The Laplace transform is a widely used integral transform, similar to the Fourier transform.

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What does “Formulate the system of equations for a finite difference discretisation of the problem” mean?

Subject: Partial Differential Equations. Here are the details of the question: $$ \frac{\partial ^2u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} = 0 $$ for $0 < x < 3, 0 < y < 1$ ...
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Laplace transform of $\sin^2(\omega t)$

What is the Laplace transform of the function $\sin^2(\omega t)$
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Expressing function in terms of unit step function

I have trouble expressing functions in terms of the unit step function, if someone could explain how it works that would be great. For example - $g(t) = t^2$ when $0 \le t < 2$ $4$ when ...
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Why does the Final Value Theorem not hold for a transfer function with more than one pole at the origin?

The Wikipedia article on the Final Value Theorem states the following for cases where it does not hold: There are two checks performed in Control theory which confirm valid results for the Final ...
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Laplace transform with initial value problem $y''+4y=12\sin(2t)$.

Using Laplace transforms solve the initial value problem. $$y''+4y = 12\text{sin}(2t); \qquad\qquad y(\pi)=-3, \quad y'(\pi)=-3$$ I have begun with writing: $\mathcal{L} (y'') = s^2y(s) -s y(\pi) ...
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Laplace transform of piecewise continuous function

$$f(t) =\begin{cases}t^2 & 0 \le t < 3,\\ 9& t \ge 3\end{cases}$$ Show that $f$ is of exponential order. Express $f$ in terms of the unit step function. Find Laplace transform of ...
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Laplace transform of convolution with no function of t

Instructions: Evaluate the given Laplace transform. Do not evaluate the integral before transforming. Problem Given: $\mathscr{L}\{\int_0^t e^{-\tau} cos\tau d\tau \}$ My Problem: To treat this as ...
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Proof that laplace's equation is rotationally invariant using chain rule

Suppose $(x, y)$ and $(p, q)$ are coordinates in the plane related by rotation around a fixed point $(a, b)$, as follows: $$\begin{bmatrix} p\\ q\end{bmatrix} = \begin{bmatrix} \cos(t) & -\sin(t) ...
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$y''+2y'+5y=0$, initial value problem with Laplace transform?

here is the question: $$ {\rm y}''\left(t\right) + 2\,{\rm y}'\left(t\right) + 5\,{\rm y}\left(t\right) = 0; \qquad\qquad {\rm y}\left(0\right) = 2\,,\quad {\rm y}'\left(0\right) = -1. $$ ...
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Find Laplace Transform of the following function

How do I find the Laplace transform for the function: $f(t)=t, 0 \leq t \leq 1$ and $2-t, t \geq 1$ I tried looking up the process online, but it remains unclear to me. Thanks in advance!
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Laplace transform, when $s \rightarrow \infty$

I'm reviewing lecture notes on Laplace Transform and there's one step that I don't understand: Find the solution to: $$x y'' + y' + xy = 0, y(0) = 1, y'(0) \mbox{ finite}$$ Taking the Laplace ...
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Solving a non-linear integro-differential equation

I am trying to solve the following equation $$ f^2(x) - g^2(x) = \alpha\int_0^x f(u) (x-u)du $$ For $\alpha=0$ we get $f=g$. I would like to see how the solution moves away from $g$ when I increase ...
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s-plane and fourier transform, together in 3d space.

I dont understand how can varying the real part in the s-plane make the amplitude in the fourier plane go to infinity. Lets say the pole is at -3 + -j for example.. Then the laplace transform is the ...
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60 views

Inverse Laplace transform of $s^{k}$

How can I find the inverse Laplace transform of $s^{k}$ where $k$ is non-integer and negative? I know that $$\mathcal{L}^{-1}[s^k] = \frac{1}{2\pi i}\int e^{st} s^k ds$$ and since we have ...
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Laplace's Method Integration

Consider the integral \begin{equation} I_n(x)=\int^2_1 (\log_{e}t) e^{-x(t-1)^{n}} \, dt \end{equation} Use Laplace's Method to show that \begin{equation} I_n(x) \sim \frac{1}{nx^\frac{2}{n}} ...
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Laplace Transform of tsin(at) using only the definition

Hello I' am stuck on how to get the final result of the laplace transform of $f(t)=tsin(at)$using (a is a constant) only the definition of $$\int_0^{\infty}f(t)e^{-st}dt$$, I know $sin(at)= {1 \over ...
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Laplace transform of a majorated function

I have the following problem. I have an analytic function and I want to show that it is majorated by a convenient function. To do that, it is very helpful to solve the transformed equation. I have a ...
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Laplace's Method (Integration)

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} Use Laplace's Method to show that \begin{equation} I(x) \sim ...
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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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Using Laplace Transform to solve a equation with piecewise function

Using Laplace Transform to solve$$y''+4y=f$$ Where $y(0)=0, y'(0)=-1,$ and:$$f(t)=\begin{cases}\cos(2t)&\text{if $0\le t \lt \pi$}\\0 &\text{otherwise}\\\end{cases} $$ Do I need to solve the ...
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Take the Laplace Transform

Take the Laplace transform of $$ \int_{0}^{t}x^2(x-t)^4 \cos(x)dx .$$ I'm not quite sure where to start...
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Deducing Laplace Formulas

I have to compute the followings integrals $\forall\; b\in \mathbb{C},\; \text{Re} \;b \gt0,p\gt 0$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x-ib}$$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x+ib}$$ ...
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Laplace transform using the definition

Find the Laplace of the given function using the definition $$f(t)=tsin(t)$$ I know what the answer is according to a sheet that I have of common transforms but I am not 100% on how to get there ...
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Laplace Transform of an integral

Find the Laplace transform of $$f(t)=t\int_0^{t} \tau e^{-\tau}$$ $L(f)(s)$= ?? My thought is that I can change the $\tau$ to $t$ by Transforming the integral to get $$t/s*L[t*e^{-t}]$$ But ...
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Laplace transform of integral equation

Use Laplace transforms to solve the integral equation $$y(t)-\frac{1}{2}\int_0^ty(t-v)~dv=1$$ First find the Laplace transform $Y(s)$ of $y(t)$
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Inverse Laplace Transform of Reciprocal Quadratic Function

Starting with the the equation: $$I(s)=\frac{6}{Ls^2 + Rs + \frac{1}{C}}$$ I need to find what $i(t)$ is by doing the inverse Laplace transform. I need to do some algebra to put it in a form that ...
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Laplace transform of $f(t)$ multiplied by $t^n$

How to prove that it is $(-1)^n\frac{d^n}{ds^n}F(s)$
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complex integral of z to the power alpha

I would like to perform an inverse laplace and at some point of the calculation I have to compute this integral $$\int_{\gamma-i\infty}^{\gamma+i\infty} z^{(1+n)\alpha-1}e^{z} \frac{dz}{2\pi i}$$ ...
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Laplace transform.

This is a past exam question. I'm having a bit of trouble at finding the inverse laplace transform of the following function. Any help would be great. $$\frac{s^2+1}{(s^2+4s+5)^2}$$ Thanks for ...
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Existence of inverse Laplace tranform

I have two questions about inverse Laplace transform. Given a function $F(s)$, does its inverse Laplace tranform always exists ? If it's not, assume $F(s)$ has an inverse Laplace tranform, does the ...
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Find the solution of the IVP using Laplace transforms

The equation is as such: $y''+y=t\sin t$; $y(0)=1, y'(0)=2$ I took the Laplace transform of both sides to yield $F(S)(s^{2}+1)-(s+2)=\frac{-2s}{(s^{2}+1)^{2}}$, and then ...
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Laplace Transform of a rounded function (or an infinitely discontinuous function)

Working on an assignment today, I thought of a problem I haven't been able to solve and haven't been able to find any solutions for online. What would the Laplace Transform of a rounded function be? ...
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Help Understanding Inverse Laplace Transforms???

I've been looking over some examples regarding inverse laplace transforms, and my textbook doesn't really go into much depth regarding examples such as this: F(s) = (2(s-1)e^(-2s))/(s^2 -2s + 2) I ...
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Laplace of $e^{-s+3}$?

I understand that the $e^{-ks}$ will time shift the function in the time domain by $k$ and will result in a time function of $u(t-3)$, but what does it mean when you have $e^{-s-k}$? How will that ...
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Help with basic Laplace Transform - unsure of procedure!!!

I am working on this Laplace Transform, and I've tried looking for a similar example off which to base my own work, but haven't been very successful. I'm confused by the formatting and don't know how ...
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Laplace transform of a piecewise function

I'd like to compute the Laplace transform of the following function: $$f(t) = \begin{cases} 0,& \mbox{if} \quad 0 \leq t \lt \pi \\ \sin(t), &\mbox{if} \quad t \geq \pi \end{cases}$$ Could ...
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properties of laplace transform

Obtain the transfer function for the following differential equation and check whether the input free solution is stable or not, $$\frac{dx}{dt} + 3x = f(t)$$ Please help, I don't even know where to ...
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Solving $y'-y=2\cos 5t$ using the Laplace Transform

Find the solution to the differential equation, using the Laplace Transform. $y'-y=2\cos 5t$, with initial condition $y(0)=0$. My attempt: First I take the Laplace Transform of each term. ...
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A question about Parseval's formula.

In operational calculus there is Parseval's theorem, which states that if $ f(t) \doteqdot F(p), \varphi(t) \doteqdot \Phi(p) $ and both $ F(p) $ and $ \Phi(p) $ are analytical in $ Re p \geq 0 $, ...
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Product of two Whittaker functions

According to 6.669.3 of Gradshteyn and Ryzhik the following identity $$ W_{a,b}(z_1)\,W_{a,b}(z_2) = \frac{2\sqrt{z_1z_2}}{\Gamma(1/2+b-a)\,\Gamma(1/2-b-a)}\int_0^\infty ...
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Is the Laplace transform additive? And why?

The first part is a simple question, but I cant find a clear answer. Does: $$\mathcal{L}(ax''(t)) = \mathcal{L}(a)\times\mathcal{L}(x''(t))$$ $a$ is a constant $x(t)$ is a variable that changes ...
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How can we take the LaPlace of a function raised to the power?

For example: $\mathcal{L}$((t-1)^1) Following simple linearity, we achieve the answer. However, following the power of theorem: (I'm not proficient enough in LaTex to write this...) I get the wrong ...
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How can we take the LaPlace transform of a piecewise function?

How can we take the LaPlace transform of a function, given piece-wise function notation? For example, $f(t)=\begin{cases} 0 &\mbox{for } 0<t<2\\ t&\mbox{ for } 2<t \end{cases}$ ...
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How to find the transfer function from the given differential equation

Question: find transfer function from differential eqn $y''(t)+2y'(t)+5=4x(t)$ I am confused about what happens to constant $5$ . will it be zero when we take laplace of whole eqn or not? Can ...
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Unit impulse / step response of a 1st order differential equation

You are given the equation $10v'(t) + 0.6 v(t) = f(t)$ $v(t)$ is the velocity of the object Determine the unit impulse response AND the unit step response. How would i approach this question? do i ...
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Laplace transform of gamma distribution

Gamma distribution has its pdf given by $f(t;k;\theta) = \frac{t^{k-1} e^{-t/\theta}}{\theta^k (k-1)!}$. Show that if the pdf's Laplace transform is $L_k (s)$, then $L_{k+1} (s) = \frac{L_k ...
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Are these Laplace transforms wrong in Stroud's Advanced Engineering Math Book?

I know that if you think a book is wrong, most probably it is your own mistake. However, I can't understand the following Laplace transforms in K. A. Stroud's "Advanced Engineering Mathematics". In ...
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Inverse Laplace Transform of $\frac{s^2+2s+2}{s+1}$

I want detailed steps of this if anyone can help.
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Zeros/poles at Laplace and at Fourier Transform

I recently started "relearning" the Laplace transform, and I noticed something. It seems to me that the intuitive idea of poles and zeros is different between these two transforms! For example, in ...
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Evaluating an integral with Laplace

We need to evaluate the following integral: $$\int_{0}^{\infty}\frac{\cos(tx)}{x^2+a^2}dx$$ There is the following note: "You may interchange taking the Laplace transform and integrating." I have ...