The Laplace transform is a widely used integral transform, similar to the Fourier transform.

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Solve a very simple second order ODE using Laplace Transforms. (5.4-9)

My question is where did I go wrong? I cannot seem to duplicate the answer given in text. The Problem: Solve the following differential equation using Laplace Transforms given that $y(0)=0$, ...
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How does the laplace transform diagonalize the derivative operator?

I was reading this post here and I got really confused at the part where the claim is that the laplace transform diagonalize the derivative operator ...
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Laplace–Stieltjes transform and renewal equation.

I'm wondering if somebody could check if I'm going about this the correct way. I have renewal equation, $$Z = z + H*Z$$ where $$Z(t) = \mathbb{P}(L > t),$$ $$z(t) = (1 - G(t))(1 - F(t)),$$ and ...
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Solve for the charge on a discharging capacitor in an RC circuit using Laplace Transforms. (5.3-61)

Please check my work. I need to solve the following problem but my answer varies from that of the book by a factor of $C$ for capacitance. A print screen of the problem is given below. Problem to ...
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Find inverse Laplace Transform using s-shifting and t-shifting. (5.3-57)

Please check my work. I need to find the inverse Laplace Transform for the following function. $$\mathcal{L}\{f(t)\}=\frac{e^{-s}}{s^2+\pi^2}$$ My solution: Recognizing the exponential factor as a ...
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Find inverse Laplace Transform having s-shifting and t-shifting. (5.3-56)

Please check my work. Did I calculate the following inverse Laplace Transform correctly? Our Laplace Transform is... $$\mathcal{L}\{f(t)\}=\frac{e^{-\pi s}}{s^2+2s+2}$$ My solution: Recognizing the ...
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111 views

Residue Theorem for Laplace Transform

I need to know what's the Residue Theorem for a Laplace Transform. Does anyone know the name or something, so I can search it? I couldn't find anything. For example, if I have this two equations: ...
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28 views

Find Laplace Transform of a windowed ramp function using unit step function and t-shifting. (5.3-44)

Please check my work. Did I calculate the following Laplace Transform correctly? Our function is $f(t)=t$ where $(1<t<4)$ and $f(t)=0$ everywhere else. My solution: First express function with ...
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Find Laplace Transform of exponential function using unit step function and t-shifting. (5.3-42)

Please check my work. Did I calculate the following Laplace Transform correctly? $$f(t)=e^{kt}u(t-a)$$ My solution: Use the following corollary from the second shifting theorem (t-shift)... ...
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Find Laplace Transform of trigonometric function using unit step function and t-shifting. (5.3-40)

Please check my work. Did I calculate the following Laplace Transform correctly? $$f(t)=sin(t)u(t-\frac{\pi}{2})$$ My solution: Use the following corollary from the second shifting theorem ...
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20 views

Laplace Transform of (t - 120) u(t - 120)

I am puzzled by what should be a simple problem. I am deriving the Laplace transform for $$ (t-120) u(t-120) $$ where $u(t)$ is the unit step signal. Using the second shift theorem, ie $$ ...
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Regarding the unilateral Laplace transform of LTI systems

Consider an LTI system described by the following differential equation, $$ \sum_{k=0}^{N}a_k\frac{d}{dt^k}y(t) = \sum_{k=0}^{M}b_k\frac{d}{dt^k}x(t) $$ With initial conditions, $$ y(t)|_{t=0}, ...
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laplace transform multiplication by power t

How to solve Laplace transform of $\displaystyle t^\frac{5}{2} e^{4t}$ . I know that this can be solve by multiplication by power of $t$ but how to differentiate $\frac{5}{2}$ part please reply
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29 views

Inverse Laplace of $\frac {s}{RCs+1} $

I was wondering how you would be able to solve the inverse laplace of $$\mathcal{L}^{-1}\left\{\frac{s}{RCs+1}\right\}\left(s\right)\tag{1}$$ where $R$ and $C$ are constants?
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How to solve this simple inverse Laplace transform

I haven't touched Laplace transforms in quite some time so I'm very rusty. I'm trying to reverse: $$ \frac{-(s+5)}{5(s+1)} = \frac{-s}{5(s+1)} + \frac{1}{(s+1)}$$ Obviously, the second term is $ ...
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Is it feasible to think of laplace transform and z transform as projections?

For Fourier transform, it has been ingrained in my head that all we are doing is projecting a function onto its Fourier basis, namely $(1, cos(t), sin(t),...cos(nt), sin(nt) ...)$ Can anyone comment ...
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Expressing a piecewise defined functions as a unit step function

I am trying to express the following function as a unit step function so that I can use Laplace: $ f(x) = \left\{ \begin{array}{lr} 0 & : t < 1\\ t^2-4t+5 & : 1\leq t ...
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Find Laplace Transform using unit step function and t-shifting. (5.3-38)

Please check my work. Did I calculate the following Laplace Transform correctly? $$\mathcal{L}\{t^2u(t-1)\}=\mathcal{L}\{(t-1)^2u(t-1)\}+\mathcal{L}\{u(t-1)\}$$ ...
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36 views

Find Laplace Transform using unit step function and t-shifting. (5.3-35, 5.3-36)

How do the Laplace Transforms vary between the two following functions? What I am really asking is if I calculated the following Laplace Transform correctly... ...
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Find Laplace Transform using unit step function given graph of a periodic impulse function. (5.3-33)

Please correct my work. The textbook answer which is expressed exactly like this $1/s(1+e^{-s})$ does not match my own. Find the Laplace Transform for one period of the perpetual periodic function ...
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Laplace Transform To Solve IVP

I need to use the Convolution theorem to solve $y'' +4y' +4y = g(t)$ with initial conditions $y(0)=2, y'(0)=-3$. This is what I have but it differs from the answer in the text so I'm wondering where I ...
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Laplace transform of x(t)^2*x'(t)

I got a result that doesn't seem correct, so I'm hoping someone can tell me if I went wrong somewhere (probably with the integration by parts or in the second to last line). $$ ...
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An Inverse Laplace Transform Problem

I am having problems solving this inverse Laplace transform: ℒ$^{-1}\Large [\frac{s-3}{s[(s-3)^2+9]}]$ I did partial fraction decomposition, but ended up with complex expressions in some ...
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69 views

The Laplace transform of the Heaviside function

I am studying complex analysis but, because I'm an engineer, I have a lot of doubts. I'm going to present my doubts and it would be nice if someone helps me to see things clearly. Let's start with ...
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Solve for a hyperbolic Laplace Transform by expressing as exponents and shiftig on s-axis (5.3-21)

I cannot get past a certain point on this problem as shall be shown. I need guidance in order to complete the problem. The exercise as stated in the text: Represent the hyperbolic function in terms ...
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Inverse Laplace Transform of $e^{c \cdot s^2}$

I am trying to find the Inverse Laplace Transform of the function $$ F(s)=e^{c \cdot s^2} $$ where $c > 0$.
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How do you find the state space representation of $G(s) = \frac {1}{s^2+s+1}$

Let $G(s) = \frac {1}{s^2+s+1}$ be the transfer function of the system Then $Y(s)(s^2+s+1) = U(s)$ Therefore $y'' + y' + y = u$ After this step, how should I set up my state transition variable $x$ ...
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44 views

Redundancy in the Laplace transform and Mellin's inverse formula

As I understand it, Mellin's inverse formula relates a sufficiently 'nice' function $f$ and its Laplace transform $F$ as follows: $$f(t)=\frac1{2\pi i}\lim_{T\to\infty}\int_{-T}^{T}e^{i\omega ...
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Example: How to find inverse Laplace Transform by integral of the function (5.2-29)

This is just a demonstration on how to solve the following type of problem. Find $\mathcal{L}^{-1}\{\frac{54}{s^3(s-3)}\}$ by the given method: $$\mathcal{L}\{ ...
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Solution of a heat transport PDE

Solve the system of partial differential equations: $$(1)\space\space \frac{\partial g}{\partial t} + v\frac{\partial g}{\partial x} = -k_1\left(g-h\right)$$ $$(2)\space\space \frac{\partial ...
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Laplace transform notation

I'm confused about the notation used, pretty much everywhere, to describe what a Laplace transform it. Wikipedia says something along the lines of "..Laplace transform of a function $f(t)$..", ...
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inverse laplace transform of F(s)

Let $f(x)$ be some arbitrary function , $F(s)$ is laplace transform of it I think inverse laplace transform of $\frac {F(s)}{s+r}$ where, r is constant may be $\int_0^t e^{-rt'}F(t') dt'$ ...
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Inverse Laplace transform of $s^{\beta-1}/(s^{\beta}+a)$ [closed]

I am stuck at calculating the inverse Laplace transform of $$\frac{s^{\beta-1}}{s^{\beta}+a}$$ where $0<\beta<1$ and $a>0$. Thanks
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Derive inverse Laplace Transform using two given trigonometric transforms (5.2-13)

I am not certain how to begin this problem. Someone please point me in the right direction. Problem Using the two given formulas ($1$ and $2$ below) show that: ...
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Why aren't my Laplace transform and Undetermind Coefficients answers matching up?

I might be losing my mind this morning (I am, for sure), but I can't these two techniques to give me the same answer to a basic differential equations problem. The problem is $y''-8y'+27y=0$ with the ...
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Methods of Solving Ordinary Differential Equations - A Small Question

I've spent some weeks now trying to learn how to solve ordinary differential equations, and I am now studying the Laplace transform and how this can be applied to solve ODEs. I feel a little bit ...
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Example: How to set up integral and find a Laplace Transform given two straigh lines (5.1-7).

This is an example on how to find the Laplace Transform for a graphical problem. The textbook solution we wish to derive is: $$F(s)=\frac{-e^{-s}}{s}+\frac{1-e^{-s}}{s^2}$$ We begin by expressing ...
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Computation of two-sided probability density functions from their cumulants using Laplace transform

The computation of one-sided probability density functions (PDFs) from their cumulants using Laplace transform is proposed by following paper: M.N. Berberan-Santos, Journal of Mathematical Chemistry, ...
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Connection between the Laplace transform and generating functions

As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've ...
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How to visualize bilinear transform of the form $S(z) = \frac {T}{2} \frac {z+1}{z-1}$

Note that this is the bilinear transform from a z-domain as appears in Z-transform to s-domain in Laplace transform Recall that bilinear transform has form $M(z) = \frac{az+b}{cz+d}$ with and has to ...
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yet another simple Laplace transform

what is $ℒ(t^2e^{3t})$ I have got this far so far: $=\int_{0}^\infty (t^2e^{t(3-s)})$ Integration by parts using: $u = t^2$ and $du = 2t$ $v = \frac{e^{t(3-2)}}{3-s}$ and $dv = e^{t(3-s)}$ Which ...
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Proving completeness of the average of a random normal sample

Suppose that $n$ is a fixed positive integer and $\theta$ is a parameter belonging to $\Theta=\mathbb{R}$. Suppose that we are given that $Y_1,\ldots,Y_n$ are i.i.d. $N(\theta,1)$. I'm trying to show ...
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Laplace Transform of $e^{a t^2}$

What is the Laplace transform of $e^{a t^2}$, for positive $a$? In order for Laplace transform to exist function must be locally integrable. Since integral of any compact set $e^{a t^2}$ is finite ...
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Why is the Z-transform of $e^{at}$, t = kT, different from Laplace transform of $e^{at}$

The Laplace transform of $e^{at}$ takes a well known form of $\frac{1}{s-a}$ The Z transform of $e^{at} = e^{akT} $ T is the sampling period takes the form of $\frac{z}{z-e^{aT}}$ Does anyone know ...
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Laplace-Fourier transform issue

Given a function $f:\mathbb{R}\rightarrow\mathbb{R}$ we take the generalised Fourier transform $\hat{f}(w)=\int_{-\infty}^{+\infty}e^{iwx}f(x)dx$ where $w\in \mathbb{C}$. Now assume, this transform ...
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relationship between laplace transform and its derivative

By definition, the Laplace transform of a function $f$ is given by, $$ L(f)(\lambda) = \int_0^\infty e^{-\lambda s}f(s) ds .$$ My question is two fold. I need help in findding the derivative of ...
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60 views

What does (0+) mean?

I'm currently learning from a script (which is written in German and not publicly available, sorry) for introduction to stochastics, where the topic is the Laplace transformed function for random ...
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Prove that the Laplace transform of $I_0\left(2\sqrt t\right)$ is $\exp\left(1/s\right)/s$

Wolfram Alpha gave me the answer to this, but unfortunately Wolfram Alpha doesn't show its work, I can't find a proof anywhere else, and my feeble attempts to show it myself went nowhere. How can it ...
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regarding the integral form of the inverse Laplace transform

The integral form of the inverse Laplace transform is given as $$f(t)=\frac{1}{2\pi i}\int_{s'-i\infty}^{s'+i\infty}e^{st}F(s)ds$$ where $s'$ is larger than the real parts of all the possible ...
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Beginner question about existence of Laplace transform

I am having problems understanding why a Laplace transform exists or not. Here is my math and logic, hopefully someone can point out where I am wrong. $$f(t)=e^{at} \implies ℒ[e^{at}] = ...