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

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Help Understanding Inverse Laplace Transforms???

I've been looking over some examples regarding inverse laplace transforms, and my textbook doesn't really go into much depth regarding examples such as this: F(s) = (2(s-1)e^(-2s))/(s^2 -2s + 2) I ...
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Laplace of $e^{-s+3}$?

I understand that the $e^{-ks}$ will time shift the function in the time domain by $k$ and will result in a time function of $u(t-3)$, but what does it mean when you have $e^{-s-k}$? How will that ...
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38 views

Help with basic Laplace Transform - unsure of procedure!!!

I am working on this Laplace Transform, and I've tried looking for a similar example off which to base my own work, but haven't been very successful. I'm confused by the formatting and don't know how ...
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Laplace transform of a piecewise function

I'd like to compute the Laplace transform of the following function: $$f(t) = \begin{cases} 0,& \mbox{if} \quad 0 \leq t \lt \pi \\ \sin(t), &\mbox{if} \quad t \geq \pi \end{cases}$$ Could ...
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33 views

properties of laplace transform

Obtain the transfer function for the following differential equation and check whether the input free solution is stable or not, $$\frac{dx}{dt} + 3x = f(t)$$ Please help, I don't even know where to ...
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114 views

Solving $y'-y=2\cos 5t$ using the Laplace Transform

Find the solution to the differential equation, using the Laplace Transform. $y'-y=2\cos 5t$, with initial condition $y(0)=0$. My attempt: First I take the Laplace Transform of each term. ...
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A question about Parseval's formula.

In operational calculus there is Parseval's theorem, which states that if $ f(t) \doteqdot F(p), \varphi(t) \doteqdot \Phi(p) $ and both $ F(p) $ and $ \Phi(p) $ are analytical in $ Re p \geq 0 $, ...
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57 views

Product of two Whittaker functions

According to 6.669.3 of Gradshteyn and Ryzhik the following identity $$ W_{a,b}(z_1)\,W_{a,b}(z_2) = \frac{2\sqrt{z_1z_2}}{\Gamma(1/2+b-a)\,\Gamma(1/2-b-a)}\int_0^\infty ...
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43 views

Is the Laplace transform additive? And why?

The first part is a simple question, but I cant find a clear answer. Does: $$\mathcal{L}(ax''(t)) = \mathcal{L}(a)\times\mathcal{L}(x''(t))$$ $a$ is a constant $x(t)$ is a variable that changes ...
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29 views

How can we take the LaPlace of a function raised to the power?

For example: $\mathcal{L}$((t-1)^1) Following simple linearity, we achieve the answer. However, following the power of theorem: (I'm not proficient enough in LaTex to write this...) I get the wrong ...
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How can we take the LaPlace transform of a piecewise function?

How can we take the LaPlace transform of a function, given piece-wise function notation? For example, $f(t)=\begin{cases} 0 &\mbox{for } 0<t<2\\ t&\mbox{ for } 2<t \end{cases}$ ...
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28 views

How to find the transfer function from the given differential equation

Question: find transfer function from differential eqn $y''(t)+2y'(t)+5=4x(t)$ I am confused about what happens to constant $5$ . will it be zero when we take laplace of whole eqn or not? Can ...
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156 views

Unit impulse / step response of a 1st order differential equation

You are given the equation $10v'(t) + 0.6 v(t) = f(t)$ $v(t)$ is the velocity of the object Determine the unit impulse response AND the unit step response. How would i approach this question? do i ...
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1answer
47 views

Laplace transform of gamma distribution

Gamma distribution has its pdf given by $f(t;k;\theta) = \frac{t^{k-1} e^{-t/\theta}}{\theta^k (k-1)!}$. Show that if the pdf's Laplace transform is $L_k (s)$, then $L_{k+1} (s) = \frac{L_k ...
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113 views

Are these Laplace transforms wrong in Stroud's Advanced Engineering Math Book?

I know that if you think a book is wrong, most probably it is your own mistake. However, I can't understand the following Laplace transforms in K. A. Stroud's "Advanced Engineering Mathematics". In ...
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Inverse Laplace Transform of $\frac{s^2+2s+2}{s+1}$

I want detailed steps of this if anyone can help.
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108 views

Zeros/poles at Laplace and at Fourier Transform

I recently started "relearning" the Laplace transform, and I noticed something. It seems to me that the intuitive idea of poles and zeros is different between these two transforms! For example, in ...
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82 views

Evaluating an integral with Laplace

We need to evaluate the following integral: $$\int_{0}^{\infty}\frac{\cos(tx)}{x^2+a^2}dx$$ There is the following note: "You may interchange taking the Laplace transform and integrating." I have ...
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73 views

Forced wave equation question?

I'm studying for my PDEs midterm and trying to do practice problems. I'm really not sure how to do this question - I've never seen anything like it. Thanks in advance for your help. Solve the ...
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26 views

How to show that the Laplace transform of $\exp(-t^2)$ is $\frac{\sqrt{\pi}}{2}\exp(\frac{s^2}{4})\rm erfc(\frac{s}{2})$

I obtained the answer from Maple. But still I want to know how it is derived.
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Undefined Laplace Transform

I'm in calculus II and our teacher gave us a problem as follows: Let f(t) be a function defined for all positive values of t. The Laplace Transform of f(t) is defined by: $$F(s) = \int_0^\infty ...
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Laplace Transform Damp Harmonic Motion

http://gyazo.com/19d18f085731c6dbc304fefdaece4f3c.png I'm currently on (a) where so far I have gotten; $ y'' + 2y' + 5y = f(t) $ Using Laplace transforms, I get; $ Y(s)$ = $ F(s) + s+2\over(s^2 ...
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36 views

The stability of a unity-feedback control system whose open-loop transfer function is $G(s)=K/[s(s+1)(s+5)]$

Question and solution from book. Regarding the solution: How do you obtain the characteristic equation? Why is it K/s(s+1)(s+5) + 1=0? Where did the 1 come from?? And then how do you go from that ...
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41 views

The Laplace Transform of nonlinear terms (eg. cos(x(t)), x(t)^2)

I've been trying to solve a differential equation of the form $ax"+bx'+cx=d$, but I do not have a constant $c$, rather I have $\cos(c*x)$. (NB: I do NOT mean to find the LT of $\cos(a*t)$, this is ...
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107 views

Obtaining fundamental solution of the heat equation (1-d) through Laplace transform

A classic problem I'm having problems with (problem requires to use Laplace transform) $\frac{\partial ^2}{\partial x^2} u(x,t)=\frac{\partial}{\partial t} u(x,t) $ with conditions: ...
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63 views

Coupled mass spring system with damping and initial values

After researching through the web, I can't figure out how to express into a differential equation a coupled mass spring system with damping and initial values. Two masses and two springs, no external ...
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22 views

Inverse Laplace transform question help

I am having a hard time finding the inverse Laplace transform of $$\frac{1}{(s^2+1)^2} - \frac{1}{s^2(s^2+1)^2}$$ and would appreciate some guidance. I have tried breaking it down to partial fractions ...
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96 views

Coupled mass spring system with damping, I need help with the equation

I know that the equation $mx''+cx'+kx=f(t)$ is used for a normal mass spring system, but I don't know how to express the differential equation for a coupled mass spring system with damping. These are ...
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42 views

Circuit RC, I need help with the equation.

A circuit RC it's described by the next equation: $\frac{1}{c} \int i(dt)+Ri=V$ Where the value of resistance is $R=10 k\omega $ , the value of the capacitor is $C=2.5 \mu F$, and the voltage of the ...
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32 views

Help with an improper integral

Can someone please help me evaluate this improper integral? $$\int_{0}^{\infty}\exp\{-au^{-a}-u\}du$$ for $a>0.$
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31 views

Laplace transform $e^{at}$

Ok the book says it is $\dfrac{1}{s-a}$ However when I evaluate $\displaystyle\int_0^{\infty}e^{-st}\cdot e^{at}=\displaystyle\int_0^{\infty}e^{-(s-a)t}$ so that the derivative is ...
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36 views

Inverse Laplace Transform, I need help

What is the ILT of $H(s)=\frac{7(3s+1)}{(s-3)(s^2+10s-13)}$ Also, if you kindly want to help with this another inverse transform, I'd really appreciate it: $H(s)=\frac{6(s+2)}{s^3(s-5)}$ Thanks!
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Laplace Transform of the Wave Equation

I am given a damped wave equation $u_{tt}(t,x)+2u_t(t,x)=u_{xx}(t,x); \forall t>0$ Now I know the laplace transform of this given the initial conditions, $u(0,x)=\sin x, u_t(0,x)=0;$ is ...
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Laplace transform to describe a bounded function

It is easy to show that if a real function $f:\mathbb{R}\rightarrow\mathbb{R}$ is contained in a strip $[a,b]$, that is if $\forall_{x}\, a\le f(x)\le b$, then its Laplace transform is bouned by ...
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Laplace s Domain Simplification involving shifting

This is probably a straight-forward question (forgive me - it has been a while) - I would like to solve the following equation for $V_c(s)$: $$ {V_{DC} \over s} + {\omega V_{AC}\over {s^2 + \omega^2} ...
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34 views

How to take the laplace of $e^{-|t|}$

I seem to be having some trouble trying to compute the laplace transform of this function. I looked on Wolfram and it said the answer was simply $$\dfrac{1}{s+1}$$ but I highly doubt that is the ...
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29 views

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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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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186 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
67 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
42 views

Find the Laplace transform of $(t-\pi/2)\sin(t-\pi/2)$ using the time shift

What is the Laplace transform of $(t-\pi/2)\sin(t-\pi/2)$? I used the relationship $\mathcal{L}((t-a)f(t-a))=e^{-as}F(s)$ Hence I get $\dfrac{2e^{-(\pi/2)s}}{s^2+1}$. Would this be correct?
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Using the Laplace transform to solve an ODE with piecewise input

I have the answer to this problem. My question is with the function $u(t)$. $u(t)$ is: $$u(t) = 2\cos(t)+2\sin(t-\pi/2)*1(t-\pi/2)$$ Why is there a $1(t-\pi/2)$ multiplying the $2\sin(t-\pi/2)$? ...
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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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51 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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72 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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39 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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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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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
103 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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37 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 ...