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

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Laplace transforms for a pharmacokinetics multi-compartmental model

I am an anaesthetist trying to write some pharmacokinetics software as a pet project. Unfortunately the maths I need is a bit too much for my rusty high school calculus, and I am out of my depth. I am ...
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Laplace inverse for Taylor expansion

By using infinite series find Laplace inverse for |1/(S^3+1)| .... I don't know what to do after using taylor expansion.. when I use it I got polynomial of $ S $ in the nominator which I can not deal ...
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The density of the distribution whose Laplace transform is the following

Is anybody aware of the density of the distribution whose Laplace transform is the following. \begin{equation} \mathbb{E}[e^{tX}] = \frac{e^{t/2}-1}{t/2} \end{equation} Note: $X$ is a continuous ...
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Inverse Laplace Transform with $e^{a s}$

How can I take the Inverse Laplace Transform of $F(s) = \frac{d}{ds}\left(\frac{1-e^{5s}}{s}\right)$? I have tried going with inverse of the derivative and convolution (even tried evaluating the ...
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21 views

Why does the laplace transform of sine and cosine looks the way they are

I always forget what the Laplace transform of sine and cosine looks like. This is because they look so similar to each other. Does there exist a good mnemonic for remembering what form each takes? Or ...
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Laplace Transform of $\cosh(bt)$

So, one of my homework assignments is to take the Laplace transform of a function such that $f(t)=\cosh{bt}$. I figured it would be equivalent to: ...
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23 views

What is the General procedure for graphing heavidside functions?

I was given an example of a second order differential equation with U1(t)-U(3t) as the forcing function. I was asked to graph the forcing function and the answer is that the function is 1 when t is ...
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Lagrangian, Kinetic & Potential energy with two masses connected to three springs [migrated]

Two masses $m_1$ and $m_2$ are on a frictionless surface. They are connected by three springs with constants $k_1,k_2,k_3$. $k_1$ and $k_3$ are attached to walls and $k_2$ is between the masses. $k_1$ ...
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28 views

Inverse Laplace transform and convolution

Suppose we have two functions of $t$, $f(t)$ and $g(t)$. Letting $\mathcal{L}\{f(t)\} = F(s)$ and $\mathcal{L}\{g(t)\} = G(s)$, I know that: $\mathcal{L}\{f(t) \star g(t)\} = F(s) \cdot G(s)$, but ...
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31 views

Finding the Laplace Transform Inverse

Solve by Laplace Transforms. So I'm stuck on how to find this $\mathcal{L}^{-1}$ $( \frac{\frac{5s}{4} + \frac{13}{4}}{s^2+5s+8} ) $ I'm not sure what t odo. I was thinking I need to use the ...
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91 views

A bessel function integral

$$\int_{-\infty}^{\infty} dy \frac{J_1 \left ( \pi\sqrt{x^2+y^2} \right )}{\sqrt{x^2+y^2}} = \frac{2 \sin{\pi x}}{\pi x} $$ How do I show this?
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Solving second order nonhomogeneous differential equation with non-constant coefficients using Laplace Transform

$ty''(t) + y'(t) -ty(t)= tf(t)$ How to solve the problem using Laplace Transform? Using Laplace transform I got $$Y(s)= C(s^2-a^2)^{-1/2} + (s^2-a^2)^{-1/2}\int (s^2-a^2)^{-1/2}F(s)\,ds$$ where ...
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29 views

Final value theorem for closed system

We have a system with output given by $\frac{Y(s)}{R(s)} = \frac{F(s)G(s)}{1+F(s)G(s)}$ where $F(s)G(s) = K\frac{s+1}{s^2+s+1}$. Let $K=4$ and $R(s) = 10/s$. Using the final value theorem, ...
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Laplace transform of ODE containing Dirac Delta Function

When solving ODE containing the Dirac Delta function by Laplace transform its impulse occurred at t=0 for example on a mass , if i assumed initial x=0 , the solution does not satisfy that condition, ...
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15 views

Laplace Transform Delay Property

I have a quick question regarding the delay property of the Laplace transform. I understand that when you have a function $$x(t-a)u(t-a),$$ the Laplace transform for that function is ...
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solve $y''-4y=\delta(t-1)$ with initial conditions $y(0)=0, \; y'(0)=1$ using Laplace transforms

I took the Laplace transform and solved for $Y$ which resulted in $Y=\frac{1+e^{-s}}{s^2-4}$. I began to break up the problem separating the result into two equations but the fact that there is a $1$ ...
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29 views

Checking region of validity for a standard Laplace transform

In these notes, http://www.math.psu.edu/papikian/Kreh.pdf, Theorem 2.14 it states that $$\mathcal{L}[J_0](s)=\frac{1}{\sqrt{1+s^2}} $$ which I suppose is equivalent to $$ ...
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28 views

Laplace Transform to solve $R\frac{dQ}{dt}+\frac{Q(t)}{C}=V(t)$

I have the differential equation $R\frac{dQ}{dt}+\frac{Q(t)}{C}=V(t)$ where $R,C\in\mathbb R$ and $Q,V$ are functions of $t$. If I take the laplace transform of the differential equations I get: ...
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25 views

Final value of 1/(( s+2 )² * (s² - s + 1)) in the time domain

The original question is given as $$\frac {d^3y}{dt^3}+y=u=(1-t)e^{-2t}$$ The initial value y(0) = 0 and the same for all derivatives of y. Determine Y(s) What happens to u(t) and y(t) when ...
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31 views

Solve ordinary differential equation using Laplace transform

I have trouble to solve the differential equation. I can write derivatives of Laplace transforms but I can't do anything $$ \ddot y(t)+3y(t)=\sin(t)\text{ with } y(0)=1,\,\dot y(0)=2 $$
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48 views

Computing the Laplace transform of $\frac {f(x)}{x}$

I am having trouble computing the following Laplace transform: $\frac {f(x)}{x}$. From Wikipedia it should be equivalent to this: $\int_s^\infty F(\sigma) \,d\sigma$ . What I've done so far is ...
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How does one find the Laplace transform for the product of the Dirac delta function and a continuous function?

As an example, what is the Laplace transform for the following: $$g(t)=\delta(t-2\pi) cos t$$ I've worked through a few examples that required finding $\mathcal{L}\{\delta(t-t_0)\}=e^{-st_0}$, but ...
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Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} ...
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40 views

The Laplace Transform of Piecewise Function

Write the following as an unit step function and find the Laplace transform. $f(t)=\begin{cases}{t}&0 \leq t < 3\\ 3&3 \leq t < 4\\ 11-2t& 4 \leq t < 5.5 \\ 0&t \geq 5.5 ...
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55 views

Use Laplace transform to solve initial value prob.

The problem is: $y" + 9y = e^t$, with the initial conditions $y(0) = 0, y'(0) = 0$. I'm stuck at the inverse Laplace transform part. Do I have to use partial fraction expansion or can I just split ...
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42 views

Solving Laplace $\nabla^2 \phi=0$ in $x,y \geqslant 0$

I'm trying to solve $\nabla^2 \phi=0$ in $x,y \geqslant 0$ $\phi(x,y)=0 $ as $x^2 +y^2 \rightarrow \infty$ $\phi_x(0,y)=0$ and $\phi(x,0)= \frac{1}{1+x^2}$ I know the solution is ...
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30 views

Why is this laplace identity true $\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$?

I was wondering why this laplace identity is true? Does it follow from definition? $$\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$$ I'm trying to understand the first ...
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Laplace Transform of an Piecewise Function

Write $f(t) = \begin{cases} 5,& \mbox{if} \quad 0 \leq t \lt 3 \\ -4,& \mbox{if} \quad 3 \leq t \lt 7 \\ 0,& \mbox{if} \quad t \geq 7 \end{cases}$ as a unit step function and find ...
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inverse laplace tranform

I have a simple question, There are some functions f(t), g(t) and lets say F(s) and G(s) for the form of Laplace transform of f(t) and g(t), respectively. While I am solving differential equation ...
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Describe the diffrence between the following two problems and give an example of a physical situation which may be modeled by each equation

$y'' + y =\mu_\pi \big(t\big)$ $y''+y= \delta (x- \pi )$ wih initial conditions: $y \big(0\big) =0$ $y' \big(0\big) =0$ It is obvious to me that the first equation is a Heaviside distribution ...
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Show that s * exp (- s * inf) = 0 ? (s complex)

Reading on control theory and the Laplace transform of the unit step function, I came upon the following in my textbook. The Laplace transform defined as: $$Y(s)=\int_{0}^{\infty}y(t)e^{-st}dt$$ s ...
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85 views

Solve the Integral Equation Involving Laplace Transforms

I want to solve $\int^\infty_0x'(T)x(t-T)dT=6t^3$ where $x(0)=0$ I did the Laplace transform to both sides, and the left side is a convolution, so I then have $X(s)x(s)=\frac{36}{s^4}$, but here I'm ...
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Laplace transform of a linear input vs output line?

I think my intuition of the Laplace transform and transfer functions is broken. Suppose I have a linear function which relates two quantities r to x as such: $$ r(x) = -100x + 25 $$ i.e. a ...
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39 views

Is multiplication commutative in the laplace domain?

I'm studying control theory and saw this picture explaining some of the basic rules. My question is if we could also say that Y(s) = (G2(s) * G1(s)) * U(s) Or Y(s) = U(s) * G2(s) * G1(s) I'm ...
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Laplace Transform to evaluate an integral

Compute $\displaystyle\int_{0}^{\infty} \frac{\cos(x)}{x^2 + a^2} \mathrm{dx}$, for $a\in \mathbb{R}$ using the Laplace Transform. I'm not sure on how to start with this problem. I tried to first ...
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Find the Inverse Laplace Transforms

Find the inverse Laplace transform of: $$\frac{3s+5}{s(s^2+9)}$$ Workings: $\frac{3s+5}{s(s^2+9)}$ $= \frac{3s}{s(s^2+9} + \frac{5}{s(s^2+9)}$ $ = \frac{3}{s^2+9} + \frac{5}{s}\frac{1}{s^2+9}$ $ ...
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113 views

Solve integral (convolution) equation

Given a function: $u(t) = \exp\left( -\frac{At^2}{1+t}\right),$ $A>0, t>0,$ and an equation: $\frac{d u(t)}{dt} = \int^{t}_0 \phi(t-\tau) u(\tau) d \tau .$ How to find a closed expression for ...
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Transfer function of differential eqaution

I'm trying to find out the transfer function of simple differential equation: $$a_0\dot y + a_1y=b_0x+b_1$$ The problem is i have no idea what to do with $b_1$. If we apply the Laplace transform ...
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Maximising a net present value function

I am looking at an equation for profit derived from fishing operations. This is defined in terms of a bounded integral (with an upper bound of $+ \infty$), so it's a Laplace transform really. It gives ...
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Using the Laplace Transform solve $y''+6y'+5y=e^t$

The initial conditions are $y(0)=0$ and $y'(0)=1$. I began the process and ended up with $Y=1/(s-1)(s^2+6s+4)$. Since the second factor in the denominator does not factor so I have a feeling I messed ...
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Laplace, Correct Use of the Second Shift Theorem

I have invested some time now trying to understand how to use the Second Shift Theorem, mostly by doing the full integration first. What threw me off at first, I discovered, is that almost all books ...
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37 views

Inverse Laplace tranform via the table formulas

In my inverse Laplace table there is this inversion "formula": $(1) \frac{1}{s-a} \rightarrow e^{at}$ I understand that $\mathcal{L}^{-1}[\frac{1}{s+4}] = \frac{1}{2}\sin(2t)$ But why can I not do ...
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21 views

Laplace Transform

Let us assume that complex-valued differential equations as follows $\dot{z}(t)=-Az(t)+Bz(t-\tau)$, $z\in \mathbb{C}$ How to find the solution of the above equation by using Laplace transform.
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Discrepancy with the book's solution and mine of Laplace transform of a piecewise defined function

Determine the Laplace transform of $f(t)$ below: $$ f(t)= \begin{cases} 0, & \text{if } t < 2 \\ (t-2)^2, & \text{if } t \geqslant 2 \end{cases} $$ So my answer is $$ 2e^{-2s}/s^3 $$ ...
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63 views

Transfer function for double cart system

System: Define X2 = Y2; I've described the system with the following diff equation: $$f_{tot} = m_1\ddot{x_1} + k(x_2-x_1)+m_2\ddot{x_2}+B(\dot{x_2}-\dot{x_1})$$ where m1, m2, k and B are Cart ...
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31 views

Exponential Order: $\forall t>M$ or $\forall t>0$?

The following comes from the discussion of Laplace transformation in ODE. Let $f(t)$ be piecewise continuous on $[0, \infty)$ and of exponential order. Prove that there exist constants $K$ and ...
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Laplace Transform for a difficult function

The Laplace Transform I'm having trouble with is: $$f(t) = 6te^{-9t}\sin(6t)$$ I'm not sure what the protocol is for multiplying t into it. The Laplace Transform for $f(t) = 6e^{-9t}\sin(6t)$ is ...
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118 views

Partial fraction expansion for non-rational functions

With regard to this answer to an inverse Laplace transform question, I derived the following result: $$\frac1{i 2 \pi} \int_{c-i \infty}^{c+i \infty} ds \, e^{s t} \Gamma(s)^2 = 2 K_0 \left ( 2 ...
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Solve for inverse Laplace transform using non-repeating complex partial fractions. (5.7-4)

Synopsis: Please check my work. I do not have a text "answers to odd problems" for reference as this is an "even" numbered problem. The following documents in good detail the steps taken to solve for ...
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Probability proof of inversion formula for Lapalce transform

Let $f:[0, \infty[\longrightarrow \mathbb{R}$ be bounded and continuous and define $L(\lambda)=\int_0^\infty e^{-\lambda x}f(x)dx$. Let $X_n$ be a sequence of independent random variables with ...