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

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finding the inverse Laplace transform of $\frac{1}{z\sqrt{z+1}}$

i know that the inverse Laplace transform is given by $$2\pi i \left\{\sum\space\text{ of the residues at the poles of}\space e^{zt}f(z)\right\}- \frac{1}{2 \pi i}\int \text{ along the branch cut}$$ ...
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Laplace transform of $f(t)=te^{-t}\sin(2t)$

I was asked to find the laplace transform of the function $f(t)=te^{-t}\sin(2t)$ using only the properties of laplace transform, meaning, use clever tricks and the table shown at ...
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23 views

Pollaczek-Khinchin formula for ruin probability - proof

I got stuck in a specific part of proof of the Pollaczek-Khinchin formula (in book "Stochastic Processes for Insurance and Finance", T. Rolski et al., section 5.3.3, theorem 5.3.4). Namely, why the ...
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27 views

Check my answer - simple laplace transform of piecewise continuous function.

I'd just like to check that I got the idea right, first exercise im doing in laplace transforms and am a bit clueless. We are given $f(t)=0$ if $0<t<2$ and $f(t)=t$ if $t>2$. We are asked to ...
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29 views

Laplace Transform of the square of first derivative

I want to compute the laplace transformation of this equation: $$4 -0.1{\operatorname{d}\!x(t)\over\operatorname{d}\!t} - 0.01\left({\operatorname{d}\!x(t)\over\operatorname{d}\!t}\right)^2 = ...
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Differential equation to space state excercise

This is a "back of chapter" excercise which im trying to solve, my answer doesnt match the solution printed on the book, I want to write the equation in state space matrix form without using the ...
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1answer
18 views

Solution to truncated renewal function

Let's begin with some theory on the renewal process. In a renewal process $N(t)$, let $t$ denote the interarrival time, and $f(t)$ and $F(t)$ denote the PDF and CDF respectively. Let $M(t)=E[N(t)]$, ...
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33 views

Inverse Laplace of $\frac 1 {(s^2+a^2)^n}$

How to compute the Inverse Laplace of $\frac 1 {(s^2+a^2)^n}$? I know that to compute Inverse Laplace $\frac 1 {(s^2+a^2)^2}$, the convolution Theorem is useful. but is there an interesting idea for ...
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28 views

Solution of given differential equation using Laplace Transforms.

I need solution of DE $$y'' + 2y' + 5y = 0$$with initial conditions $$y(0)= 1 \text{ and } y'(0)=0$$ I tried this but problem came when i started taking laplace inverse of F(s), so i need a complete ...
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18 views

Inverse Laplace Transformation

I was solving a problem but I am stuck at it. Here is the question : $\frac{7s^2+9s+3}{(s^2-12s+40)(s^2+9)}$ Find inverse Laplace transform. I performed these operation : ...
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22 views

Inverse Laplace Transformation

I have a question about laplace transformation. $\frac{8s+4}{s^2+23}$ I tried to split them. $\frac{8s}{s^2+23}$ is the image of a cosine and $\frac{4}{s^2+23}$ is the image of a sine. Here is ...
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25 views

Transforming Exponential to Ordinary Generating Functions

I am looking for a particular ordinary generating function, if it exists for the Associated Stirling Numbers of the second kind $$b(1;n,j)=b(n,j)=\sum_{k=0}^j(-1)^k\binom{n}{k}S(n-k,j-k)$$ Where ...
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24 views

Laplace transform of $\sin \sqrt x$ [duplicate]

I want to find the Laplace transform of the $\sin \sqrt x$. The first thing it came to my mind is to definition, but I think there are others solutions. Please show me this solution thanks.
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24 views

Laplace inverse transform of the complex expression branch point

I have an expression below which I need to do the inverse laplace transform Any help is highly appreicated. The expression is $$ \frac{\sqrt{s^2+1}}{\left(2+\sqrt{s^2+1}\right)^{3/2}}.$$
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39 views

How to solve a linear system in matrix form using Laplace transform?

How to solve this linear system using Laplace transform? $$\mathbf X'(t)=\left[\begin{array}{r,r,r}-3&0&2\\1&-1&0\\-2&-1&0\end{array}\right]\mathbf X(t); ~~~~~~~~\mathbf ...
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Laplace transform of the following: $f(t)=\int\limits_{0}^{t} \frac{\cosh (\tau) - 1}{\tau} d\tau$

I thought it would be a simple one, however, that integral of $f(t)$ cannot be expressed in terms of standard functions...and I'm pretty much confused. What should I do?
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28 views

Inverse Laplace transform of $se^{-s}$?

I am trying to find inverse Laplace transform of $se^{-s}$ using Laplace transform properties. But I don't know how to solve it. I would be grateful for any help.
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32 views

How to find the inverse Laplace transform?

I'm trying to calculate $$\mathcal{L}^{-1}\left(\frac{3s^3-3s^2+3s-5}{s^2(s^2+2s+5)}\right)$$ But I am not sure how to go from here. I would be really grateful for any help. Thanks.
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31 views

Tough inverse Laplace transform

I know what the solution is to this inverse Laplace transform, I just have NO idea how to get there. $$\mathcal{L}^{-1}\left(\frac{16s}{\left(s^2+4\right)^2}\right)$$ Basically, my question is what ...
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32 views

Differential equation, for which values of 'a' does this have a bounded solution?

Let $f(t) = f(t$) be the 2pi periodic("sawtooth wave"), f(t) = t for $0 \leq t \leq 2\pi$ and consider the equation $$y^{\prime \prime} + a^2y = f$$ For which values of $a$ (here $a$ >0) does this ...
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36 views

How to find solution of the integral equation?

$$y(t) + t \int_0^t y(v)dv = 1 + \int_0^t vy(v)dv$$ I found the answer to be $y(t) = \cos{t}$. I have no idea how they go this answer. I would appreciate any suggestions how to solve this.
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Inverse Laplace transform shifting error

I am doing the inverse Laplace transform of the function: $\frac{e^{-s}}{s-1}$. I am solving and receiving the answer: $e^t\mathcal{U}(t-1)$, however the correct answer is $e^{t-1}\mathcal{U}(t-1).$ ...
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41 views

How do you do this Laplace transform?

How do you do the Laplace transform of the following equation? I know there's a trig identity in here somewhere, but I've got no idea how to do/use it. $$\mathcal{L}\lbrace\cos{4t}\cos{2t}\rbrace$$
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23 views

Solve laplace equation inside a rectangular

My answer is $U = Acos(nπx/L)e^-nπy/L$ I really have no idea how to solve the particular solution. Please advise me.
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Singularities of complex exponential and asymptotic expansion

Consider the equation $$L[u(x,t)] = \tilde u(s,t) = \frac{e^{-t\sqrt{s^2-1}}}{s-2}$$ I want to find $u(x,t)$ in the form of an integral. I first need to find the poles and singularities of the ...
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9 views

Renewal Process with Pareto distributed inter-arrival time

I am analyzing a renewal process $N(t)$ whose inter arrival time $t$ conforms to the Pareto distribution, the PDF $f(t)$ of the Pareto distribution is as follows: $$f(t) = \left\{ \begin{array}{lr} ...
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1answer
36 views

What's the laplace inverse of this function?

I'm completely stuck on how to do this one. Any help is appreciated. What is the inverse Laplace transform of: $$\mathcal{L} ^ {-1} \left\{ \frac{e^{-2s}}{s-2} \right\} = f(t)$$
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21 views

How to find the Laplace transform of $\sin^2 (2t)$?

How to find the Laplace transform of $\sin^2 (2t)$? So far I have: $\sin^2(x) = \frac12 - \frac12 \cos(2x)$.
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Inverse Laplace transformation correct?

I'm actually on the way to solve a little bit complicated differential-equation. Therefore I used the Laplace transformation. I've already solved it but I am actually not sure, whether my solution ...
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22 views

Laplace transform of piecewise function

I have a piecewise function f(t), and I'm trying to get it's laplace transform. When I do it manually, i'm getting a different result than with Maple. $$ f(t)=\!\cases{{t}^{2}&$0<t$ and ...
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14 views

Approximate the inverse Laplace transform

I am struggling with an inverse Laplace transform for a long time! Assume we have a function $m(t)$ and its Laplace transform is denoted by $M(s)$. I have derived the expression of $M(s)$ by some ...
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37 views

Laplace Transformation using Heaviside functions

I'm not very familiar with Heaviside functions so I am struggling with this: I'm supposed to compute $Lu$ where $u''+4u=H(x-0)+H(x-\pi)$ and $H$ is a Heaviside function. Any suggestions are greatly ...
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Laplace transform and differential equations

Given $\frac {d^2y(t)}{dt^2} + a\frac {dy(t)}{dt} = x(t) + by(t)$ Find: a) $ H(s) = \frac{Y(s)}{X(s)}$ b) ROC of the stable function and the correspond h(t) and determine if the stable system is ...
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21 views

Approximate ways to compute Laplace transform of $t^{-a}$

I am encountered with the Laplace transform of $\frac{1}{t^{a}}(a>0)$ in a practical experiment. Obviously, the exact Laplace transform of $\frac{1}{t^{a}}(a>0)$ does not exist, but I am ...
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Use Laplace Transform to solve the following IVP:

I know that this is a somewhat simple problem but I have been having trouble coming up with the little "tricks" that help with Laplace. The problem is: $y''+2y' +5y = e^{-t}\sin(2t)$ where $y(0) = ...
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30 views

Use the chain rule to convert the Laplace equation in (x,y) coordinates into an equivilent differental equation in (r,theta) coordinates. [duplicate]

use the equations $r=\sqrt{x^2 +y^2}$ and $\theta=\arctan(\frac{y}{x})$. I was able to get the partial derivative of of $r$ with respect to $x$ and $y$ and the partial derivative of $\theta$ with ...
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Inverse laplace transform excercise

I want to find the inverse transform of $$\frac{1}{(2s-1)^3}$$ I first applied a shifting theorem to get $$(e^t)\mathcal{L}^{-1}\left( \frac{1}{(2s)^3} \right)$$ I am just wondering is it possible ...
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Frequency integration theorem (Laplace transform)

In my textbook I have the following theorem about the integration of the frequency (F(s)): Let the Laplace transform of a function $f(t)$ be $\mathscr{L}\{f(t)\}=F(s)$. If $\dfrac{f(t)}{t}$ is the ...
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Laplace Transform of $e^t\cos(3t)\operatorname{heaviside}(t)$

Find the Laplace Transform of $e^t \cos(3t) \operatorname{heaviside}(t)$ Since $\operatorname{heaviside}(t)g(t) = \mathcal{L}(g(t)) $ and $\mathcal{L}(e^t\cos(3t)) = \frac{(s-1)}{(s-1)^2+9} $ ...
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28 views

Laplace transform involving two functions of t

I need to solve the following $$ \int_0^{\infty} f(t)g(t)e^{rt} dt$$ Where $$g(t)=t^n$$ Letting r=-s we have the definition of $$ \mathcal{L} [ f(t)g(t) ]$$ and am unsure how to continue.
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How to find Laplace transform of a differential equation?

$y′′ + 3y′ + 2y = f$ , $y(0) = 0$ , $y′(0) = 1$ where $f$ is given by $f(t) = \sum_{n=1}^\infty \delta(t−n)$; find a 1-periodic function $y_*$ with $\lim_{t\rightarrow \infty} |y(t)−y_*(t)| = 0$. I ...
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How to re-write one fraction as two others.

I have the two following fractions. $$ \dfrac{A}{Bx^{\alpha+1}}$$ and $$ \dfrac{C}{Dx^{\alpha+\beta}}$$ The form i want $$ \dfrac{E}{Fx^{\alpha+\beta+1}}$$ I was thinking to do partial fractions or ...
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53 views

Trying to show $\int_0^1 e^{-xt}sin(t) dt \sim \frac{1}{x^2}$

I am using Laplace's Method and I am trying to show $$I =\int_0^1 e^{-xt}sin(t) dt \sim \frac{1}{x^2}$$ $h(t) = -t$ has a maximum at $0$ and as it is a simple function there is no need to expand it. ...
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1answer
23 views

Laplace transform of a differential equation?

$y′′ + 2y′ + 2y = δ(t − \pi) + aδ(t − T)$ , $y(0) = y′(0) = 0$ $a$ and $T$ are positive numbers and $T > \pi$. I need to find values for $a$ and $T$ such that $y(t) = 0$ for all $t \ge T$? I just ...
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34 views

Solve initial value problem with unspecified right-hand side $g(t)$

Consider the initial value problem $$y''-6y'+9y=g(t),\quad y(0)=1,\ y'(0)=3.$$ 1) Use the Convolution Theorem to find the solution to the IVP for any piecewise continuous function $g(t)$ that is of ...
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47 views

Laplace transform via complex analysis

Let $Y(s) = \frac{2e^{-s}}{s(s^2 + 3s + 2)}$. Then the inverse Laplace transform is \begin{align} y(t) &= \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{2e^{s(t - 1)}}{s(s^2 + 3s + ...
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30 views

Solution of 1d wave equation by Laplace transform

This is a homework problem that I can almost finish. I just can't invert the Laplace transform at the end. $$u_{xx}=u_{tt}, u(t=0)=u_t(t=0)=0, u(x=0)=\sin\omega t, u(x=2)=0.$$ Taking the Laplace ...
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Laplace transform for IVP not at zero in system of differential equations

Suppose we have a system $\boldsymbol X'=\boldsymbol A\boldsymbol X$. Let's denote the laplace transform of a vector $\boldsymbol Y$ as $\mathscr L\{\boldsymbol Y(t)\}(s)=\boldsymbol y(s)$. If we ...
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49 views

Understanding block diagrams

If I have block diagram with input $X(s)$ that goes to a block with $\frac{1}{s + 2}$ in it and then by way of $w(s)$ to a block with $s$ in it, and finally to the output $Y(s)$, how do I find the ...
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18 views

Hoe can I find the Inverse FourierTransform for 1/(1+w^4)?

I have the expression $S(w)=\frac{1}{1+w^4}$. I am trying to find its inverse FourierTransform. I know that I have to get a sin-cos expression, but I haven´t found the way to do it. On the tables that ...