# 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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### Burgers Equation and Coupled Burgers Equation [closed]

What is the difference between the Burgers Equation and Coupled Burgers Equation 1D, 2D, and 3D forms?
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### Calculate zeros (root of unit) by using partial fraction expansion and complex analysis, from David Kendall

I want to fill in the derivation of the following Harris's book https://www.rand.org/content/dam/rand/pubs/reports/2009/R381.pdf Specifically, I want to know how the author obtains $a_r$ after eq16.2 ...
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### Determining the Laplace Transform of a triangle signal

I have been given the following signal: I have written this using the Heaviside function. This gave me the following: $$x(t) = t[u(t)-u(t-1)] + t[u(t-1)-u(t-2)] = tu(t) - tu(t-2)$$ Applying the ...
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### General solution of the wave equation using Laplace transform in the time domain

I have been exploring different methods of solving PDEs and in general various solutions of the wave equation under more general boundary conditions (like if the boundaries are ($-\infty$, $\infty$) ...
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### finding the transfer equation of $\frac{H(s)}{Q_d(s)}$ in liquid control system with hydraulic integral controller.

On problem B-4-10 So my attempt is as follows: from the liquid control system we have $(q_d+q_i-q_o)dt=Cdh$ since $R=\frac{h}{q_0}$ and $R=0.5$ then $q_0=2h$. By the equibilirium system on the lever ...
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### On two-sided Laplace transform in probability; how to show it is injective?

I am studying a theorem in probability that the Laplace transform of a (nonnegative) random variable determines the law of that random variable, which is equivalent to its injectivity. The book (...
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1 vote
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### Laplace transform of $\sin(\omega t)$

I am learning about the Laplace transform and I know I got the answer to this example question wrong, but I'm trying to figure out if I just made a calculus or algebra type error, or if I'm ...
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### Solving 1st order PDE including convolution

I'm studying Van Kampen's "Stochastic processes in physics and chemistry" and stuck to some exercise (p.78): That is, solving \frac{\partial P(y, t)}{\partial t}=\int_{-\...
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### Using the Laplace transform, solve the system of linear differential equations with constant coefficients

Using the Laplace transform, solve the system of linear differential equations with constant coefficients $y' + z' = t,$ $y'' - z = e^{-t},$ $y(0) = 3,$ $y'(0) = -2,$ $z(0) = 0,$ where $y(t)$...
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### Calculate the Laplace transform of the function $\frac{\sin t}{t}$ and then use it to compute the integral

Calculate the Laplace transform of the function $\frac{\sin t}{t}$ and then use it to compute the integral $$\int_0^\infty \frac{e^{-at} - e^{-bt}}{t} \, dt,$$ where $a, b \in \mathbb{R}$. Attempt: ...
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### How to show the laplace trasnform of a translated piecewise function

I'm learning laplace transforms, and righ now I'm on their properties. I am given the following problem: I've done the obvious of ∫ f(t-c) exp(-st)ds but I get stuck at this point on how to integrate....
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### Why do we care about convergence of the Laplace transform?

When I took elementary differential equations, with the textbook of Boyce & DiPrima, I learned about using the Laplace transform to solve some initial value problems. I also took a course in ...
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### Laplace Transform solve ODE second order

Find ODE second order using Laplace Transform $y" + 3y' + 4y = 3t$ $; y(0) = 2$ $; y'(0) = 2$ ( This is imaginary roots) I have found the answer using undetermined coefficient but i got stuck when ...
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### Finding the function $g(t)$ given its Laplace transform $F(1/s)$

I am trying to find a function g(t) given that its Laplace transform is F(1/s), where F(s) is the Laplace transform of another function f(t). I know that if f(t) has the Laplace transform F(s), then f(...
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### Inverse Laplace Transform of $e^{-as}/s^2$ for $a>0$
I am trying to compute the inverse Laplace transform of $$F(s) = \frac{e^{-as}}{s^2}$$ for $a > 0$. I computed it as follows: \mathcal{L}^{-1}_{s\to t} \left\{\frac{e^{-as}}{s^2}\right\} = \text{...