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

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Calculate the laplace transform…

Calculate the laplace transform of $$t^2u(t-2)$$ I don't know how to manipulate t^2 in order for it to meet the form of the product between a function and a heaviside function. Number (27) on ...
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How to find $\mathcal{L}^{-1}\left\{1\right\}$?

This is probably a really simple question, but I cannot figure it out and it's not mentioned in my books. How do I find \begin{align} \mathcal{L}^{-1}\left\{1\right\}?\tag{1} \end{align} It seems like ...
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Does this strategy look correct to you (solving for probability density function with three Random Variables)

The following formula is a formula I got from a paper that deals with wireless networks specifically when calculating coverage probabilities - if needed I can provide the reference- it is powerful ...
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Help with inverse Laplace transform

Please help me to find the inverse Laplace transform of : $$ \frac{1}{S(aS^2 + bS + c)} \left( 1 - \exp (-TS) \right) $$ where a,b,c,T are constants thanks
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Laplace transform of a differential equation??

Find unique solution of $y′′ + y = f$ using $y(0) = y′(0) = 0$ and periodic function $f(t) = t$ if $0 \leq t < 2\pi$ Attempted work: $L[y'' + y ] = L[f(t)]$ $L[y''] + L[y] = L[f(t)]$ $s^2 L[y] ...
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What is the Laplace transform of this random variable?

Define a random variable that takes only one value for example $$X=c$$ where c is a positive constant. What does the Laplace of it evaluate to i.e the following $$\mathcal{L}_X(s)= ...
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44 views

Solving PDE by Laplace Transform

Use Laplace transforms to solve the boundary value problem $$Y_{xx}(x,t)-2Y_{tx}(x,t)+Y_{tt}(x,t)=0, \quad 0<x<1, t>0$$ $$Y(x,0)=Y_t(x,0)=0, \quad 0<x<1$$ $$Y(0,t)=0, \ Y(t,1)=F(t), ...
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Laplace transform of a differential equation?

Find the unique solution of $y''+ y = f$, $y(0) = y'(0) = 0$ with the $2\pi$ periodic function given by $f(t)=2\pi \sin(t)$. I am having trouble setting up and starting the the question. I would be ...
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1answer
31 views

Using complex analysis to find the Inverse Laplace transform

I have been reviewing for my comprehensive graduation exam where I have been solving the Inverse Laplace transform via complex analysis. Consider $$ H(s) = \frac{s^2 - s + 1}{(s + 1)^2} $$ Then we ...
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39 views

Laplace transform of $f(t)=\left|\sin\frac{t}{2}\right|$?

If you are given a rectified sine wave, $$f(t)=\left|\sin\frac{t}{2}\right|$$ how do you find the Laplace transform of this? I tried using the equation $$L\{ f(t)\} = \frac{1}{1-e^{-sT}} \int_0^T ...
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37 views

Laplace transform of a sawtooth wave

Find the Laplace transform of the periodic function such that $f(t) = t$ if $0\leq t < 2\pi$ I am having trouble setting up this question. Am I on the right path? $$ \mathcal{L}\{f(t)\} = ...
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1answer
38 views

How to determine $2\pi$ periodic function?

Let $f(t) = 2\pi \sin t$, and determine a $2\pi$-periodic function $y^∗$ with the property that $\lim_{t\to+\infty} |y(t) − y^∗(t)| = 0$ for every solution $y$ of $y′ + y = f$. I am having trouble ...
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Heaviside function in the function whose Laplace transformation is $e^{-(\gamma+s)}/[(s+\gamma)^2+b^2]$

This is from a homework question 13.22 part (c) from "Mathematical Methods for Physic and Engineering" by Riley et. al on p. 464 I don't understand why the heaviside function is in the solution to ...
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1answer
22 views

Laplace transform of $tf'(t)$

I know that $\mathcal{L}(tf'(t)) = -\frac{d}{ds}\mathcal{L}(f'(t))$ and that this $= -\frac{d}{ds}(sF(s) - f(0))$ but the solution says that this becomes $-F(s) - F'(s)$ and I can't figure out why ...
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IVP with Laplace Transform

My attempt: Y = Laplace $$s^2Y -sy(0) - y'(0) - 3Y = ??$$ How do I set up $$h(t)$$ in the form of laplace?
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Sign problem, Laplace transform of sin(at)

I have a problem in my integration by parts but I can't find it: $$L(\sin(\alpha t)) = \int_0^{\infty}\sin(\alpha t)e^{-st}dt$$ $$= -\frac{1}{\alpha}\left[\cos(\alpha ...
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1answer
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Difference between the Rectangular “Window” Function and the Rectangle Function

I'm getting ahead in my differential equations textbook (Fundamentals of Differential Equations by Nagle et. al) and in the chapter of Laplace Transforms it states that the rectangular window function ...
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Partial Fraction Decomposition — Inverse Laplace Transforms

I apologize if this is a rather lame question, but I've always been a little touchy with my partial fraction decompositions and I'm hoping to get better at them. Could you verify (or correct?) my ...
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Find $\mathcal{L}\left\{\cos^3\left(t\right)\right\}$

I began by breaking the problem up as follows: \begin{align} \mathcal{L}\left\{\cos^3\left(t\right)\right\}=\int_0^\infty e^{-st}\cos^3\left(t\right)\:dt & = \int_0^\infty ...
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Black's formula and feedback system stability

Consider a hypothetical system with open-loop transfer function $G(s)$. Place it in positive feedback with unit gain. (That is, take its output and directly add it to its input.) The closed-loop ...
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Laplace Transformations and Piecewise Functions

I am trying to understand why it is that Laplace transformations can simply be "added together" when finding the transform of a piecewise function. My professor has quite extensively talked about the ...
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How can i get the Transfer function of this Equation?

I have an equation with i want to simulate in Matlab. the Equation is y(t) = a/(b+x(t)) here x(t) is the input and y(t) is the output and a b are some constant. I don't know to to make its Transfer ...
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Compute the inverse fourier transform of $ e^{-Af} $ and $ e^{-A\sqrt{f}} $

I want to compute the inverse fourier transform of $ e^{-Af} $ and $ e^{-A\sqrt{f}} $, where $A$ is a constant and $f$ is frequency. In the case of $ e^{-Af} $, I tried to solve it from the Fourier ...
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Correct partial fraction construction?

Is the below the correct partial fraction decomposition? $$\frac{s^2 - 6s + 9}{(s-2)^3}=\frac{A}{s-2}+\frac{B}{(s-2)^2}+\frac{C}{(s-2)^3}$$ I can see that the numerator doesn't have a factor of ...
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22 views

Laplace transform with the Heaviside unit step function

I want to find the laplace transform for the function: $$f(t) = \left\{\begin{array}tt,\quad t\lt 2 \\ t^2 , \quad t\geq 2 \end{array} \right.$$ So I thought that the proper setup was: ...
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32 views

Find $\mathcal{L}\left\{t e^{2t}\cos\left(5t\right)\right\}$

This is what I have so far: \begin{align} \mathcal{L}\left\{t e^{2t}\cos\left(5t\right)\right\}=\int_0^\infty e^{-st}t e^{2t}\cos\left(5t\right)\:dt,\tag{1} \end{align} but notice that if ...
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Shortcut methods for Partial fraction decomposition in IVPs solved by Laplace transformation?

I have an IVP I'm trying to solve with Laplace transformations: $$y''-4y'+4y=te^{2t}$$ Given that: $y(0)=1$ and $y'(0)=0$ I've gotten to the part where I isolate $Y(s)$: ...
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Laplace Transform solution verification: $\ddot{y} + 2y = 2e^t\implies \frac13\cos(\sqrt{2}t)-\frac{2}{3\sqrt{2}}\sin(\sqrt{2}t)+\frac23e^t\,\text{?}$

Does $$\ddot{y} + 2y = 2e^t\quad y(0)=1,\dot{y}(0)=0$$ Give $$\frac13\cos(\sqrt{2}t)-\frac{2}{3\sqrt{2}}\sin(\sqrt{2}t)+\frac23e^t\,\text{?}$$ This is what I have got, and it seems to go back and ...
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Is this the correct setup for partial fractions? $\frac{1-e^{-s} + se^{-s} + s^3}{s^2(s^2+2)}=\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+2}$

I am trying to inverse laplace transform the following: $$F(s)=\frac{1-e^{-s} + se^{-s} + s^3}{s^2(s^2+2)}$$ and I believe what I do is take: $$\frac{1-e^{-s} + se^{-s} + ...
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Inverse Laplace transform for $\frac{1-e^{-\pi s}}{s(s^2 + 16)}$

I want to find the inverse Laplace transform for the following:$$\frac{1-e^{-\pi s}}{s(s^2 + 16)}$$ This was obtained from a piecewise function and required the heaviside step function to simplify. ...
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Computing transfer function then static gain

I am working on a problem for a flight controls class. I have an equation related to pure yaw. My goal is to get the transfer function associated with it, and then obtain the system static gain. The ...
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35 views

Laplace Transform…

Find the Laplace transform of $t^2e^{at}cos(bt)$ My attempt: $\large\mathit{L}\{t^2e^{at}\cos(bt)\}$ = $\large\mathit{L}\{\frac12t^2e^{at}(e^{ibt}+e^{-ibt})\}$ ... ... ...
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LaPlace transform of $t^2\sin(at)$

$$\int^{\infty}_{0} e^{-st}t^2\sin(at)dt$$ I keep running into a problem when using: $$u=e^{-st}t^2$$ $$du=2te^{-st}-st^2e^{-st}$$ $$v=\frac{1}{a} \cos(at)$$ $$dv=\sin(at)$$ Anyone have any ...
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What is the inverse Laplace of a complete square?

How do I find the inverse Laplace for something like this ? $${8 \over ( s^2 + 16 )^2 }$$ I tried using partial fraction but it didn't help any ideas on how to do it using differentiation or ...
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Evaluate $\int\limits_{0}^{\infty}\dfrac{1-e^{-t}}{t}e^{-st}\;dt$

This is laplace transform of $\dfrac{1-e^{-t}}{t}$ and the integral exists according to wolfram Do i get any help/hints about how to work this ? I have been trying integration by parts with different ...
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Additive (causal + anticausal) decomposition of a transfer function

Define the causal/anticausal decomposition of a function $F(s)$ as follows. Let $f(t)$ any function such that $$F(s) = \int_{t=-\infty}^{+\infty} f(t)e^{-st} dt.$$ Then the causal part of $F(s)$ is ...
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Laplace Transform of $\cosh(at)/(at)$

Can someone give me a clue on how to compute this Laplace transform? $$\mathcal{L}\left[ \frac{\cosh(at)}{at} \right]$$
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Convolution Theorem of a product of 3 functions of x

I am trying to evaluate the integral over a product of f(t), g(t) and h(t) using the convolution theorem. $$\int_0^\infty f(t) g(t)h(t) dt$$ So after taking the Laplace of each of f(t) g(t) & h(t) ...
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Why does the Laplace transform of $t^2 \exp(at)$ exist?

My book states a theorem : "Let $f(t)$ be a function piecewise continuous on $[0, A]$ for $ A > 0$ and have an exponential order at infinity with $|f(t)| \leq M \exp(at)$. Then the Laplace ...
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Lalplace Transform and Convolution theorem

How would one go about a problem of this nature $$\int_0^\infty f(t) g(t) dt$$ Using the convolution theorem. I have taken the Laplace transform of both f(t) and g(t) to get F(s) and G(s) however ...
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Using laplace transforms to solve a piecewise defined function initial value problem

I want to use laplace transforms to solve the following: $$\frac{d^2 y}{dt^2}+16 y = f(t) = \left\{\begin{array} 1 1&t\lt\pi\\0&t\geq \pi\end{array}\right.\text{ with } y(0)=0 \text{ and } ...
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Laplace transform of Bessel's equation

I'm working on what should be a relatively straightforward differential equation. The problem says that the Laplace transform of Bessel's equation leads to (s^2 +1)f'(x) +sf(s)=0. And asks to solve ...
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Laplace Existence of Multiplication of two functions

I have two functions, namely $f(t)$ and $g(t)$. Both of the functions satisfy the Laplace existence property, i.e. $ f(t) $ is piecewise differentiable and of exponential order and $ g (t) $ is ...
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Deconvolution of two delta functions (solving $y(t) = A x(t-a) + B x(t-b)$)

I would like to calculate $x(t)$, when only $y(t)$ with $y(t) = A x(t-a) + B x(t-b)$ is known. Since this is a linear shift invariant operation (convolution), the inverse relation must be of the ...
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Probability density function with the help of the Laplace (Fourier) transform

I am reading a paper that derives a closed form expression of the following probability using properties from Fourier transform, $$ \mathbb{P} \biggl( F \geq T \ (I+W) \biggl)$$ Assumptions: 1- ...
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Laplace transform of 1

I am getting a little confused when using the Laplace Transform. So, taking the laplace transform of 1 we get: $\mathfrak L[1](p)= \int_0^\infty e^{-pt}dt=[-\frac1pe^{-pt}]_0^\infty=\frac1p$ when ...
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Laplace transform of a function divided by t

Using the formula $$\mathcal{L}\left\{\frac{f(t)}{t}\right\}=\int_s^\infty F(u)~du$$ I'm trying to determine the transform with $f(t)=1-e^{-t}$. The formula gives me ...
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The inverse Laplace transform of an entire function

A simple calculation shows that the Laplace transform of $f(t)=e^{-t^2/4}$ is the function $F(p)=\sqrt{\pi}e^{p^2}\operatorname{erfc}(p)$. I would like to find the inverse Laplace transform of ...
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27 views

Inverse laplace transform with complex roots

hello I am having some trouble finding the solution to this inverse laplace transformation $$ I(s)= \frac{6s+24}{s^2 +4s+8} $$ The solution is solved using Euler identity and partial fractions, $$ ...
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Is $f\left(t\right)=\frac{1}{t^2+1}$ of exponential order?

I'm learning Laplace Transforms and one of the questions I'm working on is the following: $$\text{Is}\:\:f\left(t\right)=\frac{1}{t^2+1}\:\:\:\text{of exponential order?}$$ If so or if not, how do I ...