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

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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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35 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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32 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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72 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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42 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
15 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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64 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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40 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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31 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
26 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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34 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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34 views

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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12 views

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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1answer
30 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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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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143 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
64 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
55 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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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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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
57 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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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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102 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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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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54 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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96 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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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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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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84 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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220 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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75 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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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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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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50 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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61 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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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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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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62 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 ...