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

learn more… | top users | synonyms

1
vote
2answers
35 views

Inverse Laplace transform: is there a nice formula for the step response of a 2nd order system?

For the impulse response of a 2nd order system given by $$ H(s) = \frac{\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2}$$ I was surprised to see there exists a general formula found by Wolfram Alpha ...
1
vote
0answers
25 views

What is the Laplace transform of $d/dt(f * g)$ in terms of $F(s)$ and $G(s)$?

I'm not exactly sure as where to start. We know that the Laplace transform of $h(t) =$ the Laplace transform of $(f * g) = F(s)G(s)$, where $h(t) = (f * g) = \int_{0}^t f(t-T)g(T)\,dT$. Thus, I was ...
0
votes
1answer
25 views

Laplace inverse of sin/cos

how would i go about finding the Laplace inverse of: $$\frac{2s}{(s^2+5)^2}$$ Was hoping partial fractions would simplify this for me, but to no avail. I only have the Laplace transform of some ...
0
votes
1answer
16 views

Inverse Laplace Transform without using equations

I'm stuck with a question about Inverse Laplace Transform here, but the use of inverse laplace transform equation is forbidden. $Y(s)=\frac{e^{-\pi s}}{s[(s+1)^2 + 1]} $ Thank you very much!
1
vote
3answers
45 views

Exponential order in Laplace Transform:constructing a function such that it is of exponential order but its derivative is not.

I don't understand how to construct such function.I know the definition of a function being of exponential order which states that: $f(t)$ is of exponential order if there exist constants ...
1
vote
1answer
59 views

What happens to poles lying on branch cuts in contour integration?

Inverse the Laplace Transform $$\frac{1}{\sqrt{s}}\cdot\frac{1}{1 + s}$$ back to time domain requires evaluation of Bromwich integration: $$\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} ...
0
votes
0answers
21 views

easy inverse laplace transform : e^(-3s)/(s-1)^4

I am trying to solve this easy inverse laplace transform: $\mathcal{L}^{-1}\{\frac{e^{-3s}}{(s-1)^4}$ $\mathcal{L}^{-1}\ {{{e^{-3s}}\frac{1}{(s-1)^4}}}$ this is my result, which apparently is ...
0
votes
0answers
16 views

Fourrier transform

1-How to calculate the theoritical Fourrier Transform of the folling function: h(t) = 1 for t<= 0 and h(t) = 0 for t >0? Is The Fourrier transform of {Gamma(t) - signe(t)} a solution? 2- If yes, ...
0
votes
1answer
36 views

Solve differential equation using Laplace Transform and Second Shifting Theorem

Given the IVP: $$y''+2y'+5y=50t-100$$ $$y(2) = -4$$ $$y'(2)=14$$ Solve the IVP. I fairly certain that this type of problem requires using the second shifting theorem. First I apply the theorem to ...
2
votes
1answer
36 views

Laplace transform of $\cos^2(\omega t)$

Find the Laplace Transform of $\cos^2(\omega t)$, where $\omega$ is a constant. From a cosine identity: $cos^2(\omega t) = \frac{1}{2}(1+\cos(2\omega t))$. So then I get: \begin{align} ...
2
votes
2answers
31 views

Laplace Transforms

Show that ${\mathcal L} {\lbrace t^{n-\frac{1}{2}}\rbrace}=\frac{(2n-1)!!}{2^n}\frac{1}{s^n} \sqrt{\frac{\pi}{s}}$ My Attempt: ${\mathcal L} {\lbrace ...
0
votes
0answers
15 views

understanding quadratic separation inverse laplace

I am not sure about the correct english name of this mathematical technique "quadratic separation". So i am having a hard time understanding inverse Laplace transformation. I am familiar with tbe ...
1
vote
1answer
24 views

what is wrong with this easy inverse laplace transformation?

I am trying to understand the basics of easy inverse laplace transformations. On the first line is the "correct" answer. On the second line what i expected. ...
0
votes
0answers
44 views

Laplace transform of normal distribution function?

In my notes this was left as an exercise and I am a bit rusty with my calculus. Starting with the definitions: $$\mathcal{L}_X(t) = \mathbb{E}[e^{-tX}] = \int_0^\infty e^{-Xt}f(t)dt \;\;\text{ and ...
1
vote
0answers
23 views

Laplace transform of the square of Brownian motion hitting time

Let $B_{\mu}(t)$ be a one-dimensional Brownian motion with drift $\mu \geq 0.$ For $a > 0,$ let $$T_a = \inf\{t > 0: B_{\mu}(t) = a\}$$ denote the first hitting time of $B.$ The Laplace ...
3
votes
2answers
24 views

Finding the inverse laplace of this function: $ F(s)= \frac{s+8}{s^{2}+4s+5}$

Im trying to find the inverse laplace of : $ F(s)= \frac{s+8}{s^{2}+4s+5}$ I reached the following: $$ F(s)= \frac{s}{(s+2)^{2}+1} + 8 \times \frac{1}{(s+2)^{2}+1}$$ Now i have the 2nd term in the ...
0
votes
0answers
25 views

Inverse Laplace Transform by Partial Fraction Expansion

I've been trying to solve this partial fraction for a Laplace transformation but I can't. Is there any way to solve it? $$\frac{(s-t)^2}{((s-t)^2-1)((s+1)^2+4)}$$ Could somebody help, I've been ...
2
votes
1answer
36 views

inverse laplace transform of $\frac{s^3}{(s^2+4)^2}$

Using partial fractions gives $\frac{s}{s^2+4}$ - $\frac{4s}{(s^2+4)^2}$ Inverse laplace transform of the first member ($\frac{s}{s^2+4}$) is cos(2t). Can't figure out how to transform ...
0
votes
0answers
23 views

What is the Inverse Laplace of $1\over (s^2-a)$.

What is the inverse Laplace of $1\over (s^2-a)$. Since $L(sinh(at))={a\over s^2-a^2}$ can I take the inverse Laplace of $1\over (s^2-a)$ as, $sinh(\sqrt at)\over \sqrt a$
1
vote
1answer
32 views

Taking the inverse Laplace under boundary conditions

I want to solve the problem as described here in this article I am trying to solve it using Laplace Transformation. This is what I did: Taking Laplace transformation of the equation I get ...
2
votes
1answer
63 views

Laplace functional of sum of independent uniformly distributed random variables

I'm doing some of the exercises in Cinlar's "Probability and Stochastics" to better understand the material. This exercise (VI.1.17) is taken from page 247: Fix an integer $n \geq 1$. Let ...
1
vote
0answers
21 views

Problem in finding Inverse Laplace

I have the following equation But if I let $T_t(x,t)=constant$ then my equation becomes at steady state, since partial differentiation of a constant=0, right? here $\omega_bp_bc_b=M, $a ...
2
votes
1answer
40 views

Laplace transform to bio heat equation

This is the bio heat equation and I have several questions when trying to work with it. $$ \rho c \frac{\partial u(x,t)}{\partial t} = \nabla[k \nabla u(x,t)] + \omega_b \rho_b c_b [u_a - u(x,t)] ...
3
votes
2answers
56 views

Laplace inverse of $\frac{e^{-\sqrt{s+2}}}{s}$

I want to find out $$\mathcal{L^{-1}}\{\frac{e^{-\sqrt{s+2}}}{s}\}$$ How do you find the inverse Laplace? thanks
0
votes
0answers
12 views

Deriving the Laplace transform of $t$

In deriving the Laplace transform of $t$, I'm not being able to grasp why the limit term go to 0: $$\begin{align} x(t)&=t\\ \\ \therefore\; X(s)&=\int_0^\infty t\cdot e^{-st}\;\mathrm{d}t\\ ...
0
votes
1answer
26 views

Solving a differential equation that includes cosine

Anyone interested in coming up with a concise equation for $u(\tau)$ given the equation for its derivative below? \begin{align} \frac{du}{d\tau}=-\sigma u + S\bigg(1+B\cos(\tau)\bigg) \end{align}
1
vote
2answers
30 views

Find the coefficient of partial expansion $\frac{2x+3}{(1-x)(1+0.5x+0.5x^2)}$

I want to decompose the equation: $$\frac{2x+3}{(1-x)(1+0.5x+0.5x^2)}=\frac{A}{1-x}+\frac{B}{1+0.5x+0.5x^2}$$ I found $A$ by multiple both side with $1-x$ and plug $x=1$. However, it is so difficult ...
4
votes
3answers
172 views

Moment Generating Function and Inverse Laplace transform

I need to compute the inverse Laplace transform of the function $$ M(t)=e^{\frac{t^2}{2}} $$ Now, I know that this is a normal distribution with mean zero and variance 1, but how the computations are ...
0
votes
1answer
34 views

Attempting to use the Laplace transform to solve a second order ordinary differential equation with a piece wise forcing function.

Thanks to everyone who will bare with me and read this attempt and further validate it correct and improve upon it or otherwise correct its wrong. The question: $$ y''(t)+2y'(t)+y(t)=u\left( ...
0
votes
0answers
20 views

For which $s$ is the Laplace transform of Dirac delta function defined?

I have to find for which $s$ the Laplace transform of the Dirac delta distribution is defined. My idea is to start with the definition of the Laplace transform: integral from 0 to infinity. However, ...
4
votes
1answer
23 views

Show $\int_{s}^{\infty} f(x)dx = \mathcal{L} \{\frac{F(t)}{t}\}$ given $f(x) = \int_{0}^{\infty} e^{-xt}F(t)dt$

I'm trying to derive this to show that $$\int_{0}^{\infty} f(x)dx = \int_{0}^{\infty} \frac{F(t)}{t} dt$$ and use that to prove $$\int_{0}^{\infty} \frac{\sin t}{t} = \frac{\pi}{2}$$ How do I go ...
2
votes
0answers
20 views

Inverse Fourier transform using laplace

We have to solve the inverse FT of $$\frac{1}{1+4w^2}$$ I tried to do the synthesis but got mediocre results. However this term screams laplace to me. I can see a sine in there. The last lecture they ...
1
vote
1answer
91 views

Bilateral Laplace transform

My knowledge of Bilateral Laplace transform is less. Here are the few questions I need answer. What is the condition for existence of bilateral Laplace transform? How is the condition for existence ...
0
votes
1answer
69 views

Show that $Y(s)$ satisfies $(1+s^2)Y'(s) + sY(s) = 0$ for $ty'' + y' + ty = 0$

I first approached the problem by finding the Laplace transform of $ty'' + y' + ty = 0$, such that: $ts^2Y(s) - tsy(0) - ty'(0) + sY(s) - y(0) + tY(s) = 0. $ I solved for $Y(s)$: $Y(s) = ...
0
votes
0answers
32 views

laplace transform of difference between two gamma independent random variables

Knowing that the laplace transform of a Gamma distribution is given by: $$F_x(s) = \frac{\beta^a}{(s + \beta)^a}$$ and that for Z = X + Y "Sum of two independent Gamma distribution random ...
0
votes
0answers
16 views

Integral limits of Laplace transform.

In the definition of Laplace transform, limits of the integration are from 0 to infinity, Can anybody give insightful explanation for choosing these limits, How does it helps to simplify? Is there any ...
0
votes
1answer
26 views

What is the inverse Laplace transform of $\frac{p^2}{(p^2+4)^2}$

Given $$f(p)=\dfrac{p^2}{(p^2+4)^2}$$ So $$f(p)=\dfrac{p^2}{(p^2+4)^2}=\dfrac{p^2+4-4}{(p^2+4)^2}=\frac{1}{p^2+4}-\frac{4}{(p^2+4)^2}$$ I know the inverse Laplace transform of the first term but ...
0
votes
0answers
33 views

Convolution of exponential and rect functions

I have a convolution question in my signals and systems problem set that is puzzling me: $ f(t) = e^{-t/2T} u(t) $ and $ g(t) = rect(t/2T) $ find the convolution $f \ast g$ and I am assuming ...
-1
votes
2answers
90 views

Questions on Inverting Laplace transforms and Probability

From Williams' Probability w/ Martingales: Are we allowed to switch derivative and integral as follows: $$\frac{\partial}{\partial \lambda} \int_{0}^{\infty} e^{-\lambda x} f(x) = ...
-1
votes
0answers
21 views

Laplace transform for exponential cdf

I know Laplace transform of Gamma Pdf but I need transform of Cdf. Since $dCdf/dx = Pdf$ it should be that transform of Cdf is $s^{-1} L[Pdf]$ or am I wrong? Because I don't succeed in confirming ...
0
votes
0answers
21 views

Laplace Transform of the Unit Step Function

Finding the Laplace transform of: $$f(t) = \begin{cases}1 ,& 0 \leq t < 1 \\t^2 ,& 1 \leq t \leq 2 \\ 4, & t \ge 2\end{cases}$$ Heaviside /Unit step function: $$1+ (t^2 -1)u(t-1) + ...
0
votes
0answers
29 views

Laplace transform with Heaviside Unit step function

Working on a question and was wondering if this is right. Find the Laplace transform: $$x'' + 2x' + x = f(t)$$ $$x(0)=x'(0)=0$$ $$f(t) = \begin{cases}2 ,& 0 \leq t \leq 2 \\t ,& t \ge ...
-1
votes
0answers
21 views

Laplace Transform/Final value Theorem Issue

I basically have this question: 5. (a) Use the final value theorem to determine the value of y(∞), without actually solving the ODE y'+y=1, y(0) = 0 The Laplace transform of the ODE is: ...
0
votes
1answer
22 views

How do I Inverse Laplace $\frac{(s+1)^3}{s^4}$

I missed a class this week in maths and been a bit lost since with Inverse Laplace, how do I go about finding the Inverse laplace of: $$\frac{(s+1)^3}{s^4}$$ Do I simply expand the numerator? then ...
0
votes
1answer
49 views

Analytic solution of the equation $c\int_0^t x^{a-1}e^{-x}dx + (c+e^t)e^{-t}t^{a-1} = 0$

I would like to find the closed form solution of the equation in the title for the parameter $t$ when $-1<c<0$ and $0<a<1$. I tried to use the Laplace transform. The transformation of ...
0
votes
0answers
33 views

Understand first step of Laplace transform of integral

I am new to Laplace transform, and have some hard time understanding the very first step of the "preparation" before taking the laplace transform. $${f(t) =\int_0^t u \cosh(3u)\,\mathrm{d}u } ...
0
votes
0answers
42 views

The Laplace transform of an integrable function is differentiable

let $f\in L^1(0,\infty)$. For x>0, define $g(x)=\int_{0}^{\infty} f(t) e^{-tx} dt$. Prove that $f$ is differentiable for $ x>0$ and with derivative $g'(x) = \int_{0}^{\infty} -tf(t) e^{-tx} dt$. ...
0
votes
0answers
18 views

Finding Characteristic Polynomial & Minimal Polynomial of transfer function

So I seem to be lost with respect to this question. I have the following transfer function Image here: Transfer Function My boo shows that we get the characteristic polynomial of a regular matrix by ...
0
votes
0answers
4 views

System response (TF), multiplication vs substition

Let's say I have modeled a system as a transfer function: $H(s) = \frac{Y(s)}{U(s)}$ Given the question: For which values of $a$ is the input signal $e^{a t}$ absorbed by the system $H$? What is ...
2
votes
0answers
27 views

Stopping time distribution and transforms with 1-dimension B-motion.

Let $W_t$ be a 1-dimensional Brownian Motion. For $x>0$, we define: $$\tau_{x} = inf \{ t \geq 0; |W_t| = x\}$$ Compute $E[e^{-s\tau_x}]$ and prove that $\tau_x$ is equal in distribution to ...