Questions tagged [inverselaplace]

This tag is for questions regarding to "Inverse Laplace Transform" which is the transformation of a Laplace transform into a function of time.

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16 views

Solving Inverse Laplace Transform of the type: $\mathcal{L}^{-1}\left\{\left(-\frac{1}{s}\right)F\left(-\frac{1}{s}\right)\right\}\left(t\right)$

Since Laplace transforms of the type: $$\mathcal{L} \left\{g\left(t\right)\right\}\left(s\right) = sF\left(s\right)$$ can be simplified as by finding $f\left(t\right)$ such that: $$f\left(t\right)=\...
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1answer
45 views

Why does the Bilateral Laplace Transform of a constant function not exist?

I have always accepted that the bilateral Laplace Transform of a constant function $f(t) = c$ does not exist. How could the following integral possibly converge, $$\mathcal{L}[f(t)]=\int\limits_\...
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52 views

Two different ways of finding the Inverse Laplace?

I was recently given a question to solve by my professor that required me to find the Inverse Laplace. I'm sure my solution was correct but I'm wondering if my professors solution is also correct. My ...
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0answers
13 views

How to calculate the ILT of two functions with variable s? [closed]

I want to calculate the inverse Laplace transform of two expressions with variable s. I think the residue theorem is capable. But I dont' know how to do it. Would you like to help me with this problem?...
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1answer
32 views

Why do we require this condition in the Laplace inversion formula?

My question is regarding the Laplace transform and it's inversion formula given by the "Mellin", "Bromwich" or "Fourier-Mellin" integrals (found on wikipedia). Consider ...
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39 views

The Essence of Generation Functions and Coefficient Extraction

Given the roles generating functions and coefficient extraction play in solving recurrence relations, they are clearly analogous to the Laplace Transform and Inverse Laplace Transform. A hypothesis ...
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1answer
38 views

How to find the laplace transform of the function $f(t)= \sqrt{t}\sin(t) $

I would like to ask on how to find the laplace transform of the function $f(t)=\sqrt{t}\sin(t)$ when i seached on wolframmath it seems that the answer used a gamma function and has a sine of arctan of ...
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1answer
48 views

How to find the inverse Laplace $\mathcal{L}^{-1}[\ln(1+\frac{a^2}{s^2})]$

I am trying to calculate the inverse Laplace $\mathcal{L}^{-1}[\ln(1+\frac{a^2}{s^2})]$ **My attempt:** We know from a basic Laplace property that $$\mathcal{L}^{-1}[F'(s)]=(-1)tf(t) = -t\mathcal{L}[...
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1answer
46 views

Laplace transform of the improper integral of a function

i need to solve this ODE using Laplace transform. $\theta(t)$ is the Heaviside step function. $x$ and $x'$ both have a Laplace Transform. $$ x'(t) = \sin(2t)\theta(t-\pi)+\int_0^{\infty}x(t-\tau)d\tau~...
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56 views

Inverse Laplace transform of $e^{-\sqrt{s+{1\over s}}}$

I am struggling on solving a PDE group by Laplace transform method. I could not find similar inverse equations to the question I listed. I would appreciate if you could give me some suggestions.
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1answer
44 views

Inverse Laplace transform of $\ln \left( 1 + \frac{4 a}{s^2} \right)$

I would like to find the following inverse Laplace transform. $$\mathcal L^{-1} \left[ \ln \left( 1 + \frac{4 a}{s^2} \right) \right]$$ I don't know because of $4a$.
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16 views

Laplace inverse of $e^{-as-bs^2}/(as+bs^2)$

I want to know how to do the laplace inverse transform $$e^{-as-bs^2}/(as+bs^2).$$ I have tried to use convolution and any fomula I know, but I cannot deal with one part, $e^{bs^2}$. I searched ...
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1answer
16 views

Inverse Laplace of a transfer function for $\frac{K_{s}.P_{0}}{s.(s.T_{1}+1)} + \frac{V_{a}}{s}$

I have the following function in the Laplace domain: $$\frac{K_{s}.P_{0}}{s.(s.T_{1}+1)} + \frac{V_{a}}{s}$$ And I want to do the inverse Laplace transform.So, this is my result: $$L^{-1} (V(s))=K_s....
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23 views

Inverse Laplace Transform with frequency change

I have two functions $f,g$ in $t$ s.t. their Laplace Transforms are $F,G$ in $s$. I have also, for some constant $k$, $$F(s)-1=(s+k)G(s)$$ If I had instead $$F(s)-1=(s+k)G(s+k)$$ I'll get $$f(t)-...
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36 views

Solution for y for $ y''+4y=g\left(t\right),\ y\left(0\right)=3,\ y'\left(0\right)=-1 $ using Laplace transformation?

How to find the solution for y for this: $$ y''+4y=g\left(t\right),\ y\left(0\right)=3,\ y'\left(0\right)=-1 $$ using Laplace transformation? Since $ 0 \neq y\left(0\right) \neq y'\left(0\right) \neq ...
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1answer
36 views

What is the inverse Laplace of $(e^{-sx/c})(-f_0/s^3)$

Where $c$ and $f_0$ are constants. I know it should be of the form $H(t-a)f(t-a)$ but I got lost a bit
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34 views

How to proceed with inverse Laplace of $s\log\frac{s+4}{s-4}$ [duplicate]

I have found the inverse Laplace transform of $\log\frac{s+4}{s-4}$, but do not know how to proceed
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1answer
68 views

Any good math calculators online to find inverse laplace transforms? Specifically $s\log\frac{s+4}{s-4}$

I think I have found the final answer $\left(4 \left[\frac{2t \cosh(4t) - \sinh(4t)}{t^2}\right]\right)$ but need to verify it.
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2answers
56 views

Using Laplace transform method, solve $y'''- 3y'' + y' - y = t^2e^{2t}$

Using Laplace transform method, solve $$\dfrac {d^3y}{dt^3} - 3\dfrac {d^2y}{dt^2} + \dfrac {dy}{dt} - y = t^2e^{2t}$$ given $y (0) = 1, y′(0) = 0, y′′(0) = –2$. I'm not able to factorize once ...
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0answers
15 views

How the following Approximation is done to obtain the Inverse Laplace Transform?

The following highlighted part from a textbook is where I am stuck. The Laplace transform is independent of the derivations made prior to equation $(2.77)$. Please help me to understand how they have ...
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19 views

physical significance of dirac in inverse laplace

i was working through this old electronics SE question. Initial conditions have the capacitor voltage and inductor current both zero. Using Laplace we can find the voltage across the inductor, ${30s/...
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1answer
22 views

Solve using the Laplace transform of the initial value problem

The given equation and initial values are: $$\frac{d^2x}{dt^2}+25x=50e^{5t}$$ $$x(0)=0 \space, x^{'}(0)=0$$ Then taking the Laplace transform of the given: $$\mathscr{L}\left[x^{''}+25x \right]=\...
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1answer
37 views

Finding an Inverse Laplace Transform

Given the differential equation $y'' +2y'+5y = 3e^{-x}\sin(x)$, with $y(0) = 0$, $y'(0) = 3$, I'm asked to use Laplace transforms to find the solution. So I used the tables to get the following: $\...
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4answers
60 views

Different methods to find Inverse Laplace Transform

Find Laplace Transform of:$$F(s)=\frac{1}{s^4(s^2+1)}$$ It was the Bonus point question in my exam. I solved it with this Lemma : Let $F(s)=\mathcal{L}\{f(t)\}$, we have $\frac{F(s)}{s}=\mathcal{L}\{...
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0answers
30 views

Solve using Laplace Transform $2y''+5y'+2y=e^{-2t}$

My teacher gave this question to solve using Laplace Transform \begin{cases} 2y''+5y'+2y={e^{-2t}}\\ y(0)=1, y'(0)=1 \end{cases} Taking Laplace transform on both sides we get: $$L\{2y''(t)+5y'(t)+...
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0answers
29 views

Inverse Laplace transform of $s^c/log(s)$

I'm trying to solve for the inverse Laplace transform of $\frac{s^c}{\log(s)}$ where $c$ is some constant. Mathematica is apparently unable to solve it, and while I know there's a running joke that ...
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2answers
34 views

I know it uses convolution theorem of inverse Laplace function but wasn't able to apply it .

Find inverse laplace for: $$f(s)=\dfrac 1{(s-2)(s+2)^2} $$ I know it uses convolution theorem of inverse Laplace function but wasn't able to apply it .
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73 views

Dirac-delta in real exponential form?

While I was studying about the spin-glass model based on replica theory (Spin-glass theory for pedestrians, page 16, equation 49), I have encountered something like an exponential form of Dirac-delta ...
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1answer
32 views

Solving IVP with Laplace transform involving step function and summation

Given the IVP $$ y'' + y = f(t) , \qquad\quad y(0) = 0 , \quad y'(0) = 0 , \tag{1}$$ where $$ f_{k} (t) = u_{0} + 2 \sum_{k=1}^{n} (-1)^{k} u_{k \pi}(t). \tag{2}$$ We want to find the solution. My ...
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24 views

Solution for genera second order PDE

I appreciate if you can help me to solve the below PDE: $$\frac{\partial f}{\partial t}-\frac{\partial ^2 f}{\partial x^2}+\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}=0$$ I took the ...
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0answers
19 views

What is the inverse Laplace transform of $\sin\left[\sqrt{\frac{r-s}{D}}x\right]$, from $s$ to $t$

I was doing a exercise and I couldn't finish it because I needed to do this inverse Laplace transform, and I got no idea how to do it. I tried convolution, I expanded the sin to complex exponential ...
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1answer
30 views

Impulse/Delta Function--homework help

I need to solve the initial value problem: I took the Laplace transform of both sides and this is what I have thus far: I now need to take the inverse Laplace transform to find x(t). I can't ...
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0answers
31 views

Simplifying function to take inverse Laplace transform

I have the following function: $100/(50 + \sqrt{(a + sb)/sc}) $ where a, b, and c, are constants and $s=j\omega$. I have tried using inverse Laplace calculators online, but I get that there are no ...
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1answer
44 views

How to find the inverse laplace transform of $\frac{48s+36}{s(6s^2+11s+6)}$? [duplicate]

How to find the inverse laplace transform of this function? $$F(s) = \displaystyle \frac{48s+36}{s(6s^2+11s+6)}.$$ It has been hard to break down the denominator part. Any help from you guys would ...
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1answer
51 views

What is the inverse Laplace transform of $\frac{1}{1+ks^{\alpha}}$?

What is the inverse Laplace transform of F(s)=$\frac{1}{1+ks^{\alpha}}$, $k$ is the constant and $\alpha$ is another constant. Especially when $\alpha$ is not an integer.
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1answer
100 views

Integration $\int_{0}^{\infty}\frac{b^2 \sin(ax)}{x^2+b^2}\,dx$

$$\int_{0}^{\infty}\frac{b^2 \sin(ax)}{x^2+b^2}\,dx$$ Integration of such function using inverse Laplace transform is possible or other approaches should be applied?
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0answers
20 views

Bromwich integral transformed to an integral on the real axis

I am new in complex integration and inverse Laplace transforms. The author of a textbook claims that the inverse Laplace transform has expression $$ f(t) = \frac{2\exp(bt)}{\pi}\int_0^\infty\Re\bigl(\...
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1answer
64 views

$\mathcal{L}^{-1}\left\{W(x)\right\}$

I've been looking at the Laplace Transform of the Lambert W Function. While attempting to answer it, I wondered what the inverse Laplace Transform would be: $$\mathcal{L}^{-1}\left\{W(t)\right\}$$ I ...
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2answers
52 views

Inverse Laplace transorm of $\sqrt s/(s^2+1)$

Can anyone calculate inverse Laplace of $$F(s) = \frac{\sqrt{s}}{{s^2+1}} $$ ?
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3answers
33 views

Estimation of the upper bound of the integral

I studied the of @Ron Gordon answer to the question How to find inverse laplace transform of 2s√2s√+1 where is estimation of the magnitude of the integral over $C_2$ $$\oint_{C_2} dz \frac{e^{z t}}{...
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1answer
38 views

How to pass from Laplace transform $\hat{\Psi}_z$ to $\Psi_z$?

Let's define the Laplace Transform $$\hat{\Psi}_z = \int_{\mathbb{R}} e^{-izt}\Psi_t \,dt$$ and the Laplace Transform $$\hat{\Phi}_z=\int_{\mathbb{R}} e^{-izt} \Phi_z \,dt$$ If I know that $$\hat{\Psi}...
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0answers
64 views

Mellin/Bromwich Integral (Inverse Laplace Transform) problem

I have a solution in laplace images: \begin{align} &p_f(x,s) = - \frac{1}{s}\frac{b}{a} \frac{1}{\sqrt{\frac{s}{a}+ \frac{b}{a} \frac{\sqrt{ s}}{1+c\sqrt{s}}}} e^{-x \sqrt{\frac{s}{a}+ \frac{b}{...
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1answer
182 views

How do Integral Transforms work

It's been a while since I have learnt Laplace's Transform and I am taking a look at Fourier's. But I feel I know nothing about them, just how to use in calculations. So I would like to have Any ...
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1answer
125 views

Does the inverse Laplace transform of $F(s)=\frac{\sin(\xi\sqrt{s})}{\sqrt{s}}$ exist for some $\xi\in\mathbb{C}$?

In particular, I am interested in $\xi = z\sqrt{i}$, with $z>0$. To begin with, formal considerations, such as $$\begin{align*}\mathcal{L}^{-1}\!\left[F(s)\right]=\frac{1}{2i}\,\mathcal{L}^{-1}\!\...
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2answers
19 views

Laplace inverse of given question

$$\mathcal{L}^{-1}\left(\frac{2 s - 1}{s^{4} + s^{2} + 1}\right)=~~?$$ I have done the $~\dfrac{2s}{s^4+s^2+1}~$. But what to do with the $~\dfrac{1}{s^4+s^2+1}~$ ? I don't get any idea after ...
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1answer
21 views

Laplace inverse of the question on unit step function

$$L^{-1} \left\{\dfrac{e^{-1/s}}{s}\right\}=?$$ First it use the integral property of Laplace and unit step function but I don't how to do it.
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32 views

How should I do to inverse Laplace transform without table?

Can I use integration method like fourier transform? How should I integral $$ f(p)=\frac a {p^2+a^2}$$ to inverse it to $\sin(at)$ ?
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0answers
36 views

Converting complex functions to $a + ib$ form and extracting real component for an algorithm

I'm preparing a bunch of Laplace $s$-space equations for numerical iteration in a Bromwich integral/series to invert and return it to $t$-space (or in my case, from $p$- to $u$-space for a non-time ...
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1answer
46 views

Inverse Laplace transform of $\frac{2n!}{2s\cdot(2s+1)\cdot(2s+2)\cdots(2s+n)}$

Find the original f function if it's Laplace transform $F(s)$ is equal to : $$\frac{2n!}{2s\cdot(2s+1)\cdot(2s+2)\cdots(2s+n)}$$ This is looking tricky and I don't know how to start it The correct ...
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1answer
30 views

How to make this Laplace function work? (Inverse Laplace transform)

In terms of force (F), acceleration (a), velocity (v), and position (y), with variables of impedance (R) and mass (m), I have a system where: $y(t) = \frac{v(0) m}{2 R} (1-e^{\frac{-2 R t}{m}})$ $v(...