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

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Solving PDE by Laplace Transform

Use Laplace transforms to solve the boundary value problem $$Y_{xx}(x,t)-2Y_{tx}(x,t)+Y_{tt}(x,t)=0, \quad 0<x<1, t>0$$ $$Y(x,0)=Y_t(x,0)=0, \quad 0<x<1$$ $$Y(0,t)=0, \ Y(t,1)=F(t), ...
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33 views

Laplace transform of a differential equation?

Find the unique solution of $y''+ y = f$, $y(0) = y'(0) = 0$ with the $2\pi$ periodic function given by $f(t)=2\pi \sin(t)$. I am having trouble setting up and starting the the question. I would be ...
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1answer
60 views

Using complex analysis to find the Inverse Laplace transform

I have been reviewing for my comprehensive graduation exam where I have been solving the Inverse Laplace transform via complex analysis. Consider $$ H(s) = \frac{s^2 - s + 1}{(s + 1)^2} $$ Then we ...
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44 views

Laplace transform of $f(t)=\left|\sin\frac{t}{2}\right|$?

If you are given a rectified sine wave, $$f(t)=\left|\sin\frac{t}{2}\right|$$ how do you find the Laplace transform of this? I tried using the equation $$L\{ f(t)\} = \frac{1}{1-e^{-sT}} \int_0^T ...
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93 views

Laplace transform of a sawtooth wave

Find the Laplace transform of the periodic function such that $f(t) = t$ if $0\leq t < 2\pi$ I am having trouble setting up this question. Am I on the right path? $$ \mathcal{L}\{f(t)\} = ...
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1answer
46 views

How to determine $2\pi$ periodic function?

Let $f(t) = 2\pi \sin t$, and determine a $2\pi$-periodic function $y^∗$ with the property that $\lim_{t\to+\infty} |y(t) − y^∗(t)| = 0$ for every solution $y$ of $y′ + y = f$. I am having trouble ...
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Heaviside function in the function whose Laplace transformation is $e^{-(\gamma+s)}/[(s+\gamma)^2+b^2]$

This is from a homework question 13.22 part (c) from "Mathematical Methods for Physic and Engineering" by Riley et. al on p. 464 I don't understand why the heaviside function is in the solution to ...
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1answer
22 views

Laplace transform of $tf'(t)$

I know that $\mathcal{L}(tf'(t)) = -\frac{d}{ds}\mathcal{L}(f'(t))$ and that this $= -\frac{d}{ds}(sF(s) - f(0))$ but the solution says that this becomes $-F(s) - F'(s)$ and I can't figure out why ...
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36 views

IVP with Laplace Transform

My attempt: Y = Laplace $$s^2Y -sy(0) - y'(0) - 3Y = ??$$ How do I set up $$h(t)$$ in the form of laplace?
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23 views

Sign problem, Laplace transform of sin(at)

I have a problem in my integration by parts but I can't find it: $$L(\sin(\alpha t)) = \int_0^{\infty}\sin(\alpha t)e^{-st}dt$$ $$= -\frac{1}{\alpha}\left[\cos(\alpha ...
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21 views

Difference between the Rectangular “Window” Function and the Rectangle Function

I'm getting ahead in my differential equations textbook (Fundamentals of Differential Equations by Nagle et. al) and in the chapter of Laplace Transforms it states that the rectangular window function ...
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13 views

Partial Fraction Decomposition — Inverse Laplace Transforms

I apologize if this is a rather lame question, but I've always been a little touchy with my partial fraction decompositions and I'm hoping to get better at them. Could you verify (or correct?) my ...
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Find $\mathcal{L}\left\{\cos^3\left(t\right)\right\}$

I began by breaking the problem up as follows: \begin{align} \mathcal{L}\left\{\cos^3\left(t\right)\right\}=\int_0^\infty e^{-st}\cos^3\left(t\right)\:dt & = \int_0^\infty ...
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89 views

Black's formula and feedback system stability

Consider a hypothetical system with open-loop transfer function $G(s)$. Place it in positive feedback with unit gain. (That is, take its output and directly add it to its input.) The closed-loop ...
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1answer
23 views

Laplace Transformations and Piecewise Functions

I am trying to understand why it is that Laplace transformations can simply be "added together" when finding the transform of a piecewise function. My professor has quite extensively talked about the ...
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18 views

Correct partial fraction construction?

Is the below the correct partial fraction decomposition? $$\frac{s^2 - 6s + 9}{(s-2)^3}=\frac{A}{s-2}+\frac{B}{(s-2)^2}+\frac{C}{(s-2)^3}$$ I can see that the numerator doesn't have a factor of ...
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33 views

Laplace transform with the Heaviside unit step function

I want to find the laplace transform for the function: $$f(t) = \left\{\begin{array}tt,\quad t\lt 2 \\ t^2 , \quad t\geq 2 \end{array} \right.$$ So I thought that the proper setup was: ...
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1answer
33 views

Find $\mathcal{L}\left\{t e^{2t}\cos\left(5t\right)\right\}$

This is what I have so far: \begin{align} \mathcal{L}\left\{t e^{2t}\cos\left(5t\right)\right\}=\int_0^\infty e^{-st}t e^{2t}\cos\left(5t\right)\:dt,\tag{1} \end{align} but notice that if ...
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2answers
46 views

Shortcut methods for Partial fraction decomposition in IVPs solved by Laplace transformation?

I have an IVP I'm trying to solve with Laplace transformations: $$y''-4y'+4y=te^{2t}$$ Given that: $y(0)=1$ and $y'(0)=0$ I've gotten to the part where I isolate $Y(s)$: ...
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Laplace Transform solution verification: $\ddot{y} + 2y = 2e^t\implies \frac13\cos(\sqrt{2}t)-\frac{2}{3\sqrt{2}}\sin(\sqrt{2}t)+\frac23e^t\,\text{?}$

Does $$\ddot{y} + 2y = 2e^t\quad y(0)=1,\dot{y}(0)=0$$ Give $$\frac13\cos(\sqrt{2}t)-\frac{2}{3\sqrt{2}}\sin(\sqrt{2}t)+\frac23e^t\,\text{?}$$ This is what I have got, and it seems to go back and ...
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Is this the correct setup for partial fractions? $\frac{1-e^{-s} + se^{-s} + s^3}{s^2(s^2+2)}=\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+2}$

I am trying to inverse laplace transform the following: $$F(s)=\frac{1-e^{-s} + se^{-s} + s^3}{s^2(s^2+2)}$$ and I believe what I do is take: $$\frac{1-e^{-s} + se^{-s} + ...
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34 views

Inverse Laplace transform for $\frac{1-e^{-\pi s}}{s(s^2 + 16)}$

I want to find the inverse Laplace transform for the following:$$\frac{1-e^{-\pi s}}{s(s^2 + 16)}$$ This was obtained from a piecewise function and required the heaviside step function to simplify. ...
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Computing transfer function then static gain

I am working on a problem for a flight controls class. I have an equation related to pure yaw. My goal is to get the transfer function associated with it, and then obtain the system static gain. The ...
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28 views

How to justify this complex substitution using contour integration

I tried to solve the laplace transform of $\cos(at)$ and $\sin(at)$ using Euler's formula. That is, $$\int^\infty_0e^{-(s-ia)t}dt\color{red}{=}\frac{1}{s-ia}\int^\infty_0e^{-t}dt=\frac{1}{s-ia}$$ ...
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1answer
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Laplace Transform…

Find the Laplace transform of $t^2e^{at}cos(bt)$ My attempt: $\large\mathit{L}\{t^2e^{at}\cos(bt)\}$ = $\large\mathit{L}\{\frac12t^2e^{at}(e^{ibt}+e^{-ibt})\}$ ... ... ...
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LaPlace transform of $t^2\sin(at)$

$$\int^{\infty}_{0} e^{-st}t^2\sin(at)dt$$ I keep running into a problem when using: $$u=e^{-st}t^2$$ $$du=2te^{-st}-st^2e^{-st}$$ $$v=\frac{1}{a} \cos(at)$$ $$dv=\sin(at)$$ Anyone have any ...
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What is the inverse Laplace of a complete square?

How do I find the inverse Laplace for something like this ? $${8 \over ( s^2 + 16 )^2 }$$ I tried using partial fraction but it didn't help any ideas on how to do it using differentiation or ...
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Evaluate $\int_{0}^{\infty}\frac{1-e^{-t}}{t}e^{-st}\;dt$

This is laplace transform of $\dfrac{1-e^{-t}}{t}$ and the integral exists according to wolfram Do i get any help/hints about how to work this ? I have been trying integration by parts with different ...
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Additive (causal + anticausal) decomposition of a transfer function

Define the causal/anticausal decomposition of a function $F(s)$ as follows. Let $f(t)$ any function such that $$F(s) = \int_{t=-\infty}^{+\infty} f(t)e^{-st} dt.$$ Then the causal part of $F(s)$ is ...
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Laplace Transform of $\cosh(at)/(at)$

Can someone give me a clue on how to compute this Laplace transform? $$\mathcal{L}\left[ \frac{\cosh(at)}{at} \right]$$
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Convolution Theorem of a product of 3 functions of x

I am trying to evaluate the integral over a product of f(t), g(t) and h(t) using the convolution theorem. $$\int_0^\infty f(t) g(t)h(t) dt$$ So after taking the Laplace of each of f(t) g(t) & h(t) ...
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Why does the Laplace transform of $t^2 \exp(at)$ exist?

My book states a theorem : "Let $f(t)$ be a function piecewise continuous on $[0, A]$ for $ A > 0$ and have an exponential order at infinity with $|f(t)| \leq M \exp(at)$. Then the Laplace ...
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Lalplace Transform and Convolution theorem

How would one go about a problem of this nature $$\int_0^\infty f(t) g(t) dt$$ Using the convolution theorem. I have taken the Laplace transform of both f(t) and g(t) to get F(s) and G(s) however ...
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Using laplace transforms to solve a piecewise defined function initial value problem

I want to use laplace transforms to solve the following: $$\frac{d^2 y}{dt^2}+16 y = f(t) = \left\{\begin{array} 1 1&t\lt\pi\\0&t\geq \pi\end{array}\right.\text{ with } y(0)=0 \text{ and } ...
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1answer
42 views

Laplace transform of Bessel's equation

I'm working on what should be a relatively straightforward differential equation. The problem says that the Laplace transform of Bessel's equation leads to (s^2 +1)f'(x) +sf(s)=0. And asks to solve ...
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Deconvolution of two delta functions (solving $y(t) = A x(t-a) + B x(t-b)$)

I would like to calculate $x(t)$, when only $y(t)$ with $y(t) = A x(t-a) + B x(t-b)$ is known. Since this is a linear shift invariant operation (convolution), the inverse relation must be of the ...
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1answer
141 views

Probability density function with the help of the Laplace (Fourier) transform

I am reading a paper that derives a closed form expression of the following probability using properties from Fourier transform, $$ \mathbb{P} \biggl( F \geq T \ (I+W) \biggl)$$ Assumptions: 1- ...
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Laplace transform of 1

I am getting a little confused when using the Laplace Transform. So, taking the laplace transform of 1 we get: $\mathfrak L[1](p)= \int_0^\infty e^{-pt}dt=[-\frac1pe^{-pt}]_0^\infty=\frac1p$ when ...
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Laplace transform of a function divided by t

Using the formula $$\mathcal{L}\left\{\frac{f(t)}{t}\right\}=\int_s^\infty F(u)~du$$ I'm trying to determine the transform with $f(t)=1-e^{-t}$. The formula gives me ...
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The inverse Laplace transform of an entire function

A simple calculation shows that the Laplace transform of $f(t)=e^{-t^2/4}$ is the function $F(p)=\sqrt{\pi}e^{p^2}\operatorname{erfc}(p)$. I would like to find the inverse Laplace transform of ...
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Inverse laplace transform with complex roots

hello I am having some trouble finding the solution to this inverse laplace transformation $$ I(s)= \frac{6s+24}{s^2 +4s+8} $$ The solution is solved using Euler identity and partial fractions, $$ ...
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Is $f\left(t\right)=\frac{1}{t^2+1}$ of exponential order?

I'm learning Laplace Transforms and one of the questions I'm working on is the following: $$\text{Is}\:\:f\left(t\right)=\frac{1}{t^2+1}\:\:\:\text{of exponential order?}$$ If so or if not, how do I ...
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1answer
26 views

Laplace transform convolution

$x(t) = cos(3πt)$ h(t) = $\exp(-2t)u(t)$ Find y(t) = x(t) * h(t) (ie convolution) Y(s) = X(s)H(s) and then take inverse laplace tranform of Y(s) $ L(x(t)) = \frac{s}{s^2+9π^2} $ $ L(h(t)) = ...
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Laplace Transform..

How do I solve the Laplace transform of $ f(t)=ta^t $ and $ f(t)=tsin(at) $? Has some property that can help me solve or I can just revolve using the definition? I can not solve... Help me please. ...
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the continuity theorem with respect to Laplace transform

Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of probability measures on $\mathbb{N}$, such that the Laplace transform $\phi_n(\lambda)=\int e^{-\lambda x}\mu_n(dx)$ converges pointwise to a limit ...
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Laplace transformation of $u(t-1)e^{2-2t}$

I actually need to refer to the table of transforms but I cannot recognize anything. Instead I directly applied the definition of the Laplace transformation and got $$F(s)=\frac{e^{-s}}{s+2}$$ Is ...
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26 views

Finding the Inverse laplace transform .

Can anyone give me some hint how can I find the inverse Laplace Transform of : $$\textrm{log}\bigg\{\frac{s+1}{s-1}\bigg\}$$
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36 views

Inverse Laplace Transform of $s^n$

I want to calculate Inverse Laplace Transform of $s^n$. I have an idea, but I do not know if it works? We have a formula to compute inverse laplace transforms of functions as below, ...
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48 views

Closed form of certain integral.

I am solving the following problem in heat transfer using the Laplace transform $$\rho\,c{\frac {\partial }{\partial t}}T \left( x,t \right) =k{\frac { \partial ^{2}}{\partial {x}^{2}}}T \left( x,t ...
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2answers
92 views

Find the leading order asymptotic behaviour of the integral

$$I(x) = \int_0^{\infty}e^{-t-\frac{x}{t^2}}dt \mbox{ as } x \mbox{ tends to infinity} $$ I know this has a moveable maximum so you need to make a substitution which transforms it into the integral: ...