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

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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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2answers
345 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
75 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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221 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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2answers
55 views

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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1answer
288 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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1answer
110 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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1answer
113 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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1answer
31 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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2answers
77 views

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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1answer
77 views

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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1answer
130 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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1answer
130 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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94 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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1answer
26 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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3answers
131 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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1answer
48 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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35 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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1answer
38 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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1answer
38 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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216 views

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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1answer
85 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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30 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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36 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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242 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
71 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
122 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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1answer
48 views

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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1answer
73 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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54 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
80 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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1answer
87 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
105 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
39 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
79 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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1answer
109 views

Inverse Laplace transform of $(2s+4)/(s^2+4s+5)^2$

Out of many transformation shortcuts in Laplace table I still find difficulty in finding the inverse laplace transform of $\displaystyle \frac{2s+4}{(s^2+4s+5)^2}$. I tried partial fraction and its ...
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1answer
60 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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382 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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3answers
51 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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167 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 ...
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2answers
153 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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138 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
43 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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1answer
94 views

Inverse laplace transform of a hard partial fraction, $1/[s^2(s^2+\omega^2)]$

So the question is find the inverse of $\dfrac{1}{s^2(s^2+\omega^2)}$. And here is the solution. I have no idea why its done this way. I would think to take a partial fraction of the form ...
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2answers
564 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
395 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
41 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
95 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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117 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 $(p+2)^2 +4$ ?