# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 ...
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(...