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

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A question of multi-dimensional integral

Consider the function $$\Omega(N,E)=\int dE_1 \int dE_2 \cdots \int dE_N \Omega_1(E_1)\Omega_2(E_2) \cdots \Omega_N(E_N)\delta(E-E_1-E_2\cdots -E_N)$$ Is there a sufficiently condition on the ...
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Conditions for Laplace Transform

Consider the Laplace transform: $$\mathscr{L}(1) = \int_0^\infty e^{-st}dt = -\left.\frac{1}{s}e^{-st}\right|_0^\infty = \frac1s$$ Math textbooks usually state that this is only valid for the ...
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Multivariate Laplace transform of independant Poisson processes

Let $W_1$ and $W_2$ be independant Poisson processes on $S$ with mean measures $\rho_1$ and $\rho_2$. Let $$w_1^\ast=\sum_{X\in W_1} X$$ and $$w_2^\ast = \sum_{X\in W_2} X$$ Reference: Kingman's ...
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Question regarding $\mathcal{L}\{t*\mathcal{U}(t-2)\}$

I'm working on a problem for homework (* is multiplication not convolution): $\mathcal{L}\{t*\mathcal{U}(t-2)\}$ I understand that $\mathcal{L}\{(t-a)\mathcal{U}(t-a)\}=e^{-as}F(s)$ The first step ...
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Laplace transform using the convolution theorem

(This question is about laplace transforms) By making use of the convolution theorem show that the solution $y(t)$ to the ODE $$\ddot{y}(t)+4\dot{y}(t)+5y(t)=u(t), \quad y(0)=0,\quad \dot{y}(t)=0,$$ ...
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Laplace transform of multiplication of two terms with different arguments

What is the Laplace Transform of the product of two functions with different arguments? The function is: $\mathcal{L}( \sin({3 t}) \cos({5 t}) )$
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Doubt in laplace transforms

Let $f(t)=e^{t^2}$. Now the laplace transform of $f(t)$ is $$\int_0^\infty e^{-st}e^{t^2}dt=\int_0^\infty e^{-st+t^2}dt$$ But after this.. How can I proceed? Help me..
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Finding a basic laplace transform

Find the laplace transform of the function $f(t)=t^3e^{4t}$. The solution I am presented is Now $\mathcal{L}(e^{4t}f(t)) = F(s-4)$ and $\mathcal{L}(t^3) = 6/s^4$. So the Laplace transform of $f$ is ...
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Solving IVP using Laplace Transform

Let $$g(t) =\begin{cases} t & \text{if $t \leq6π$} \\ 6\pi & \text{if $t>6\pi$} \end{cases} $$ Solve $y''+ 16y = g(t)$ where $y(0) = 9$ and $y'(0) = 4$ using Laplace transforms. I got ...
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Question regarding $\mathcal{L}^{-1}\{\frac{s}{s^2+4s+5}\}$

The book asks for: $\mathcal{L}^{-1}\{\frac{s}{s^2+4s+5}\}$ So I can see: $\frac{s}{s^2+4s+4+1} = \frac{s}{(s+2)^2+1}$ From the properties of the inverse Laplace transform: ...
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The inverse Laplace transform of $ s^{3/2}-a-bs \over s^{3/2}+a+bs$

How can I solve the inverse Laplace transform as below: $$\mathscr{L}^{-1}\left( s^{3/2}-a-bs \over s^{3/2}+a+bs \right) $$ where a and b are constants. Hint: we can consider $${ s^{3/2}-a-bs ...
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What is my error in applying this Laplace Transform?

So, our book has the seemingly innocuous problem: $y''-y'-6y=0$. I was able to solve by hand, and come up with $${\scr L}(y)=\frac{s-2}{s^{2}-s+6}$$.That completed, I factored the bottom to ...
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Is $\cosh(t^2)$ of exponential order?

Is $\cosh(t^2)$ of exponential order? I know that it isn't, but I am unsure as to why. Also why is $\cosh(t) $ of exponential order?
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Solving 2nd Order ODE w/Laplace Transforms + Heaviside

This is a similar problem to the one I posted earlier with some differences. Attempt at solution: Write g(t) as a heaviside function. Take Laplace transform of LHS and RHS. Solve for Y. Take ...
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Solving 2nd Order ODE w/Laplace Transforms

I am having difficulty with this problem: *Note: The Delta3(t) is the delta dirac function, also the answer in the image is WRONG. Attempt at solution : Let Laplace{y(t)}=Y Take Laplace of LHS ...
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Laplace transform nonlinear equation

How can I apply the Laplace transform on a the following nonlinear PDE $$ \frac{\partial y}{\partial t}=\frac{\partial y^n}{\partial x}$$ where $n$ is a natural number? When I say apply the Laplace ...
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Find the distribution with the following Laplace transform.

Is anybody aware of the distribution whose Laplace transform is the following. \begin{equation} \mathbb{E}[e^{-tX}] = \frac{e^{-t}}{(1+2t)} \end{equation} Note: The Laplace transform of the ...
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35 views

Inverse Laplace Transform of $\ln[\frac{s^2+a^2}{s^2+b^2}]$

How does one find $\mathcal{L}^{-1}\{\ln[\frac{s^2+a^2}{s^2+b^2}]\}$? I've tried splitting it up into $\mathcal{L}^{-1}\{\ln(s^2+a^2)\}-\mathcal{L}^{-1}\{\ln(s^2+b^2)\}$. However, I can't think of ...
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How to find the Laplace Transform of two (independent) functions multiplied together?

How does one find the laplace transform for an equation consisting of two trig functions multiplied together, when it is not possible to use any trig identities? For example, take a function say; ...
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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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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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22 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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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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74 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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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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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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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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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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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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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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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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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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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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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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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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21 views

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 ...