# Tagged Questions

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

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### Laplace transform problem [closed]

This is the problem. I know ℒ (t^n) = ℒ (t^n-1) x n/s, which should give us -2/sqrt(s). That's as far as I've come and I don't get the hint either. Where does the pi even come from?
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### Cross-correlation, Fourier transform and Laplace transform: measure of how much signal are alike?

I'm studying electrical engineering and use correlation, Fourier transform and Laplace transform a lot. I know how and when to use them, however, the interpretation I've seen in the lectures still ...
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### Circuit Analysis problem (find the problem)

In this question, I know that $\text{C},\text{R},\text{T},\text{A}\in\mathbb{R}^+$ I've this circuit (the bottom of the resitor is connected to earth ($0$)): When I use Laplace transform I can find ...
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### Expressing the Weierstrass transform in terms of the unilateral Laplace transform

I was looking for a way to express the Weiestrass transform of $f(t)$, $$\mathcal{W}\{f(t)\}(s)=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^{\infty}f(t)\exp\left(-\frac{(s-t)^2}{4}\right)dt$$ in terms of the ...
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### Laplace transform ODE

Use laplace transform to solve the ODE $y''(t)+4y(t)=4u(\pi-t)cos(t)$,,,,,,, $y(0)=y'(0)=0$ u is the unit step function (heaviside function) I use: $u(\pi-t)=1-u(t-pi)$ By inserting this and ...
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### How does multiple integral change into terms multiplying each other in convolution theorem of Laplace?

In the steps of the proofs highlighted below, how does a multiple integral changes in to multiplication of two integral. This is only possible if V is independent of u, but as it turns out V = t - u, ...
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### Laplace transform of $\cos(2t-(\frac\pi3))$

Problem I need to find the Laplace transform of $\cos(2t-(\frac\pi3))$ Attempt I've tried to look up some relevant formulae in my book, but I can't find anything that looks useful. I suspect there ...
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### How do you integrate $e^{-st}t\cos(t)$?

I'm doing differential equations and specifically studying Laplace Transformations, where of course the Kernel is: $K(s,t) = e^{-st}$ And the Laplace Transformation $\mathcal{L}$ of a function $f(t)$...
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### Is it possible that a PDE solved by two different analytical methods with same Initial and boundary values give different results?

I have developed two models of same scenario. Both models involve a PDE which is solved with same Initial and Boundary conditions. In one model it is solved with Laplace transform and in other with ...
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### Laplace transform and inverse laplace transform

1- Find laplace transform for $4e^2t-3\cos^2(2t)+2\cosh(3t)$ My answer $L(4e^2t-3cos^2(2t)+2cosh(3t))=4L(e^2t)-3L(cos^2(2t))+2L(cosh(3t))$ $=\frac4 {s-2}-3L(\cos^2(2t))+\frac{2s}{s^2-9}$ But how ...
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### Compute Heaviside Laplace transform, then use this to solve initial value problem

I've been stuck on this problem for a while, and can't really seem to find where I should go with it, or where I went wrong if I made a mistake. Let $L(x)$ denote the Laplace transform of x. Q: ...
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### Dirac Delta Function Problem

In my Differential Equations class, we had the following equation on a test today: $y''+6y'=2\delta(t)$, $y(0)=0$, $y'(0)=1$. I got the following using Laplace Transforms (the only way I know how to ...
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### Laplace pairs - proof of summation transform

I am studying this question for my finals revision and I'm lost on how to start it, can anyone suggest something? Probably pretty simple but I've hit a dead end. Here's the question: If $F_i(t)$ ...
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### Solving integral equation with Laplace transform.

The equation in question is: $y(t)+\lambda \int_{0}^{t}y(\tau)d\tau=t$ My work so far is first to Laplace domain:  Y(s)+\frac{\lambda Y(S)}{s}=\frac{1}{s^2} \longrightarrow Y(S)=\frac{1}{s^2} \...
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### Laplace Transforms Property

Please see image. I'm confused by how the went about to get d[exp(-st)] in the second line Thanks Laplace Transform
During the past several hours, while studying the Laplace transform, I've started wondering what $$\mathcal{L} \{ \cos^n(at)\}(s)$$ would look like – since it won't ...
We have to show that $\mathcal{L}\{t^2e^{-3t}\cos{at}\}=-\frac{s^3-9s^2+(3a^2+9)s+5a^2+33}{(s^2+2s+a^2+9)^3}$. Steps : \$\mathcal{L}\{t^2\cos(at)\}=(-1)^2\frac{d^2}{ds^2}\left(\frac{s}{s^2+a^2}\...