# Tagged Questions

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### Proof of modifed Final Value theorem

Let $f:[0,\infty] \longrightarrow C$ be a continuous and bounded function such that the limit $\lim_{t\to \infty } \frac1T \int_0^T f(t) dt = d$ exists. Let $F(s)$ be the Laplace transform ...
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### Solve second order differential equation with Heaviside function using Laplace transform

The equation is: $$y'' + 3y = u_4(t)\cos(5(t-4)), \quad y(0) = 0, \quad y'(0) = -2$$ Here $u_4$ is the Heaviside function with activation switch at $t=4$. I can get all the way to the partial ...
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### Coupled mass spring system with damping and initial values

After researching through the web, I can't figure out how to express into a differential equation a coupled mass spring system with damping and initial values. Two masses and two springs, no external ...
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### Coupled mass spring system with damping, I need help with the equation

I know that the equation $mx''+cx'+kx=f(t)$ is used for a normal mass spring system, but I don't know how to express the differential equation for a coupled mass spring system with damping. These are ...
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### Circuit RC, I need help with the equation.

A circuit RC it's described by the next equation: $\frac{1}{c} \int i(dt)+Ri=V$ Where the value of resistance is $R=10 k\omega$ , the value of the capacitor is $C=2.5 \mu F$, and the voltage of the ...
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### Inverse Laplace Transform, I need help

What is the ILT of $H(s)=\frac{7(3s+1)}{(s-3)(s^2+10s-13)}$ Also, if you kindly want to help with this another inverse transform, I'd really appreciate it: $H(s)=\frac{6(s+2)}{s^3(s-5)}$ Thanks!
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### Laplace transform of $g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$

Find Laplace transform for this function "$g$" $$g_n(t)=\begin{cases}\frac{(1-e^{-t})^n}{t^n}&:t>0,\\0&:t\le0.\end{cases}$$ Then Take advantage of it to calculate the following ...
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### Laplace Transformations of $\frac{1}{t}$

what is the laplace transform of $\frac{1}{t}$? I tried different ways like integrating by parts from the general form of laplace but it's getting more complex as my solution goes by.
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### Find the Laplace transform of the following hard equation

Ok so the objective is to factor this into something that resembles the Laplace tables. Give me some help pls. thx
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### Laplace transform with time shift property

ok so i have no idea how the inverse laplace went from $F(s)$ to $f(t)$. I understand $\frac{c}{s^2}$ => $ct$, and $\frac{b}{s}$ => $b$, but the $e^{-as}$ is what gets me. In my Laplace tables I ...
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### Laplace question

How do you express Laplace transform $\mathcal{L}(g)(z)=\int_{0}^\infty e^{-zt}g(t)dt$ with Fourier transform? And how do you form the reverse formula for Laplace transform using Laplace transform ...
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### Find the inverse Laplace transformation of $\frac{(s+1)e^{-s}}{s^2}$. [closed]

Find the inverse Laplace transformation of $\dfrac{(s+1)e^{-s}}{s^2}$.
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### $\mathcal {L}(\operatorname e^{-6t}\cos(5t))=?$ [closed]

Find the laplace transform of the following equation $f(t) =\operatorname e^{-6t}\cos (5t)$
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### Laplace transform of $(t-2)^2u_2$

The homework problem is $$f(t) = \begin{cases} 0 & t < 2\\ (t-2)^2 & t\geq 2\end{cases}$$ $f(t)$ as a step function $$f(t) = (t-2)^2u_2(t)$$ Using what we learned in class ...
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### Inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$

Trying to find the inverse Laplace transform of $\frac{s}{\sqrt{(s+a)^3}}$. So solving $\oint_B dz \: \frac{z}{\sqrt{(z+a)^3}} e^{z t}$ (Bromwich contour). I tried doing a u-substitution with $u=z+a$ ...
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### $f(t)=1+t-\dfrac{8}{3}\displaystyle\int_{0}^{t}(\tau-t)^3f(\tau) \ \mathrm d\tau \quad , f(t)=?$

$$f(t)=1+t-\dfrac{8}{3}\displaystyle\int_{0}^{t}(\tau- t)^3f(\tau) \ \mathrm d\tau$$ According to the convolution theorem, $\displaystyle\int_{0}^{t}(\tau- t)^3f(\tau)d\tau$ = $f(t) * t^3$ (I ...
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### What is the easiest way to find the inverse Laplace of F(s)?

$$F(s)= \frac{1}{(s-1)^2(1-1/s^2)}$$ Do I have to multiply by $s^2/s^2$ and then use partial fractions or is there a way to use the convolution theorem?
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### How do i find the lapalace transorm of this intergral using the convolution theorem?

$$\int_0^{t} e^{-x}\cos x \, dx$$ In the book, the $x$ is written as the greek letter "tau". Anyway, I'm confused about how to deal with this problem because the $f(t)$ is clearly $\cos t$, but ...
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### Solve for z(t) from the simultaneous equation using Laplace transform

Solve for $z(t)$ from the simultaneous equation using Laplace transform $$y' + 2y + 6 \int\limits_0^t z \mathrm{d}t = -2 u(t) \\ y' + z' + z = 0$$ subject to $y(0) = -5$ and $z(0) = 6$.
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### determine the locations of all the poles and zeros (including zeros at s = infinite). Make an S-Plane plot of the infinite poles and zeros

Determine the locations of all the poles and zeros (including zeros at $S = \infty$). Make an $S$-Plane plot of the infinite poles and zeros. $$G(s) = \dfrac{5S^2 + 20S + 15}{S(S + 3)(S^2 + 4S + 4)}$$ ...
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### How do i find the inverse laplace?

$$F(s) = \frac{2s-1}{s^2(s+1)^3}$$ If I try to use partial fractions, I end up with 8 constants to solve for! Is there some shortcut I'm not seeing? Am I supposed to simplify it first? Am I even ...
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### Laplace Transform using t-shift

$$f(t)=\begin{cases}cos(Ï€t), & 1\leq t < 4 \\ 0, &elsewhere \end{cases}$$ Okay, I attempted to write it in terms of step functions and I got $$f(t) = cos(Ï€t)u(t-1)-cos(Ï€t)u(t-4)$$ But ...
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### Laplace Transform using t-shift (second shift)

$$f(t) = tu(t-Ï€)$$ I know I have to get t in terms of $$(t-Ï€)$$ and to do that I have done $$t = a(t-Ï€) + b$$ $$t = at-aÏ€ + b$$ $$t = (a-Ï€)t + b$$ $$(a-Ï€) = 1$$ and $$b = 0$$ Then I think I ...
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### How to use Complex Inversion Theorem to find the Inverse Laplace Transform?

How to use Complex Inversion Theorem to find the Inverse Laplace Transform for the given $F(t)=L^{-1} \{s^{-1/2} e^{-1/s}\}$ ? Hint: make the radius $\epsilon$ of the inner circle $t-1/2$ rather than ...
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### Finding Inverse Laplace Transform using Taylor Series

Find the inverse Laplace transform $F(t)=\mathcal{L}^{-1}(s^{-\frac{1}{2}}e^{-\frac{1}{s}})$ using each of the following techniques: Expand the exponential in a Taylor series about s=âˆž, and take ...
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### inverse of laplace transform

How to compute this inverse Laplace transform ? $$\displaystyle{ \mathcal{L^{-1}} \left\{ \frac{1}{s(\exp(s)+1)} \right\} }$$ Thanks.
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### Find the Laplace transform from given graph

![an image of the fuction][1] How do I find the function from its graph here to find its Laplace transform?
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### Inverse Laplace Transform of s/(s+1)

What is the inverse laplace transform of $\frac{s}{s+1}$? My work was: $$X(s)=\frac{s}{s+1}\\ X(s)=s\frac{1}{s+1}\\ x(t)=\frac{d}{dt}e^{-t}=-e^{-t}$$ My only issue is that when I check my answer ...
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### Laplace transform of $\cos(at)$

I need to find the Laplace transform of $\cos(at)$ I know that $L\{\cos(at)\}= \int_{0}^{\infty} e^{-st} \cos (at) dt$ but I am having trouble finding the integral Thank you
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### Inverse Laplace Transform of a polynomial fraction

How do I find the inverse Laplace transform of $\;\;\large\frac{4s}{(s^2+4)^2}\;\;$?
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### Solving Differential equation with laplace transformation

Solve the differential equation with the laplace transformation. $$y''-4y'+9y=9\quad,\quad y(0)=0 \quad ,\quad y'(0)=-8$$ I will solve this question to this state, but I cannot continue. ...
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### Functions without Laplace transform?

We have just started working with Laplace transformations at our university course. One of the I came across as following: Provide three examples of functions for which the Laplace transform does ...
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### Inverse Laplace Transform involving $\cosh$.

While doing an assignment on solving a PDE I stumbled into the following inverse Laplace transform question (involving $\cosh$? I can't believe it). Mathematica gives no solution and I have no idea ...
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### How to Find Inverse Laplace Transform of $F(s)=\frac{1}{\pi} \cot^{-1}(\frac{10s}{\pi})$

$$F(s)=\frac{\cot^{-1}(\frac{10s}{\pi})}{\pi}$$ $$f(t) = ?$$
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### Did I use Laplace correctly?

I haven't done Laplace transforms in a while and I wanted to know if I did this right. I start out with the expression $$\tau\frac{dT}{dt}+T(t)=T_{a}$$ I took the Laplace of this expression and got ...