Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [laplace-transform]

The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform.

0
votes
0answers
16 views

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?
1
vote
0answers
13 views

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 ...
0
votes
0answers
5 views

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 ...
3
votes
2answers
95 views

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 $\...
0
votes
0answers
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.
0
votes
0answers
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) ...
5
votes
3answers
100 views

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 ...
0
votes
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 ...
0
votes
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 ...
4
votes
0answers
31 views

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 ...
0
votes
0answers
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 ...
0
votes
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 ...
3
votes
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-...
0
votes
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, ...
0
votes
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://...
0
votes
0answers
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 $1$, ...
1
vote
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 ...
1
vote
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 (\...
0
votes
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)...
2
votes
1answer
153 views

How to prove the following identity regarding Laplace transforms?

I tried solving it by integrating by parts but i was unsuccessful. $${\cal L}\left[\int_0^xf(x-t)g(t)\ dt\right]=F(p)G(p)$$
1
vote
0answers
41 views

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: $4-3i, h(x)=\frac{1}{6}(72-36x+6x^2+54i-18ix)=12+x^2+9i-6x-3ix$...
1
vote
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 \...
1
vote
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 ...
0
votes
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} $...
1
vote
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 ...
0
votes
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 \...
-1
votes
1answer
34 views

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)}-\...
1
vote
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[...
0
votes
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 ...
1
vote
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 ...
0
votes
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
votes
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}\...
0
votes
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})...
0
votes
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 ...
0
votes
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}\...
1
vote
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 ...
-1
votes
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 $( ℒ_t[f(t)] = ??)$
0
votes
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}{\...
0
votes
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
votes
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 $*$ ...
1
vote
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 ...
0
votes
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.
1
vote
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
votes
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 ...
0
votes
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 $m>0$.
0
votes
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{...
0
votes
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
votes
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
vote
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?
1
vote
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 ...