Questions tagged [laplace-transform]
The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.
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Can you find the Laplace transform of a function multiplied by a non-linear exponential function?
I am considering the Laplace transform of the function:
$$g(x,t)=f(x,t)\exp(-(c+N(x,t))t).$$
Can you use the shift theorem in this case, or does the dependence of $N$ on time prevent this?
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Laplace transform heat equation $u(0,t)=f(t),u(x,0)=u_0$
Solve the heat equation $$\frac{\partial u}{\partial{t}}=\nu\frac{\partial^2{u}}{\partial{x}^2},t>0,x>0$$ where $u(x,0)=f(t),u(x,0)=u_0$ using Laplace transforms.
My trouble is dealing with the ...
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Deriving the discrete-time lowpass filter from the Laplace equation
I was trying to demonstrate the discrete-time expression of a lowpass filter:
$ y_i = \alpha x_i + (1-\alpha) y_{i-1} $
However my result is completely off the target so I am wondering which of my ...
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2answers
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Integral of $\int_0^{\infty} \frac{\sin^2(x)}{x^2+1}dx$ using Feynman integration.
Using $$I(t) = \int_0^\infty \frac{\sin^2(tx)}{x^2+1}dx$$ I want to know how to get an answer using Feynman integration and the Laplace transform of a differential equation. The correct answer is $\...
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31 views
Compute the inverse Laplace transform of $\frac{(s^2+3)}{(s^2+2s+2)^2}$.
I tried a few things, like completing the square in the denominator and going from there, but nothing really gets me anywhere or matches the answer that has been posted.
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18 views
Laplace Transform of $f(x) \theta(x) \star f(x) \theta(-x)$
Is there a simplified result for Laplace Transform of $f(x) \theta(x) \star f(x) \theta(-x)$, where $f(x)$ is an even function, $\theta(x)$ is Heaviside function and $\star$ is convolution?
For $f(x) ...
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3answers
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Using Laplace Transforms to solve $\int_{0}^{\infty}\frac{\sin(x)\sin(x/3)}{x(x/3)}\:dx$
So, I've come across the following integral (and it's expansion) many times and in my study so far, Complex Residues have been used to evaluate it. I was hoping to find an alternative approach using ...
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2answers
32 views
Can Laplace transformation be used here?
I would really appreciate it if someone can tell me what i got wrong below or any other method that could solve this easier.
The problem:
$(1-x)y''+xy'-y=0$
$y(0)=2$
$y'(0)=-1$
I tried solving it ...
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1answer
30 views
Inverse Laplace transformation - Bessel function
How to find $f$ using Laplace transformation?
$f = J_0 * J_0$ where * is a convolution. According to the Convolution theorem it is $$(J_0 * J_0)(t):= \int_0^t J_0 (t - \tau) J_0 (\tau)\mathop{\mathrm ...
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Transformation from ODE or PDE via operator to “easier form” - what is he background?
I know a bit about how integral transformations can be applied to differential equations. ODEs can be transformed under some circumstances to algebraic equations and PDEs can be transformed under some ...
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25 views
Uncommon Inverse Laplace Transforms
I'm interested in calculating inverse Laplace transforms of functions such as
$\frac{1}{(1-s^{2})\log(s)}$, $\frac{1}{arccosh\left(1+\frac{s^{2}}{4}\right)}$
and more exotic examples.
In general ...
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1answer
80 views
Using Laplace Transforms to solve $\int_{0}^{\infty} \sin\left(x^2\right)\:dx$
I recently came into the following definite integral
$$ I = \int_{0}^{\infty} \sin\left(x^2\right)\:dx$$
To solve this, I used a composition of both the Feynman Trick with Laplace Transforms as ...
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2answers
27 views
Find the inverse Laplace of $ \ \frac{2}{(s-1)^3(s-2)^2}$.
Find the inverse Laplace of $ \ \frac{2}{(s-1)^3(s-2)^2}$.
Answer:
To do this we have to make partial fractions as follows:
$ \frac{2}{(s-1)^3(s-2)^2}=\frac{A}{S-1}+\frac{B}{(s-1)^2}+\frac{C}{(s-...
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1answer
64 views
Have no idea how to solve this differential equation
An arbitrary signal $v(t)$ pass through the following system,
$w'(t) + 5 w(t) = v'''(t) + 320v''(t) + 40 v' (t) + 40v(t)$
How to determine the coefficient of the following differential equation, ...
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2answers
24 views
Explanation of Laplace transform
In my Differential equations course I have topic on Laplace transform, I didn't understand the concept of Laplace transform. I searched it but I don't understand. I referred to one video
https://...
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22 views
Distribution of a stable and exponential r.v
I am given an absolutely continuous positive r.v $X$, such that $\mathbb{E}[e^{-sX}]= e^{{-s}^\alpha}, \ s>0.$
The goal is to prove that $(Y/X)^\alpha$ is an Exponential r.v with intensity $1$, ...
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0answers
19 views
Applying Fubini to $\int_0^\infty \int_0^\infty |f(t)|^2 e^{-st} \sqrt{\frac{t}{s}}\;\mathrm dt \, \mathrm ds, $
Consider the double integral
$$ \int_0^\infty \int_0^\infty |f(t)|^2 e^{-st} \sqrt{\frac{t}{s}}\;\mathrm dt \, \mathrm ds, $$
where $f \in L^2(0,\infty)$.
This double integral appears when computing ...
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1answer
55 views
Asymptotics inversion of Laplace transform
I've got the following problem:
Let's imagine we've got a laplace transform
$$\hat f(\lambda ) = {{(1 - \beta ){{1 - \hat \varphi
(\lambda )} \over \lambda }} \over {1 - (1 - \beta )\hat \varphi (\...
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0answers
19 views
Linear systems and validity interval
Suppose we have a linear system represented by a differential equation like this:
eq.(1) : $y'(t) + 2*y(t) = x(t)$
consider these two conditions:
validity interval of eq.(1) is just for t>0 and y(0)...
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1answer
153 views
How to prove the following identity regarding Laplace transforms?
I tried solving it by integrating by parts but i was unsuccessful.
$${\cal L}\left[\int_0^xf(x-t)g(t)\ dt\right]=F(p)G(p)$$
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Convolution of two functions. What am I doing wrong?
Let f(x) be 1/6 for $0\leq x \leq 6$ and 0 elsewhere. Let g(x) be $x^2-3ix$
Find h(4) where h is the convolution of f and g.
Solution: $4-3i, h(x)=\frac{1}{6}(72-36x+6x^2+54i-18ix)=12+x^2+9i-6x-3ix$...
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3answers
73 views
Seeking methods to solve $ I = \int_{0}^{\infty} \frac{\sin(kx)}{x\left(x^2 + 1\right)} \:dx$
I am currently working on an definite integral that requires the following definite integral to be evaluated.
$$ I = \int_{0}^{\infty} \frac{\sin(kx)}{x\left(x^2 + 1\right)} \:dx$$
Where $k \in \...
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0answers
17 views
how to derive Laplace transform, delta function?
This is an awfully specific question that is extremely similar to previous question (linked below). This problem utilized a quality of the delta function that allows an indefinite integral to be ...
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1answer
41 views
Laplace Transform of $sin(at)$ using Different Method
I am trying to understand a separate way of showing $\mathcal{L}\{sin(at)\}$ using the the following method:
What I have so far is
$$\text{Let $g(t)=f(at)$ where we substitute $u=at$. Then we have} $...
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0answers
26 views
“Time”-Laplace transforms of Brownian motion transition and first hitting time laws
I was having a look at Girsanov's theorem and some of its applications. One of them is the possibility of affirming the following theorem.
Given a probability space $(\Omega, \mathcal{F},P)$ equipped ...
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3answers
33 views
Partial derivative of nabla operation
Given that
$$\nabla^2f = \frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} = \frac{\partial^2f}{\partial x'^2} + \frac{\partial^2f}{\partial y'^2}$$
$$ x = x' \cos \theta - y'\sin \...
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1answer
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Solving Differential equation with g(t) involving Dirac Delta function
$y"+2y'-15y=6\delta(t-9), y(0)=-5, y'(0)=7$
Solutions to this differential equation is
$f(t)=\frac18e^{3t}-\frac18e^{-5t}, g(t)=\frac94e^{3t}+\frac{11}{4}e^{-5t},Y(s)=\frac{6e^{-9s}}{(s+5)(s-3)}-\...
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3answers
59 views
Laplace transform of Bessel's equation: $xy'' + y' + xy = 0$
page 178, "differential equations demystified", 2004:
Use the laplace transform to analyze Bessel's equation:
$$xy'' + y' + xy = 0$$
$$y(0)=1$$
We know that:
$$ L[xy] = -\frac{d}{ds}Y(s)$$
$$ L[...
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0answers
22 views
laplace transformations one after another in order to make pde of dimension 4 to ode
for a second order system of PDE (having 4 equations), can we change this PDE system to ode system by taking Laplace 3 times repeatedly before taking inverse Laplace of first?
can this will be helpful ...
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1answer
32 views
Laplace transform of $\sin(wt)\cdot u(t-t_0)$
I need to find the laplace transform of $\sin(wt)\cdot u(t-t_0)$
The answer is $$\frac{e^{-st_0}}{\sqrt{s^2+w^2}}\cdot \sin\left(wt_0 + \tan^{-1}\frac{w}{s}\right)$$
I tried using the general ...
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1answer
28 views
Derivative of nabla operation
$$ x = x' \cos \theta - y'\sin \theta \\
y = x' \sin \theta + y'\cos \theta
$$
$$ \frac{\partial f}{\partial x'} = \frac{\partial f}{\partial x} \cos \theta + \frac{\partial f}{\partial y} \sin \...
3
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1answer
42 views
Finding the laplace transform of $\delta(t^2-3t+2)$
I have to find the laplace transform of $\delta(t^2-3t+2)$.
The answer is : $e^{-s}+e^{-2s}$
I tried using the definition of laplace transform :
$$\int_{-\infty}^{\infty}\delta(t^2-3t+2)e^{-st}\...
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0answers
23 views
inverse laplace transform of $ \frac{s^2 +1}{s^2+s-2} {e^{-2s}}$
inverse laplace transform of $$ \frac{s^2 +1}{s^2+s-2} {e^{-2s}}$$
I tried solving it like that, but I'm not sure:
1- Apply partial fraction to the f(t) part: $$(\frac{2/3}{s-1} + \frac{-5/3}{s+2})...
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0answers
14 views
Fourier transform symbolic expression from Laplace transform symbolic expression with “$s=j2\pi f$”
I just wanted to know under which conditions can we have the Fourier transform symbolic expression from The Laplace transform symbolic expression by replacing "$s$ with "$s=j2\pi f$" ?
I mean from a ...
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1answer
32 views
Laplace transform and general method of solving DE.
$y"+3y'+2y=g(t), y(0)=0, y'(0)=-2,$where,
$g\left( t \right) = \left\{ {\begin{array}{*{20}{l}}2&{\hspace{0.25in}t < 6}\\t&{\hspace{0.25in}6 \le t < 10}\\4&{\hspace{0.25in}t \ge 10}\...
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1answer
36 views
Solving $f(t)=t+\frac{1}{6}\int_{0}^{t} (t-u)^3f(u) \ du$ Using Laplace Transforms
I am trying to use the Laplace Transformation to find the unknown function $f:[0,\infty)\rightarrow\mathbb{R}$ in the integral equation, $$f(t)=t+\frac{1}{6}\int_{0}^{t} (t-u)^3f(u) \ du.$$
I start ...
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1answer
21 views
I need to find laplace transform for this $ℒ_t[\sqrt{\sin(2 t) + 1}]$ [closed]
if $f(t) = \sqrt{\sin(2 t) + 1}>> I$ need to solve this problem and find Laplace transform for it $( ℒ_t[f(t)] = ??)$
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1answer
39 views
Nabla operation to cos and sin replacement
There are given equations,
$$ x = x' \cos \theta - y'\sin \theta \\
y = x' \sin \theta + y'\cos \theta
$$
$$ \frac{\partial f}{\partial x'} = \frac{\partial f}{\partial x}\frac{\partial x}{\...
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1answer
47 views
Trying to Find a Function whose Laplace Transform is $\frac{z}{z^3-1}$
I am trying to find a function whose Laplace transform is $$\mathcal{L}(f)(z)=\frac{z}{z^3-1}.$$
I first simplify$\mathcal{L}(f)(z)$ by solving for the cube roots of unity. Hence, $$\mathcal{L}(f)(z)=...
2
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0answers
25 views
Laplace Transformation to Solve $f(t)=t^2+\int_{0}^{t} (t-s)f(s) \ ds$
I am trying to solve the Volterra integral equation $$f(t)=t^2-\int_{0}^{t} (t-s)f(s) \ ds,$$ using the Laplace transformation.
I first rewrite the equation as $$f(t)=t^2-(f*t)(t),$$ where $*$ ...
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1answer
69 views
Laplace Transform of Lambert W function
Does there exist a Laplace transform of the Lambert W function (evaluated at $at$, where $a$ is a constant) that can be expressed in terms of elementary functions and the product log ($W(x)$)?
The ...
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1answer
29 views
Find a Laplace transform of $\mathcal{L}\{t^nf(t)\}$ with full solution
I need to find a full solution to this transform: $\mathcal{L}\{t^nf(t)\}$.
I know the result, but don't know how to solve it.
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1answer
31 views
Solve $u''(t)+2u'(t)+2u(t)=f(t), \ t\geq 0, \ u(0)=1, \ u'(0)=0$ using Laplace Transforms.
Use Laplace Transforms to give a solution formula for the initial
value problem
$$u''(t)+2u'(t)+2u(t)=f(t), \ t\geq 0, \ u(0)=1, \ u'(0)=0. \tag1$$
where $f$ is a function such that $|f(t)...
0
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1answer
18 views
How to find the constants of this transfer function?
A stabil, casual and continuous system have the transfer function $$H(s)=\frac{a}{s+b}$$ If the input signal to the system, $x(t)=\sin(t)$, become the output signal $y(t)=\sqrt{2}\sin(t-\pi/4)$ when ...
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0answers
44 views
How can I calculate this integral?
How can I find the answer of this integral
$$\int_1^\infty \frac{(x^2-1)^{\frac{3}{2}}}{e^{mx}-1}\, dx$$
where $m>0$.
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1answer
39 views
Heavyside function and Laplace transform
How to calculate the Laplace Transform of the following
$f(t)=-t^3u_3(t)+\cos{(3t)}u_6(t)$ ...(1)
Solution:- The Laplace Transform of $\cos{(3t)}u_6(t)$ can be calculated using $\cos{(at+b)}=\frac{...
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0answers
15 views
Laplace transform of Laplacian
I have seen the laplacian being used in the partial differential equations involving quantum mechanics, particularly some form of the Schrodinger equation. My question is simple: What is the Laplace ...
2
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1answer
41 views
Laplace Transform of Vector Valued LTI system [closed]
I'm not really interested in proofs, but I have a LTI system that is vector valued and want to take the laplace transform of it and the inverse laplace transform of the transfer function. I'm ...
1
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2answers
33 views
Laplace Tansform of $\frac{1-\cos(2t)}t$
What is the Laplace Transform of $\frac{1-\cos(2t)}t$ and is there a general formula for such situations with $\cos$, $\sin$ and others?
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0answers
24 views
Laplace transform of product of functions
I was asked to give this function $G'(t) = -G(t)(S+X(t)) + S\cdot H + R(t)/V$ in terms of $G(s)/R(s)$ and $G(s)/X(s)$ by using the Laplace transform. I know $G(0)=A$ and $X(0)=0$, and all the other ...
