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

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Having trouble finding inverse Laplace Transform

I have this Laplace transform $$X(s)=\frac{1}{s\cdot(s^{2}+0.2s+1)}$$ I want to find its inverse transform. I did the following. First I decomposed that into partial fractions $$X(s)=\frac{1}{s}-\...
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25 views

Laplace and unit step- multiplication vs convolution

Please be gentle if the question is stupid. When using the laplace transform, you often multiply the function of interest by a shifted unit step function to operate on the positive portion of the ...
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Prove this inequality using laplace transform

Let $n$ be a strictly positive integer, and $a_1,\cdots a_n,b_1,b_2 \cdots b_n$ strictly positive real numbers. Prove that $$\sum_{i=1}^n (\frac{a_i}{b_i})^2 \le 2\sum_{1\le j,i \le n} \frac{a_ia_j}{(...
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52 views

Pde using laplace transform

Could you help me to find a solution for this partial differntial equation by using laplace transform $$u_{t} - u_{xx} = xt$$ where $$u(0,t)=t, \quad u(1,t)=t^2, \quad u(x,0)= \sin \pi x$$ I tried ...
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60 views

Is the Laplace transform essentially a generalized version of the Fourier transform?

My current understanding of the two concepts is far from perfect, and I am essentially just a beginner. But it seems to me that while the Fourier tries to decompose functions as a composition of ...
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Find f(x) $\int_0^x f(u)du - f'(x) = x$

Find f(x) $$\int_0^x f(u)du - f'(x) = x$$ I was not given f(0) which makes it difficult for me to find f(x). This is what I have thus far: $$\frac{F(p)}{p}-pF(p)+f(0)=\frac{1}{p^2}$$ $$\frac{F(p)}{...
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Laplace Transform: $g(x)=a\sin(x)+\int_0^x \sin(x-u)g(u) du$ [closed]

$$g(x)=a\sin(x)+\int_0^x \sin(x-u)g(u)du$$ I need to find $g(x)$ I believe I need to use Laplace Transform with this in mind (Convolution Thm): $$(f*g)(x)= \int_0^x f(x-t)g(t)dt$$ However I don't ...
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48 views

Two approaches to problem give different answers — which one is correct?

I approached this problem in two ways and arrived at different answers. Both ways seem logical to me. Are they both correct, or is one flawed? This is the original problem: $$\mathfrak L^{-1} \lbrace{...
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63 views

Laplace transform of $\cos(at)/t$

If someone could help me solve for $$\mathcal{L}\left\{\frac{\cos(at)}{t}\right\}$$ it would be great. Step-by-step I have so far: $$\begin{align}\int_0^\infty \frac{\cos(at)\space e^{-st}}{t}\space\...
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35 views

Complex inversion of a function

I am trying to find the function whose laplace transform is below using the complex inversion formula: $$ f(s)= \frac{se^s}{(s-2)^3}$$ My attempt below seems to be giving me the wrong answer but I'm ...
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Existence/Uniquness/Solution of a countably infinite system of linear ODEs.

Consider the following system of ODE's. \begin{align*} \frac{\mathrm{d}^2}{\mathrm{d}t^2} x_0 &= F(t) + x_1\\ \frac{\mathrm{d}^2}{\mathrm{d}t^2} x_i &= x_{i+1}-2x_i+x_{i-1}\, \quad \mathrm{...
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43 views

Laplace transformations on a homogeneous ODE

$$y^{\prime\prime} - 3y^{\prime} + 2y = 0$$ $y(0) = 14$, $y^{\prime}(0)=0$, and using the Laplace transformation I'm trying to solve this IVP
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Matched pole zero discretization

There are several techniques to discretize continuous-time transfer functions to discrete-time transfer functions. Some of them, such as, zero-order-hold, forward euler or Tustin, are well known. ...
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46 views

Solving Second-order non-linear ODE, with fractional expansions

I am solving a differential equation related to fluid mechanics, a rigid air bubble rising towards the surface of a liquid. Doing all of the maths, I have come to this differential equation, which I ...
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16 views

Proving that if $f \in \mathcal{E}$, then $f' \in \mathcal{E}$ (same for $\mathcal{E}_q$)

In the context of ordinary differential equations, I'm trying to prove that if some function $f$ is an element of $\mathcal{E}$, which is the function space of all exponential polynomials, then $f'$ ...
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27 views

Laplace transformation on an exponential

Using the definition of Laplace transformation (and without using a table), how to find the Laplace transformation of $$ g(t)= \begin{cases} 0,&\text{if }0\leq t\leq 4;\\ e^{3t}&\text{if }4\le ...
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46 views

inverse Laplace transform of gamma function

My problem is to get the inverse Laplace transform of the following equation. $$\hat{P}(s) = \frac{\Gamma(p+1+s T)}{p! N^{s T}}$$ $p$, $T$ and $N$ are positive constants. The denominator $N^{-s T}$ ...
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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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50 views

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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28 views

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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30 views

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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55 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) = \frac{1}{1-...
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30 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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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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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[ \frac{\sinh(\alpha\sqrt{...
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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 $L\{(u(t-\pi/6)f(t-\pi/6)\}=...
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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 g)...
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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 $\mathcal{L}[e^{at}f(t)]...
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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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42 views

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 t}+bg(x,t)\left[g(x,t)-f(x,t)+C\...
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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] {y=n\left(\frac{e^{-n(t-t_0)}}{n^2+\omega^2}+\...
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36 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) = \frac{F(s)}{E_c(s)}=\frac{100}{s}-\frac{...
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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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29 views

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 gotten. ...
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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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25 views

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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26 views

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
24 views

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: $$\mathcal{L}_{t}\...