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

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Did I Inverse Laplace correctly?

$$L^{-1}\frac{4s}{(s-6)^{3}}$$ $$4L^{-1}\frac{s}{s^{3}}|s=s-6$$ $$4L^{-1}\frac{1}{s^{2}}|s=s-6$$ $$4L^{-1}\frac{1!}{s^{1+1}}|s=s-6$$ $$4te^{6t}$$ Is this correct? symbolab and Wolfram are giving me ...
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Laplace Transform method to solve a differential equation

I'm trying to solve this diffrential equation using Laplace Transform method but have trouble finishing it (x(0) = 0): $$\frac{dx}{dt}+2x=15e^{-2t}$$ $$L\left(\frac{dx}{dt}+2x=15e^{-2t}\right)$$ ...
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29 views

Finding Inverse Laplace Transform (Using the table of Laplace transform)

I want to take the inverse laplace transform of $$\frac{e^{-s}}{s(s^{2}+1)}$$ So I separate the equation into $$e^{-s}\times\frac{1}{s(s^{2}+1)}$$ Now, I take the partial fraction of ...
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14 views

Replicate Matlab integrator block in MS excel

I created a simple diagram to solve ordinary differential equation as shown below. Simple ODE I was trying to compute the result of xf_dot manually in Ms Excel but I did not get the same answer ...
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37 views

How to obtain this partial fraction decomposition?

I am studying Laplace transforms right now and got stuck at this step that involves a weird partial fraction decomposition. It looks like the instructor skipped a bunch of steps and assigned ...
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49 views

Solving a differential equation with the heaviside unit step function

I am having trouble figuring out what exactly to do for this question. Given the initial conditions y(0)=1 and the equation y'-2y=4-3u(t-2) where u is the heaviside unit step function. I took the ...
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1answer
36 views

Restrictions on repeated use of initial conditions in ODE

It seems to be common practice when solving ODE's to keep a count of what conditions you have used. I was under the impression that once a condition has been used it cannot be used again. However, I ...
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1answer
51 views

Inverse Laplace transform seems to be always vanishing but it couldn't!

Let's consider $x\in (0,1)$ and the distribution $p(x)=\lambda x^\lambda$, $\lambda>0$. I would like to find the pdf of the sum. The characteristic function of the $N$ sum reads: \begin{equation} ...
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1answer
30 views

Inverse Laplace transform of convolution

$$\int_{0}^t \sin(2\pi(t-T)) \delta(t-5) \, dt$$ Wouldn't you just replace the $T$ in $\sin(2\pi(t-T))$ with a $5$ and that would be the answer?
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23 views

Laplace transformation of a piecewise function?

So I know in general how to do the laplace transformation of piecewise functions, but I ran into a different kind of piecewise than I have been doing so far. So I know for a function like: I just ...
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35 views

Find the Fourier transform for this function

Find the Fourier transform for this function $$f(x)=e^{x-e^x}$$ My Solution:- $T[f(x)]=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx} f(x)dx$ $=\frac{1}{\sqrt{2\pi}} ...
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27 views

Solve this integral equation using Laplace transform

Solve this integral equation using Laplace transform $$f(x)=x^2 + \int_{0}^{x}f^{\prime}(x-t) e^{-at} dt ,f(0)=0 $$ Please Help see mu answer below Thank you for your participation
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20 views

Find the Laplace transform for this function

Find the Laplace transform for this function $$f(x)=(1+2ax)x^{-\frac{1}{2}}e^{ax}$$ Please, help me see my answer below Thank you for your participation
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1answer
34 views

Laplace Transform Question

I was looking in my differential equations textbook and I found an interesting problem and I have no idea on how to approach it. I am supposed to let $F(s) = \mathcal{L} \{f(t) \} $ where $f(t)$ is ...
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1answer
27 views

Find the Laplace transform of the Gamma pdf

Per wikipedia the Laplace transform of the gamma distribution is $$L_X(s) = (1+\theta s)^{-k} = \frac{\beta^\alpha}{(s+\beta)^\alpha}$$ As an exercise I would like to show this.The definition I have ...
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35 views

laplace of piecewise (possibly dumb question but should have quick answer)

Find the Laplace Transform of $$f : (0, +\infty) \rightarrow \mathbb{R}, \quad f(t) = \left\{\begin{array}{lr} \sin(t), & \text{for } 0 < t < 2\\ 1+2t^3, & ...
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32 views

Sum of random variable and Laplace transform

Let $\tau$ be a r.v. $\in(0,\infty)$ with PDF $f_\tau(\tau)=\lambda e^{-\lambda \tau}$. How do I find the PDF of $f_{\sum_{i=1}^NX_i}(\sum_{i=1}^NX_i)$ where $X=e^{-\tau}$? I can easily find the PDF ...
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28 views

What is wrong with my Laplace Transform?

I need to find the Laplace transform of sine(t), and it is proving rather difficult. I integrate by parts, then integrate by parts again so that the original integral is on both sides of the ...
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1answer
27 views

A system of diferential equations solved by Laplace Transform

I have this system of diferential equations $$ \begin{cases} q'+q+i=50e^{-t}u_1(t) \\ i'+i-q=0 \end{cases}$$ $q(0)=i(0)=0$ ** $ u_1(t) $ is the Heaviside step function Solution Rewriting: $$ ...
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35 views

Laplace Transform of equation

I'm having trouble with the laplace transform: $\mathcal{L} \lbrace \sqrt{\frac{t}{\pi}}\cos(2 t) \rbrace$ The problem gives me the transform identity $\mathcal{L} \lbrace \frac{\cos(2 t)}{\sqrt{\pi ...
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1answer
32 views

Inverse Laplace Transform of $\frac{1}{\sqrt{s+a}+\sqrt{s+b}}$

I need to calculate the inverse laplace of: $$F(s)=[\frac{1}{\sqrt{s+a}+\sqrt{s+b}}] \qquad \qquad (s>-a\quad ;\quad s>-b;\quad a\neq b) $$
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54 views

Inverse Laplace Transform and error function

Express your answer in terms of the error function: $$L^{-1}\left[\frac{1}{\sqrt{s^3+as^2}}\right]$$ Clue: $\qquad L\left[\frac{1}{\sqrt{t}}\right]=\sqrt\frac{π}{s} \qquad , \qquad s>0$ Error ...
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solve this partial integration with step function

I would like to know how to solve this partial integration. The equation I got is based on the following convolution: $$t^2e^{-2t} * te^t$$ The part I am having a hard time with is the (t-u) ...
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34 views

How to obtain Laplace transform of {f(t-a)U(t-b)}

$f(t)=g(t-10)U(t-15)-g(t-10)U(t-20)$ The above $f(t)$ contains terms of the form $g(t-a)U(t-b)$, where $a$ doesn't equal $b$. Describe the form that $L\{f(t-a)U(t-b)\}$ takes. [Hint: The formula for ...
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21 views

How to find the Laplace Transform of $\sin(2t) u(t-\pi)$

Using a translation theorem to determine that $ℒ{f(t-a)u(t-a)}$ is equivalent to $F(s) e^{-as}$, I determined that $ℒ\sin(2t)u(t-\pi)$ is equal to $ℒ\sin(2t) * e^{-(\pi)t}$ and found the Laplace ...
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On Laplace transform of periodic functions

I recently bumped into this theorem regarding the Laplace transform of periodic function: Given a periodic function $f(t)=f(t+p)$, where $p$ is its period, then its Laplace transform is given by: ...
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1answer
39 views

differential equation problem with laplace - calculators cant solve

I am trying to solve a third order differential equation problem with laplace transform. But I am stuck since 3 days... Could someone tell me what I did incorrectly? I transformed my equation in the ...
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42 views

Dirac Bra-ket notation and Laplace Transformations

I had a hard time developing an intuition for the Fourier Transform until I was introduced to Bra-ket notation in Quantum Mechanics (I come from physics). With this notation, many problems make ...
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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 ...
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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 ...
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1answer
26 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 ...
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1answer
18 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!
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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 ...
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1answer
73 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} ...
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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 ...
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42 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 ...
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1answer
45 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} ...
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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 ...
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16 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 ...
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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. ...
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147 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 ...
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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 ...
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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 ...
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30 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 ...
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1answer
39 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 ...
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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, $sinh(\sqrt at)\over \sqrt a$
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33 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 ...
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1answer
69 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 ...
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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 ...
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43 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)] ...