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

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Is this a correct way of thinking of Fourier transforms

I am working on my understanding of various transforms and I have been thinking about the Fourier transform, what i "does" to the function it is applied to. The way I see it: The function $f$ that ...
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Use the transform rule $\mathcal{L}^{-1}\{\frac{F(s)}{s}\} = \int_0^t{f(\tau)d\tau}$ to find the inverse transform of $F(s) = \frac{6}{s(s+3)(s+3)}$

How does the transform rule help us solve this problem? Does this just mean I can rewrite the problem as: $$\mathcal{L}^{-1}\left\{\frac{6}{(s+3)(s+3)}\right\} = \int_0^t ...
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Laplace transform in solving 2d wave equation

I have the following wave equation $\dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}=\dfrac{1}{c^2}\dfrac{\partial^2 u}{\partial t^2}$ with boundary conditions at $x=0,\ ...
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Finding Laplace transform of two functions multiplied together

How can I find the Laplace of $f(t) = \cos^2(t)$ $f(t) = \sin3t\cos3t$ $f(t) = te^{t}$ $f(t) = t\cos(2t)$ What about inverse transform for $F(s) = \frac{5-3s}{2^s+9}$ $F(s) = ...
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Finding Laplace transform of $f(t) = t^2$. Where's my mistake?

$\mathcal{L}\{f(t)\} = \int_0^\infty{e^{-st}t^2}dt$ Integrating by parts: $u = t^2$ $du = 2tdt$ $v = -\frac{1}{s}e^{-st}$ $dv = e^{-st}dt$ $\int_0^\infty{e^{-st}t^2}dt = -\frac{t^2}{s}e^{-st} + ...
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Usage of inverse Laplace transform

At my current study level in college, use of inverse Laplace transform is not mentioned well - textbooks say "use tables." So, can anyone show me how to use inverse Lapalce transform? And also proof? ...
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203 views

Convolution Laplace transform

Find the inverse Laplace transform of the giveb function by using the convolution theorem. $$F(x) = \frac{s}{(s+1)(s^2+4)}$$ If I use partial fractions I get: $$\frac{s+4}{5(s^2+4)} - ...
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191 views

Expressing exponential functions

Can someone provide me a list of the important complex exponential functions. For example, $\cos (at) = \large \frac{e^{iat} +e^{-iat}}{2}$. I am trying to find the laplace transformation for a given ...
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795 views

Find the Laplace transform of $g(t) = 1+cos^2(2t)$ by direct integration

I'm having trouble finding out how to directly integrate the function $f(t)$ because of the $\cos^2(2t)$ term. I understand that $\cos^2(2t) = \frac{1}{2} + \frac{1}{2}\cos(4t)$ but I don't ...
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501 views

Laplace Transform Dirac Delta Function

Find the solution of the initial value problem. $y'' +2y' +2y = \delta(t - \pi)$ with initial conditions $y(0) =1, y'(0) =0$. What I did was take the Laplace and got: $(s^2Y(s) - s) + ...
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Invers laplace transform of resolvant using residues

I got that the inverse laplace of $\frac{1}{(s-λ)}$ is $e^{λt}$. Does this look correct?
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about discrete versions of “inverse laplace transform”

Is it possible to have general formulae to calculate the discrete versions of "inverse laplace transform" ? For example, do $F(s)=\sum\limits_{t=0}^\infty f(t)e^{-st}$ and ...
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309 views

Computing inverse two-sided Laplace transform symbolically

How can I compute the inverse two-sided Laplace transform symbolically? I know MATLAB has ilaplace[1], but that's just for a one-sided transform. [1] ...
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Trying to find laplace transform $x(s)$ that satisfies $x'(t)=x(t)-t$?

Taking the Laplace transform of the equation $$x'(t)=x(t)-t,$$ we get $$sx(s)-x(0)=x(s)-\frac{1}{s^2},$$ right? So if $x(0)=1$, don't you get $$x(s)=\frac{1-\frac{1}{s^2}}{s-1}?$$ When I take the ...
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Inverse Laplace transform of fraction $F(s) = \large\frac{2s+1}{s^2+9}$

Is there a general method used to find the inverse Laplace transform. Are there any computational engines that will calculate the inverse directly? For example, can a procedure be followed to find ...
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74 views

Finding the transform of a function with a picture of the graph?

How can we find the transform of a function just given its graph? I know the definition implies some type of differentiation (which should be easily obtained from the graph), but I'm still having ...
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353 views

Does the Laplace transform biject?

Someone wrote on the Wikipedia article for the Laplace trasform that 'this transformation is essentially bijective for the majority of practical uses.' Can someone provide a proof or counterexample ...
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388 views

Integrating using Laplace Transforms

$$\int_{0}^\infty {\cos(xt)\over 1+t^2}dt $$ I'm supposed to solve this using Laplace Transformations. I've been trying this since this morning but I haven't figured it out. Any pointers to push me ...
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108 views

Inverse Laplace Transform Assistance

How do I compute the following transform? $$\frac {s-1}{2s^2+s+6}$$ I've gotten this far: $$\frac {1}{2}\cdot \frac {s-1}{\left(s+\frac{1}{4}\right)^2 + \frac{47}{16}}$$
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Laplace Transform Help

The Laplace Transform of $\frac{3}{(2s+5)^3}$ is given as $\frac{3 t^2}{16}e^{-\frac{5}{2}t}$ Can someone please walk through how this was obtained? Especially the $\frac{3}{16}$?
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limit propert of Laplace transform

consider $f\in L^1(R_+)$ and define Laplace transform $$\mathcal{L}f(z):=\int_0^{\infty} f(s)e^{-zs}\mathbb{d}s. $$ How can I prove $$\lim_{\mathbf{Re}z\rightarrow\infty}\mathcal{L}f(z) = 0?$$ ...
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Laplace transform exercise

I found this on Priestley's Complex Analysis in the Laplace transforms bit. Suppose $f$ satisfies $f'(t)=f(kt)$ for $t>0$, where $0<k<1$ and $f(0)=1$. Prove that ...
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45 views

Did I use Laplace correctly?

I haven't done Laplace transforms in a while and I wanted to know if I did this right. I start out with the expression $$\tau\frac{dT}{dt}+T(t)=T_{a}$$ I took the Laplace of this expression and got ...
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174 views

Find the inverse laplace transform

How to find the inverse laplace transform of $$\frac{s(c-F(s))}{s-a}$$ where $a$ , $c$ are constants and $L^{-1}\{F(s)\}=f(t).$ I have found different answers by different approaches, where is the ...
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459 views

Physical meaning behind Frequency domain?

I understand its usage and why is it important because It transforms differential equations to algebraic ones.. But I can't get the physical meaning of the new form of the equation and the meaning of ...
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349 views

Solve differential equations using Laplace transform..

Solve each of the following differential equations with initial values using the Laplace Transform. $(b)\space y''-4y'+4y=0$ Where $y(0)=0$ and $y'(0)=3$ What I have so far: ...
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Selberg trace and Riemann zeros

Let us suppose we got a modification of the Selberg trace as follows $$ \sum_{n=0}^{\infty} h(r_n) = \frac{\mu(F)}{4 \pi } \int_{-\infty}^{\infty} r \, h(r) \tanh(\pi r) dr + \sum_{ \{T\} } \frac{ ...
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Unclear step on proof of Laplace transform of a derivative

Reading the article on the Laplace Transform in Wolfram MathWorld, I found the proof that $\mathcal{L}[f'(t)] = sF(s) - f(0)$. I understand the first and second steps, but I don't understand the ...
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Calculate the inverse Laplace transform of $\frac{\exp\big(\frac{-1}{s}\big)}{\sqrt{s}}$

If, $$\mathcal L \left\{ \frac{\cos(2\sqrt{3t})}{\sqrt{\pi t}} \right\}= \frac{\exp\big(\frac{-3}{s}\big)}{\sqrt{s}}$$, $$\mathcal L^{-1} \left\{ ...
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Laplace transform, proof that $L \{ \frac{1}{k}f(\frac{t}{k}) \}= F(ks)$

Let $L \{ f(t)\}=F(s)$, show that for all $k \in \mathbb{R}$, $k \neq 0$ $$L \{ \frac{1}{k}f(\frac{t}{k}) \}= F(ks)$$ if, $u=\frac{t}{k}$ $L \{ \frac{1}{k}f(\frac{t}{k}) \}= \int_0^{\infty} ...
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How can I find the time constant of a first order system transfer function?

How can I obtain the time constant of the transfer function of a first order system, such as the example below? $$ \frac{C(s)}{R(s)} = \frac{2}{s + 3}$$ Where $C(s)$ is the output of the system and ...
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How to find the inverse laplace transform of an arbitrary function

How to find $$\mathcal{L^{-1}} \left[ \frac{F(s)}{s+a} \right]$$where $F(s)$ is the Laplace transform of $f(t).$
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Does a piecewise-continuous function need to be defined at its points of discontinuities?

Is the following function considered piecewise-continuous?? I'm reading conflcting definitions in different places: some highlight that that the function need not be defined at the (jump/removable) ...
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How to find Laplace Transform

How to evaluate $\int_t^\infty e^{-sx}f(x)dx$ using laplace transform properties?
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the inverse laplace transform of an exponential function

I was solving differential equations by using laplace transform, but I am stuck here trying to find the laplace inverse below, can anyone help? $$ ...
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How to solve differential equations of type $x' = x^3 + x^2 + x$ using Laplace Transform?

How do i solve equations like, $f'(x) = f^3 + f^2 + f$ using laplace transforms? Any help would be appreciated.....
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Inverse Laplace transform - using the table

I am trying to find the inverse Laplace transform $(g(t))$ of $$ G(s) = \frac{2s}{(s+1)^2+4}$$ I know about the inverse transforms $e^{a t}\cos(\omega t)$ and $\mathrm{e}^{at}\sin(\omega t)$ ...
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How to find the laplace transform of $ |\sin (t)| $?

How to find the Laplace transform of $ |\sin (t)| $?
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Inverse Laplace transform calculation of $\frac{s+1}{s(s-4)}$

Find the inverse Laplace transform of $$\frac{s+1}{s(s-4)}$$ Can anybody help me, please.
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Inverse Laplace transform of $e^{-\pi s}/(s^2+3)$

Find the Inverse Laplace transform of $e^{-\pi s}/(s^2+3)$ Can anybody solve the above question please
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How to check solution of Laplace Transform ODE problem

$$\frac{\mathrm dw(t)}{\mathrm dt}+2w(t)=y(t)$$ $$\frac{\mathrm dy(t)}{\mathrm dt}+3y(t)=2w(t)+f(t)$$ The input to the system is $~f(t)$ and the output is $~y(t).$ The initial conditions are ...
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Solving the DE for a two-body system

I am interested in solving a differential equation from the two-body problem. This is not homework, and I just thought of this on my own during physics class one day. However, after many attempts, I ...
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Laplace transform of a Gaussian function with complex variable

The bilateral Laplace transform of a Gaussian function could be established as: $$e^{x^2/2}=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}e^{-xy}e^{-y^2/2} dy$$ Then what should be a similar relation ...
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What exactly is Laplace transform?

I've been working on Laplace transform for a while. I can carry it out on calculation and it's amazingly helpful. But I don't understand what exactly is it and how it works. I google and found out ...
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Fundamental matrix and exponential of matrix using Laplace Transform

I'm trying to work out how to find $$\exp(At)$$ for a system of linear differential equations $$x'=Ax.$$ I know that the solution is a fundamental matrix of the system such that $$\exp(At)=I$$ at ...
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Approximated Laplace transform of a non-linear system

Assume a system with dynamics: $\dot{\omega}(t) = \alpha \omega^2(t) + \beta i(t)$, where $\dot{\omega}(t), \omega(t)$ are system's states and $i(t)$ is the system's input. I'd like to approximate ...
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Laplace transform of a product of Modified Bessel Functions

Working with a scalar field in 2 dimensions I've come to the following integral, from which I can extract the proper ultraviolet behavior ($a \ll 1$) of the theory: $\int_0^\infty ...
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What kind of book would show where the inspiration for the Laplace transform came from?

I'm trying to find out where to learn about integral transforms and inversions like the Laplace transform and the Bromwich integral. I'm looking for a book that describes how you can find (derive) ...
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Help with a partial fraction decomposition

One of my homework problems last week was to find the inverse Laplace transform of the following: $$F(s)=\frac{2s+1}{s^2-2s+2}.$$ The answer is $f(t)= 2e^t \cos t + 3e^t \sin t$. Obviously once ...
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Laplace transform of $ \int_1^\infty\frac{\cos t}{t}dt$

Is the result of the of Laplace transform of $\int_1^\infty\frac{\cos t}{t}dt$ equal to $\frac{\int_1^\infty\frac{\cos t}{t}dt}{s}$?