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

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applying two Laplace properties on same function

For example, the Laplace transform of $(t - 3)\cdot u(t-3)$ I'm confused about how to apply the two Laplace properties (multiplication of t and time shift). Do I apply one property first then the ...
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0answers
29 views

Figuring out impulse response

I need a little help with figuring out this problem. I understand most of it but the main part I don't understand is: The signal $h''(t)+2*h'(t)+2*h(t)$ is of finite duration. In the problem we are ...
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131 views

Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this: The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The ...
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1answer
63 views

Bromwich integral of $1/s^k$ with k real (non integer) and $1<k$

Is there a simple way to compute the inverse laplace transform of $1/s^k$ with k non integer using Bromwich integral (basically without using the known laplace transform of $t^n$)?
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1answer
54 views

What is the solution of this differential equation? / How to solve it?

I have the following problem : $$m\ddot{x} + c\dot{x} + kx = f_f\delta(t-t_0) + f_c \sin(\omega t) + f_h \theta (2t_0-t)$$ where $x(t)$ is a function of time, $t>0$ and $t_0>0$ and where ...
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44 views

Why does this phase calculation go to 180 instead of 90?

This is all coming from the following video I am studying from http://www.youtube.com/watch?v=XSS6L42ce88 So I am working from this system $$ G(s)\,=\,\frac{4}{s^{2}+s+2}$$ and the video states the ...
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1answer
64 views

Inverse Laplace transform of $\large \frac{1}{s^2-As^{1.5}}$

Title says it all. How do I go about finding inverse Laplace transform of that expression? If it were complete exponents, I would have used partial fractions. But what to do with non integer ...
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30 views

Inverse laplace transform of complicated function

I have a function: $f(s)=\dfrac{(-HT/s)e^{-x*\sqrt{a/s}}}{\sqrt{a/s}+He^{-x*\sqrt{a/s}}}$ where s is frequency domain variable and H,T,a,x can be regarded as constants. How do I find inverse Laplace ...
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56 views

What is the laplace transform of $e^x$

The Laplace transform of $e^{at}$ is $\frac{1}{(s-a)}$. But what is the Laplace transform of $e^x$.
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1answer
76 views

Laplace transform of $g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$

Find Laplace transform for this function "$g$" $$g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$$ Then Take advantage of it to calculate the following ...
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1answer
54 views

Simple inverse using Laplace transform

I have the following excercise. I looked at the Laplace transform table for said transform, but I can't find any that looks similar to this. Help, please? $$ \mathcal{L} ^ {-1} \left[ \frac{s} {((s + ...
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1answer
31 views

Simple inverse laplace transform problem

I have the follow excercise. I am aware of partial fraction expansion, but the roots are imaginary in this problem. Does somebody know how to solve it? Thanks. $$ \mathcal{L} ^ {-1} (1 / (s ^ 2 + 4 s ...
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1answer
40 views

Finding Laplace Transform of a Function

Firstly, I believe there is an error with the function G on the problem. It should be: $ G(x) = \sum_{k = 0}^{\infty} P[N=k]\cdot F^{k*}(x)$, where $F^{k*}(x) = P[X_1+...+X_k<x]$. Because we ...
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39 views

Inverse Laplace Transform of …

I am trying to find the inverse Laplace transform of $$\frac{1}{pe^{p}}\int_2^p\frac{e^{q}q}{q-1}dq.$$ This function decays as $p$ goes to infinity, so it's reasonable trying to find it's Laplace ...
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1answer
54 views

Laplace Transformations of $\frac{1}{t}$

what is the laplace transform of $\frac{1}{t}$? I tried different ways like integrating by parts from the general form of laplace but it's getting more complex as my solution goes by.
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1answer
87 views

Laplace transformation $t$-shifting proof $L(f(t-a)) = \exp (-as) \cdot F(s)$

The property says: $$L[f(t-a)] = e^{-as} * F(s)$$ Standard proof goes as: $$L[f(t-a)] = \int_{0}^{\infty}f(t-a)*e^{-st}dt$$ Now, we make a change of variables, assume $u=t-a$ then $t=u+a$ and ...
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48 views

Inverse Laplace of $F(s) = \frac{3s}{(s^2+9)^2}$

Can somebody please show how to go about answering the following; ${\scr L^{-1}}(F(s)) $ where $F(s) = \dfrac{3s}{(s^2+9)^2}$ I know the ${\scr L}\left(\dfrac{3}{s^2+9}\right)=\sin(3t)$ and that ...
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0answers
53 views

Contour integral (inverse Laplace transform) with arctan

I have what I think is a relatively simple contour integral involving arctan, but it is giving me difficulty. I would really appreciate any help. The integral itself is, with τ, λ, and k all real and ...
2
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2answers
93 views

2nd order differential equation: $y''+y=xe^x\cos(x)$

How can I solve the following differential equation? $$y''+y=xe^x\cos(x)$$ I've studied the (1) Undeterminate Coefficients, (2) Variation of Parameters and (3) Laplace Transforms methods, but I ...
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41 views

Finding the inverse of Laplace–Stieltjes transformation and Convolution related to Probability

I would like to ask you something I do not understand from my book. If I have G following exponential distribution with $G(t)=1-e^{-\lambda t},t\geq 0$ then ...
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0answers
51 views

What is a transform?

I've been working in vain to find a way to find the integral of an intractable function. It's great practice anyway. I thought about using intergration by parts with three functions to solve it and ...
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1answer
35 views

Find the Laplace transform of the following hard equation

Ok so the objective is to factor this into something that resembles the Laplace tables. Give me some help pls. thx
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76 views

Laplace transform with time shift property

ok so i have no idea how the inverse laplace went from $F(s)$ to $f(t)$. I understand $\frac{c}{s^2}$ => $ct$, and $\frac{b}{s}$ => $b$, but the $e^{-as}$ is what gets me. In my Laplace tables I ...
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1answer
204 views

Laplace Transform $f(t)=2\cos(3t)$

Determine the laplace transform of the function $f(t)=2\cos(3t)$, without using the table of Laplace transforms. I use by part integration to solve it, with $u=e^{-st},\, du/dt=-se^{-st}$ and ...
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2answers
32 views

how to find inverse laplace transform of

how to find the inverse laplace transform of $\frac{s}{s^4+s^2+1}$. I tried to do it via partial fraction and reached $\frac{s}{(s^2-s+1)(s^2+s+1)}$
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1answer
74 views

Find the inverse Laplace transform in special case

How to find the inverse Laplace transform of: $$g(x,p,x') = \begin{cases} - \dfrac{e^{\sqrt{p} x'} \sinh({\sqrt{p}x})}{ {\sqrt{p}}} & 0 < x < x' \\ -\dfrac{\sinh({\sqrt{p}x'}) ...
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3answers
60 views

Inverse Laplace Transform question

How do you work out the inverse laplace transform of $$\dfrac{p+2}{16((p+2)^2 + 4)} $$ I know the $p+2$ is $e^{-2x}$ but what is the inverse of $(p+2)^2 +4$ ?
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1answer
69 views

What is a solution of v(t) in the differential equation?

I has Vs(t) as well as a wave form in picture. So what is a solution of v(t) in the differential equation? I used Laplace transform for solving but it is incorrect. Next picture, it is a wave form of ...
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1answer
49 views

Laplace question

How do you express Laplace transform $\mathcal{L}(g)(z)=\int_{0}^\infty e^{-zt}g(t)dt$ with Fourier transform? And how do you form the reverse formula for Laplace transform using Laplace transform ...
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60 views

solve this differential equation using laplace transform and the series method :

Problem : $y''+8ty'-16y = 3 , y(0) = y'(0) = 0 $ I am supposed to use the series method to get F(s) , then get the inverse laplace transform to get f(t). I got the Laplace transform : $(s^3 - 24s) ...
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1answer
40 views

Laplace transform and inverse $\coth$ function

What is laplace inverse of $\coth{\pi s/2w}$.Laplace transform of coth function.and how to evaluate it.I tried but unable to get the correct solution.
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30 views

The Laplace transform - does it have an associated differential operator, if the kernel is to be viewed as a Green's function?

I've begun learning about Green's functions, and if I understand correctly, the Green's function for a linear differential operator $L$ with appropriate boundary conditions is the kernel for the ...
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31 views

Inverse laplace transform issue

I'm stuck trying to perform an inverse laplace transform of $\dfrac{2}{4s^2+s}$. MathCad said that the original is $2 -2e^{-\frac{1}{4}t}$, and I know that this is the right solution, but I can't ...
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1answer
64 views

Inverse Fourier Transform of the output, Y(f)

A linear system is defined by the differential equation: $$ y''(t) + 4y'(t) + 25y(t)= x(t) $$ The transfer function of this system is: $$ H(f) = \frac{Y(f)}{X(f)}= \frac{1}{(2\pi fj)^{2}+ 4(2\pi ...
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1answer
61 views

Laplace Transform and Fourier Transform of a function

I have this transfer function: $$ h(t)= -\frac{1}{16}te^{-2t} $$ and the Laplace Transform is: $$ H(s) = \frac{-\frac{1}{16}}{(s+2)^{2}} $$ I know that to find the Fourier Transform, I would ...
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183 views

A convolution like integral equation

I would like to solve the following integral equation for $g(z)$. $$\int_z^\infty g(\zeta)(\zeta-z)^{\alpha-1} d\zeta = e^{-bz}, \tag{1}$$ where $\alpha$ and $b$ are constants. I would also like to ...
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88 views

Laplace Heaviside function,how can I continue?

I have to find the Laplace transform of the function $H(t-a)t^{n}$. That's what I have done so far: ...
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1answer
54 views

Find the inverse Laplace transformation of $\frac{(s+1)e^{-s}}{s^2}$. [closed]

Find the inverse Laplace transformation of $\dfrac{(s+1)e^{-s}}{s^2}$.
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55 views

Representation of heaviside step functions

Can the heaviside step function, $u(t)$ be represented like so: $$u(t)=\frac{1}{2}\left(\frac{|x|}{x}+1\right)$$
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92 views

Is it possible to solve the following differential equation using Laplace Transforms?

With initial condition $x(1)=\frac{1}{2}$, solve the differential equation $t\frac{dx} {dt}+x=t$ $t\frac{dx}{dt}+x=t$ we know that, $L[t\frac{dx}{dt}]=-\frac{d(sX(s)-x(0))}{ds}$ ...
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0answers
37 views

Transfer Function

Vehicle dynamic system (Bicycle model) is given by the following state space model (which also includes Road Bank angle): $\begin{Bmatrix}\dot{x_1}\\\dot{x_2}\end{Bmatrix}=\begin{bmatrix}a_{11}& ...
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1answer
69 views

Using Laplace transform method

Using Laplace transform method, solve $$u_t=3u_{xx}$$ Subject to the boundary conditions $u(π/2)=0 $ , $∂u/∂x (0,t)=0$ , And initial condition $u(x,0)= 30\cos{5x}$
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1answer
74 views

Inverse Laplace

Hi how to verify the following I tried substitution and integration by parts but can bot figure it out.. $$\int_0^{\infty} \exp(- \lambda t ) \frac{x}{\sqrt{2\pi t^3}}\exp(-\frac{x^2}{2t}) dt = ...
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38 views

Particular integral equation

Let $a,\sigma, n>0$ be some parameters and define the conditional probability density function $$ p(x,y):= \frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-n-x)^2}{2\sigma^2}\right). $$ Is it ...
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1answer
58 views

Inverse Laplace Transformation Simple Question

I'm new to Inverse Laplace Transformation so, I need to know whether the following question is correctly solved or not.. Question: 1/(s+2)^3 Solution: Multiplying and dividing by 2; = 2/2(s+2)^3 ...
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35 views

Calculate Laplace transformation of the product

Laplace transformation of the derivative of the product equals? $$L\left(\frac{d(f_1(y)f_2(y))}{dy}\right) = sf_1(s)f_2(s) - f_1(0)f_2(0) ?$$ I have meant not the convolution of the functions under ...
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1answer
39 views

Z-transform a transfer function

Could someone help me invers Z-transform of this transfer function. $H_k(z) = \frac{Y_k(z)}{X(z)} = \frac{1}{1-cos(\frac{2·\pi ·k}{N})·z^{-1}+z{^-2}}$
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74 views

Laplace Transform & Initial Value Problem

$$ y'' + 4y = \begin{cases} t, & 0 \leq t < 3\\ 1, & 3 \leq t <\infty \end{cases} $$ $$y(0)=0, y'(0)=0$$ I need to find the Laplace transform of the solution of the given IVP above. ...
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50 views

Calculating convolutions of probability density functions

I have a PDE: $$\frac{\partial N (x,u)}{\partial x}=\int _0^uN(x,u)f(u-u')du'$$ $$N(0,u) = \delta (u)$$ Here $f(u)$ is a probability density function for $0 \le u \le u_{max}$, $\int _0 ^ {u_{max}} ...
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1answer
126 views

Extracting moments from a specific Z-transform

Suppose I have a sequence of positive continuous random variables $\{X_k\}_{k=1}^\infty$ with MGF's $M_{X_k}(s)$. Furthermore, it is known that ...