Dynamic real-time system problem

I am struggling with a systems theory problem, the task is as follows: u(t) -> H(s) -> y(t) H(s) being the transfer function $$ H(s) = H(s) = \frac{s+1}{s(s+2)^{2}} $$ $$ u(t) = e^{-5t} $$ So ...
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22 views

Why M.G.F transform is injective a.s.?

We always use the theorem that If we know a random variable's MGF, we can determine its Pdf, which means the map from Pdf to Mgf is injective almost surely. And I just wanna know why this is ture.
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What topics should I study to understand Laplace transform?

If I'm a beginner to start understanding Laplace transform, from where should I start to understand Laplace Transform?
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How to evaluate integral $\int_0^{\infty} e^{-x^2} \frac{\sin(a x)}{\sin(b x)} dx$?

I came across the following integral: $$\int_0^{\infty} e^{-x^2} \frac{\sin(a x)}{\sin(b x)} dx$$ while trying to calculate the inverse Laplace transform $$ L_p^{-1} \left[ ...
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What is the laplace transform of $\delta(t-\pi /6)\sin (t)$

What is the laplace transform of $\delta(t-\pi /6)\sin (t)$ I know that $L\{\delta(t-\pi/6) \}=e^{-s\pi/6}$ I also know that $L\{\sin (t) \}=1/(s^2+1)$ I also know that ...
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Does $e^{1/t}$ have a Laplace Transform?

I'm having a little trouble understanding why some functions have a Laplace transformation and others don't. The definition I was given in class last week was "Given a suitable function $F(t)$ the ...
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Proving that the Laplace Transform is an isomorphism with convolution

My question is primarily more about the convolution integral/theorem than the proof in question, but I wanted to give some idea of why I'm asking. The Laplace transform of the convolution $$(f\star ...
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If $\mathcal{L}[f(t)] = \hat f(p)$ then $\mathcal{L}[e^{at}f(t)] = \hat f(p+a)$

Let $f:[0,\infty) \rightarrow \mathbb{R} $ be continuous, with the property that $f(t)e^{-pt} \rightarrow 0$ as $t \rightarrow \infty$. If $\mathcal{L}[f(t)] = \hat f(p)$ then ...
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Inverse Laplace Transform of $e^{\frac{1}{s}-s}$

doing some work on a PDE system I have stumbled across a Laplace transform which I'm not sure how to invert: $$ F(s) = e^{\frac{1}{s}-s} $$ I can't find it in any table and the strong singular growth ...
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Solution of partial differential equation - modified heat equation

I want to solve the "modified" heat equation $$ \frac{\partial y}{\partial t}=a\frac{\partial^2 y}{\partial x^2} +b\frac{\partial y}{\partial x} +cy+d $$ I assumed that a, b, c and d are all constant ...
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How to solve nonlinear partial differential equation with two variables

somehow, I got this partial differential equation but I don't know how should I start. $$ a\frac{\partial f(x,t)}{\partial x}\left[ \frac{\partial g(x,t)}{\partial ...
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In the context of Laplace transforms, what does the subscript in $h(t) = f(t)\cdot u_3(t)$ signify?

Problem Note: I do not need help solving this problem (yet), but I'm unsure about notation. Find the Laplace tranform of the function $h(t) = e^{2(t-3)}u_3(t)$. Question What does the subscripted ...
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How do you find the Inverse Laplace transformation for a product of $2$ functions?

If $$\mathscr{L}(y)=\frac{ne^{-pt_0}}{n^2+\omega^2}\left(\frac{1}{p+n}+\frac{n}{p^2+\omega^2}-\frac{p}{p^2+\omega^2}\right)$$ show that $$\bbox[yellow] ...
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35 views

Find a Lebesgue integrable function which satisfies a convolution equation

Let $f:\mathbb{R}^n \to \mathbb{R}$ be a non-negative Lebesgue integrable function with integral on $\mathbb{R}^n$ equals to 1. Let $\tilde{f}(x)=f(-x)$. Suppose $f$ satisfies the following equation: ...
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Interchanging integral and derivative operations in the context of Duhamel's formula

I'll give you the whole context: In solving the heat equation $u_t = ku_xx$ with bounds $u(x,0)=0, u(0,t)=0, u(l,t)=f(t)$, let $v(x,t)$ be the solution for the special case $f(t)=1$. Use the Laplace ...
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Transfer function of controller

I am solving this question given in book (Automatic control system). As asked in (a) part $G_c(s)$ of the controller. I solved it and getting answer$$G_c(s) = ...
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56 views

Why is it justifiable to use contour integration to find the inverse Laplace transform?

I asked this on Quora, but I want to see what the answers here will be. I've always wondered why it is possible to represent the inverse Laplace transform as a contour integral.
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Operations with Fourier Transforms

If I have a expression, say $\frac{\partial }{\partial x}\frac{\partial A(x)}{\partial x}$ and I applied the derivative theorem to the second term, such that it becomes $\frac{\partial }{\partial x} ...
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Differential Equation Theory and Laplace Transform

I have this problem and I'm stuck with what to do. I know solutions to the homogeneous equations are $$y_1(t)=C_1e^{\alpha_1 t}$$ $$y_2(t)=C_2e^{\alpha_2 t}$$ but that's about as far as I've ...
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Region of Convergence of a finite signal [closed]

How would I go about proving that the ROC of any finite duration signal consists of the entire complex S plane?
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Find the Inverse Laplace Transform of the following

I have a Laplace tranform in the form given below $\mathcal{L}_I(s)=\text{exp}(-\pi\lambda \Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})P^{2/\alpha}s^{2/\alpha})$ Can some one help me to find ...
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How to find the Inverse Laplace Transform of the following?

I have a Laplace tranform in the form given below $\mathcal{L}_I(s)=\text{exp}(-\pi\lambda \Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})P^{2/\alpha}s^{2/\alpha})$ Can some one help me to find ...
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1answer
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Laplace Transform of a this definite integral

What is the Laplace Transform of, with $t\in\mathbb{R}^+$: $$\int_{0}^{t}\text{U}_{\text{in}}(t)\space\text{d}t$$ I know that the Laplace Transform of a indefinite integral is: ...
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Transform with tensor product

I'm new to Laplace and Fourier transforms when convolution is involved, and I've never seen an example involving a tensor product. I'd like to see how the Fourier transforms of the following would ...
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Solving a electronics circuit with Laplace Transform

I've the following problem, it's maybe a electronics problem but I've to make a mathematical model so that's way I paste it here. Find the transfer fuction of the following circuit ...
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Find the original function by using convolution theorem

Seems like I don't know how to apply convolution theorem on this problem properly, I would appreciate some help and a brief explanation how did you solve it if you do it. ...
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Find the inverse Laplace transform of $L(s)= \frac{s}{s^2 + 25} e^{-\pi s}$

$$L(s)= \frac{s}{s^2 + 25} e^{-\pi s}$$ I never seen such function. Can exponential function appear in Laplace transform? Help required
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Laplace Transform Injectivity

Intuitively how can the Laplace transform be injective? You are taking an integral with limits $0$ and $\infty$. So you don't care about the function before $0$. Define $g(x)=x^2$ for $x>0$ and ...
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How and why an integral Transform is created?

I don't know if what I'm going to ask will make any sense, but I was just wondering about integral transforms. I am talking about, for example, Mellin Transform, or Laplace Transform or Hilbert ...
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Problem with convolution, insecure

$$f(t)= t^2\cdot u(t),\quad g(t)=t^4\cdot u(t)$$ I know that I need to use convolution theorem to solve this problem, but I really don't know what to do with step functions. Do I need to include ...
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How to calculate inverse laplace of $e^{a\sqrt s}$?

I was using Laplace to find solutions for $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions $$u(0,t)=1 \\ u(1,t)=1 \\ u(x,0)=1+ \sin \pi x$$ I used ...
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Laplace transform second shifting rule

I have found seccond shifting rules with and without the Heaviside function. Even my professor taught us the one without Heaviside. For example if $f(t)=2e^-e(t-3)$ 1)Rewrite this expression as ...
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Polar System with Short Answers, How $U(0, \theta)=\pi$ will be calculated?

I read some notes on Laplace. I ran into a short answer question as follows. We have a Laplace equation in Polar Systems: $\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial u}{\partial ...
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Laplace Transform of $e^{t^3}$

I have to find the Laplace transform of $$e^{t^3} u(t)$$ and I know that $u(t)$ will just change the integral from negative infinity to positive infinity to $0$ to positive infinity, but I'm stuck ...
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Laplace transform for $-t\cos(2t)$

This Laplace transform exercise is giving me a headache. I was trying to use the definition of the Laplace transform but when I make the $u$ and $dv$ substitutions for the integration by parts I never ...
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Different answer when using the 'method of undetermined coefficients' compared to Laplace transform

I have an ordinary differential equation: $$ \frac{\mathrm{d}^2u}{\mathrm{d}t^2} + u = \mathrm{e}^{-t}\cos(t)$$ with $u(0) = u_0$ and $\dot{u}(0) = v_0$, when using the method of undetermined ...
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Initial value Laplace Transform exercise

I'm having trouble with the following exercise $$ y'' +4y - (4/e^x) = 0 $$ with the initial values: $$ y(0) = 1 y'(0)=5$$ I used the formula $$ y'' = s^2Y(s) − s*f(0) − f'(0)$$ and got to: $$ ...
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Why is Laplace Transform used for ODEs

This part is taken from differential equations with applications and historical George simmons. According to the given information , there are another integral transformation.I wonder why is the ...
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Moment generating function and convergent random variables

denote by $X$ and $X_n$, $n\in \mathbb{N}$, random variables and $r\in\mathbb{R}_+$ with $E=\mathbb{E}\left[ e^{rX} \right] < \infty$ and $E_n=\mathbb{E}\left[ e^{rX_n} \right] < \infty$ for all ...
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Laplace transform and value in x(0)

Somebody told me that if i have something like this: $x''(t) + x'(t) = -2x(t) + u$ $x(0) = 7$ and use laplace transform on it i will get $s^2X(s) + sX(s) = -2X(s) + U(s)$ next i'm getting ...