# Questions tagged [laplace-transform]

The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.

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### How would I simplify this Laplace transform?

I was reading an article about using a Laplace Transform on a fractional differential equation and attempted to derieve a formula that was just skipped over to. The Laplace transform is as follows: ...
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### Solve Laplace Transformation using Convolution

Find $\displaystyle \mathcal{L}\left (\int\limits_0^t e^{-2\tau}\sin(3\tau)\;d\tau \right )$. I know we can solve this problem by solving the integral and then finding the Laplace Transformation. Is ...
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### to find inverse laplace transform of $f(s)=s\ln\left|\frac s{\sqrt{s^2+1}}\right|$

I tried the differential property of the Laplace transform to the logarithm component, then I tried the “multiplying by s” property, but the answer would be infinity. What do you think is the solution?...
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### Solve $f(t)=e^t+e^t\int_0^te^{-\tau}f(\tau)\mathrm{d}\tau$

The question: $$\begin{equation*} f(t)=e^t+e^t\int_0^te^{-\tau}f(\tau)\mathrm{d}\tau \end{equation*}.$$ I believe we need to take the Laplace transform of all terms. I am getting stuck with this part: ...
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### Commutative law in convolution of laplace transfer

This question needs to use t-shifting and convolution in inverse laplace transfer. I tried to consider t-shifting first, then did convolution as shown in below. Since there is u(t-a) function, I ...
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### Solving nonhomogeneous ordinary differential equation by use of Laplace transform

Given the differential equation $$𝑦′′ − 5𝑦′ + 6𝑦 = e^{−𝑡}, \quad 𝑦(0) = 𝑐_1, \, 𝑦’(0) = 𝑐_2$$ how can I use the Laplace transform to solve this nonhomogeneous differential equation ?
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### Is there a 3D version of the convolution and cross-correlation theorems for the Laplace transform?

The Laplace-transform version of the convolution and cross-correlation theorems is essentially the same as the "usual" (Fourier-transform) version: if $\mathcal{L}[f(t)]$ is the Laplace ...
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### Laplace transform of a PDE

I am learning how to use Laplace transforms to solve PDEs. In case of an expression like \begin{equation} \int_{r}^{\infty} f(x,t) g(x) dx \end{equation} the resulting Laplace transform (with respect ...
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### Laplace transform of normal distributions and their convolution

Hi (apologies for my english), i have been interested in the CLT for the past weeks, and already have proved that the convolution of 2 normal distributions is another normal distribution using the ...
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### Solutions and graphs of partial integro differential equations

Consider $$\partial_tu+ \partial_xu-D\partial_{yy}u=0,$$ subject to the initial condition $u(x,y,0) = v(x,y)$ and the boundary conditions at the infinities, $u(0,0,t) = u(\pm\infty,\pm\infty,t) = 0,$ ...
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### Verifying the inverse Laplace transform for a production-inventory problem: total expected backlogs when demand is Poisson

I am entirely self-taught when it comes to Laplace transforms, and I am seeking an independent opinion on my attempt to work out how to arrive at the below expression (note: I am interested ...
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### First-passage time distribution in Laplace space?

I'm struggling to understand the reasoning between moving between two steps in a reaction scheme for a paper I am reading. For this (from the description), the probabilities over different paths are ...
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### Sum the inverse of the successor of the square of the natural numbers

I was thinking if I could solve the sum $\sum_{n=1}^\infty \frac{1}{n^2+1}$, and I reached using the laplace transform of the sine at the integral $\int_0^\infty \frac{\sin(t)}{e^t-1} dt$. I entered ...
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### Using Laplace- and Fourier-Transformation for solving dynamic beam equation

let's assume we have a cantilever beam of length L, which is clamped at left end (x=0) and is loaded with a time-varying force $F(t)$ at the free end (x=L). Further, the bending stiffness is $EI=const$...
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### Incorrect Transfer Function - Control Systems

I want to find the transfer function for a system described by this non-linear state-space model: \begin{equation} X(t) = \begin{bmatrix} y(t)\\ v(t)\\ ...
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I would like to solve the IVP $$y'' + 2y' + 3y = 3t$$ with $y(0)=0, y'(0)=1$ using transform methods. Could someone check my work is correct, as my solution differs from that in the text? Solution. ...