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

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ODE using Laplace transform

[ I got my Y(t) to be : $$12 \, e^{-4} \, e^{-2s} \, [\frac{1}{12(s+2)} + \frac{1}{4(s-2)} - \frac{1}{3(s-1)}] + \frac{1}{(s-2)} - \frac{1}{(s-1)}.$$ so i assume I need to use t shifting for the ...
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limitation of initial value theorem

I am student and stuck in this question , this question was asked to me on exam , what is the limitation of initial value theorem ,but i was not able to think of limitation Since the time was running ...
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Choosing between SOV/Green's functions/Laplace transform for solving PDE - Guideline for choosing the most appropriate method?

Forgive me if this questions seems silly, but I have a question which is keeping me busy. I'm not really looking for a mathematical proof (but it is welcome), however I'm more looking for guided ...
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Inverse Laplace Transform with a functional

I don't know if it is possible, but I would appreciate if someone help me to obtain the inverse Laplace transformation of the function $F(s)=hs/(s^2+w(s)^2)$. Where $s$ is a complex number and $w$ is ...
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solving 2nd order pde with dirac delta

I want to find the functional form of the Green function G(x,t) for a parabolic differential equation: $$ ...
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Still getting wrong answer after trying to solve $x''(t)+4x(t)=t^2$ where $x(0)=1$ and $x'(0)=2$

I am trying to solve this differential equation: $$x''(t)+4x(t)=t^2,x(0)=1,x'(0)=2$$ The answer should be: $$x(t)=\frac{1}{4}t^2-\frac{1}{8}+\frac{9}{8}\cos{2t}+\sin{2t}$$ Which is also verified ...
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40 views

Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...
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Laplace transform of the square root of a generic function

Let $f(t)$ be a function (for example of time $t$). Is there a general expression of the laplace transform of $\sqrt{f(t)}$ ? Same question for the inverse Laplace transform : Let $f(s)$ be the ...
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Why the Laplace transform of u(-t) is 1/s?

Yesterday I had my first contact with the Laplace transform, in an Electric Circuits class. $\mathcal{L} \left\{ u \left( -t \right) \right\}$ showed up. Our teacher said it was equal to ...
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39 views

Different proofs of uniqueness of the Laplace transform

How many different types of proof do you know for the so-called Lerch's theorem, i.e., uniqueness of the Laplace transform? I have found the following references for proofs. New books, in general, do ...
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inverse Laplace transfrom of a product of integrals

in this imgur gallery there are some equations to explain my problem: http://imgur.com/a/DxYli . Sorry if this is not so comfortable for you, but I can't link too many pictures in one question. The ...
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Solving general linear ODE $\sum_{k=0}^n y^{(k)}=0$

Is there a way to solve this general linear ODE: $$\sum_{k=0}^n y^{(k)}=0$$ For the first few $n$ here are the solutions: $$\begin{array}{c|c} n & y \\ \hline 0 & 0 \\ 1 & c_1 e^x \\ 2 ...
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1answer
21 views

Help with two functions - continuity, Laplace transform and Fourier series [on hold]

I've been practicing for my exam lately, and there are two function that I've had a real trouble analyzing. 1.$f(x) = \sum_{n=1}^{\infty} \frac{\sin(nx)}{10^n \sin(x)}$, for $x \neq k\pi$ $f(x) = ...
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1answer
33 views

Inverse Laplacetransform of rational function with multiple pole

I have to calculate the inverse Laplacetransorm of this function using Residue calculus $$ \frac{s^4 + 6s^3 - 10s^2 + 1}{s^5} $$ but I can't find any Residue rule that would solve this. Can you show ...
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Solving Heat Equation with Laplace Transform

I am trying an alternative method to separation of variables to the following equation $$ \begin{cases} u_{xx} =4u_t , 0 < x < 2, t>0\\ u(0,t)=0, u(2,t)=0, t>0\\ u(x,0)=2\sin(\pi x), 0 ...
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Doubts relating to Spaces of type $\mathcal{S}$

I have doubts in the following two questions : 1) What is the Laplace transform of $[x^k\varphi(x)]^{(q)}$, where $\varphi\in \mathcal{S}_\alpha^\beta$ and $-\infty<x<\infty$ , ...
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Laplace vs Fourier density representation of a positive rv

Given a general random variable $X$ with density function $f(x)$ and characteristic function $\phi_X(u)$ we can go back and forth between the density and the characteristic by using the Fourier ...
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What is the mapping of Z-transform?

Recall that given a series $x(k)$, the Z-transform $\mathcal{Z}$ is defined as: $$\mathcal Z(x(k)) = \sum_{k =0}^{\infty} x(k) z^{-k}$$ where $x(k)$ satisfies $|x(k)| \leq M\rho^k$, $M, \rho$ real ...
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A discussion on fourier and laplace transforms and differential equations …?

i have read many of the answers and explanations about the similarities and differences between laplace and fourier transform. Laplace can be used to analyze unstable systems. Fourier is a subset of ...
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To find Inverse Laplace of $\,F(s)=\log\dfrac{s+1}{(s+2)(s+3)}$

To find Inverse Laplace of $$F(s)= \log\frac{s+1}{(s+2)(s+3)}.$$ I have tried to use shifting theorems, but of no use. Should I apply series for log and take inverse laplace of individual terms, if ...
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How to find laplace transform of $\,\sinh(ct)\int_a^te^{au}\sinh(bu)\,du$

How to find laplace transform of $$\sinh(ct)\int_a^te^{au}\sinh(bu)\,du.$$ I tried to integrate inner function and then do it, but it became way more tedious. So I was thinking there should be good ...
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matrix elementary column operations

Till now i was using the elementry row operations to do the gaussian elemination or to calculate the inverse of a matrix. As i started learning the Laplace's transformation to calculate the ...
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Two different answers with Laplace

Find the solution for the equation $$ -u'' + u = \delta'(t)$$ for which it "disappears" for $t<0$ By using residuals! So I used Laplace transformation for this. $$Y(-s^2 + 1) = ...
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Inverse Laplace Transform of s/(s+1)

What is the inverse laplace transform of $\frac{s}{s+1}$? My work was: $$ X(s)=\frac{s}{s+1}\\ X(s)=s\frac{1}{s+1}\\ x(t)=\frac{d}{dt}e^{-t}=-e^{-t} $$ My only issue is that when I check my answer ...
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transforming ordinary generating function into exponential generating function

I have seen a post here that says that you can convert an exponential generating function into an ordinary one with the aid of the Laplace transform. Is it possible to do the reverse transformation? ...
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Unit Impulse response vs Impulse response in ODE

I'm was watching MIT OCW lectures for Differential Equations and in lecture 23, the professor goes over impulse inputs where impulse is $\int_a^b{f(t)dt}$ where $f(t)$ can vary or be constant. He ...
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Where is my error in solving $y'' + y' + y = 0, y(0) = 1, y'(0) = 0$ with Laplace transform?

Im trying to solve a laplace transoform question, but i am stuck. The question is $y′′(t) + 2ζy′(t) + y(t) = 0, y(0) = 1, y′(0) = 0$ and $ζ = 0.5$. I have so far done: Laplace transform which gives ...
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35 views

Solve 2nd order ODE using Laplace transform

Im trying to solve a laplace transoform question, but i am stuck. The question is $y''(t)+2\zeta y'(t)+y(t)=0$,$y(0)=1$,$y'(0)=0$ and $\zeta=2$. I have so far done: Laplace transform which gives ...
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Convolution of two piecewise functions using Laplace transform [closed]

I'm practicing Laplace transforms and I stumbled upon one question which I am not exactly sure how to tackle. The question is: Using Laplace transforms (or otherwise) calculate the convolution of ...
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Scaling property of Laplace transform

I am not sure how to do the following problem: Let $$\hat{F}(s)=\mathfrak{L}(f(t))$$ be the Laplace transform of $f(t)$. Show that: $$\mathfrak{L}(f(at))=\frac{1}{a}\hat{F}\left(\frac{s}{a}\right) ...
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A very simple question: what spaces of function does the laplace transform map from and into?

Given a function $f$, we can write $f:\mathbb{R} \to \mathbb{R}$ to denote that $f$ takes a number from $\mathbb{R}$ into $\mathbb{R}$. Easy enough. Given the laplace transform operator ...
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Piecewise continuity hypothesis in Laplace transform theorems

f is a real values function of a real variable t. "f is piecewise continuous in every finite closed interval [0,b], for every b>0." Differential Equations book of SL Ross uses the expression in ...
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Partial fraction decomposition with exponential in denominator

Can the fraction below be decomposed ? $s$ is the Laplace variable and $T$ is a constant. $$\frac 1 {s (1-e^{-s T})}=?$$
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inverse Laplace transform of $e^\sqrt{as}$

I am trying to find the inverse Laplace transform of $e^\sqrt{as}$ for $a>0$. So we need to solve $\oint_B dz \: e^\sqrt{az} e^{z t}$ (Bromwich contour), but not sure how to start. How do we even ...
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Zeros/poles at Laplace and at Fourier Transform

I recently started "relearning" the Laplace transform, and I noticed something. It seems to me that the intuitive idea of poles and zeros is different between these two transforms! For example, in ...
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ODE $x'' + 2x' +5x = \sin3t$, $x(0)=1,\ x'(0)=-1$, Solve using Laplace Transform

While solving the differential equation $$x'' + 2 x' + 5 x = \sin3t, \quad x(0) = 1, \quad x'(0) = -1$$ by use of Laplace transform I got to $$X(s^2 +2s+5)=\frac{(3)}{s^2 +9} +s +1$$ ...
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Why does the Laplace transform of a matrix exponential $f(t) = e^{At}$ satisfy $sF(s) = AF(s) + I$

Where $A$ is some $n \times n$ matrix Suppose I am given $f(t) = e^{At}$, then $\dot f(t) = Ae^{At}$, so $L(\dot f(t)) = sF(s) = AF(s)$ Why does $sF(s) = AF(s) + I$ hold true. (more specifically, ...
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PDE: Fokker-Planck equation with time-dependent boundary conditions

We have the following PDE: \begin{equation} \frac{\partial p(x,t)}{\partial t}= - a\frac{\partial p(x,t)}{\partial x} + \frac{D}{2} \frac{ \partial^2 p(x,t) }{\partial x^2}, \quad0<x<L, \quad ...
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Simplifying transfer functions in Z domain

I have difficulties to check whether the below transfer function is recursive or non-recursive: $$H(z)=\frac{1-z^{-1}+z^{-2}-3z^{-3}}{z^{-2}(1-z^{-1})}$$ I know that I have to multiply the num and ...
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Solving differential equation using Laplace transform

Can this DE be solved using Laplace transform? $\frac{\mathrm{d} y}{\mathrm{d} x}\cos x=y\sin x+\cos ^{2}x$
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Laplace Transform of a Heaviside function

Find the Laplace transform. $$g(t)= (t-1) u_1(t) - 2(t-2) u_2(t) + (t-3) u_3(t)$$ I understand that the $\mathcal{L}\{u_c(t) f(t-c)\} = e^{-cs}*F(s)$ Finding $F(s)$ is the hard part for me. My ...
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Changing a heaviside function into a one line function

$$h(t) = \left\{\begin{array}{l}1,\, \pi\leq t<2\pi\\ 0,\, 0\leq t<\pi\text{ and }t\geq2\pi\end{array}\right.$$ I need to change $h(t)$ into a one line function. I believe it to be ...
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An inverse Laplace transform I

While viewing the problem "Find the inverse Laplace transform" the solution given by Amir Alizadeh can be reformulated into the form \begin{align} \mathcal{L}^{-1}\left\{ \frac{s \, (a - f(s))}{s-b} ...
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What does it mean to take the Laplace transform of a non-periodic position function?

What I'm trying to get through my head here is how taking the Laplace transform of a system with a position function like $X(t)=t$ is possible. To my current (admittedly incomplete) ...
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Inverse Laplace

I want to calculate the inverse laplace of $$F(s)=e^{-3s}\frac{1+s}{s^3+2s^2+2s}$$ And i'm wondering if applying the derivative theorem is correct. To keep it simple it split them up: ...
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What is Fourier transform of space variable? on the similar grounds what is the Laplace transform of the same?

I understand that the transform of time domain is frequency domain and the transformation of time to frequency domain is done by Fourier/Laplace transforms. I am confused about the transformation of ...
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riemann zeta function : entire and even Laplace transforms

$$\xi(s) = s(s-1)\pi^{-s/2}\Gamma(s/2) \zeta(s)$$ $$\xi(s) = \xi(1-s)$$ thus $\Xi(s) = \xi(1/2+s) = \Xi(-s)$ is even, and furthermore it is an "entire and even Laplace transform" : $$\Xi(s) = ...
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Laplace transform of the logarithmic integral function

What is the Laplace transform of the logarithmic integral function $\text{li}(t)$. Meaning, how to compute the integral : $$\int_{0}^{\infty}\text{li}(t)e^{-st}dt$$
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Moving a limit inside an integral

Under what conditions does $$ \lim_{a \to 0^{+}} \int_{0}^{\infty} f(x) e^{-ax} \, dx = \int_{0}^{\infty} f(x) \, dx \ ?$$ For example, for $a>0$, $$ \int_{0}^{\infty} J_{0}(x) e^{-ax} \, dx = ...
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Laplace Transform for solve ODE (RLC circuit)

I have an RLC circuit and I want to know the charge on the capacitor $q(t)$ using Laplace transform: The diferential equation is: $$ Lq'' + Rq' + \frac{1}{C}q = E(t),$$ where $L = 1H , R = 20 ...