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

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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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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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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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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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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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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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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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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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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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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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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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33 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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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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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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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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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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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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38 views

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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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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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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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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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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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 ...
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Discrete PID controller Laplace formula

I saw the following formula: the transfer function is: $$Gr(s) = K_p \bigg(1 + \frac{1}{T_n s}+ \frac{T_v s}{1 + T_d s}\bigg) $$ From my understanding: $K_p$ is the proportional gain $T_n$ is the ...
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Relation Fourier/Laplace Transform

I have a question about the relation between Fourier and Laplace transforms. I have seen in some places that the transfer functions in the Laplace space are represented as $G(s)$ where $s$ is the ...
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Find the Laplace inverse of the following.

$$ \frac{2s+5}{s^2+6s+34} $$ I am stuck on this part: Wolfram has the step by step showing that you simply split up the original fraction into $$ \frac{2s}{s^2+6s+34} + \frac{5}{s^2+6s+34} $$ and ...
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help with laplace [closed]

I have to get to the second part of the ecuation with laplace and I don't know how to do it step by step, help please! thanks!! $$\int\limits_{-\infty }^{+\infty }{\left( \frac{\sin x}{x} ...
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Laplace Transform conundrum

consider, x1(t) + constant = x2(t) => w/ laplace, X1(s) + c/s = X2(s) but, take the time derivative of the first equation, x1dot = x2dot => sX1(s) = sX2(s) => X1(s) = X2(s). Which is correct?
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Why can't Wolfram calculate the Laplace transform of $\sinh(t)\sin(t)$ correctly?

Question Show that the Laplace transform of $\sinh(t)\sin(t) = \frac{2s}{s^4+4}$. Wolfram can't calculate this as is, so I tried to simplify it a bit. I defined $\sinh(t)$ as $e^t-e^{-t}$ and split ...
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Use the Laplace transform to solve the initial value problem.

$$ y''-3y'+2y=e^{-t}; \quad\text{where}~ ~ y(2)=1, y'(2)=0 $$ Hint given: consider a translation of $y(x)$. I am stuck on this problem on our homework. I don't understand what they mean by a ...
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Numerical Laplace Transform

I want to compute the Laplace transform of data vectors. I have tried the usual numerical software and I'm surprised to see that does not have this operation available. I wonder if there is a straight ...
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Convergence of Laplace transform and its inverse

There is a sequence of functions $F^{\epsilon}(\lambda)$ which converges to 0 as $\epsilon \rightarrow 0$. Assume that each $F^{\epsilon}(\lambda)$ has a inverse Laplace transform f(s) such that ...
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Computing the Laplace transform of $\tan(pt)$

I've been thinking of using complex number approach , what's your view guys ?.
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Is it possible to say that $L(f^n)=s^nL(f)$ when the differential equation is not in the rest condition?

Question Use the Laplace transform to solve the following equation: $y'+2y=\cos(3t)$ ; where $y(0)=1$ In class our teacher wrote that "When in rest condition: $L(f^n)=s^nL(f)$", but I want to use ...
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Find the inverse Laplace transform of: $\frac{1}{(s^2+a^2)(s^2+b^2)}$

I'm having trouble doing this homework problem because I'm not sure how to deal with the $a$ and $b$. I did it the usual way we were taught - use partial fraction decomposition and then try to solve ...
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Solve the following initial value problem: $2y''+y'-y=e^{3t}$

$$ 2y''+y'-y=e^{3t}; \text{ with } y(0)=2,\ y'(0)=0 $$ I got to this point: $$ L(y)=\frac{1}{(s-3)^2}\cdot\frac{1}{(2s-1)(s+1)} $$ but now I'm not sure what to do with these polynomials. I know ...
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Inverse laplace transform with square completion

I need to find the inverse laplace of this : $$\frac{s+2}{s^2+2s+5}$$ I know that completing the square should help me to solve this so I get $$\frac{s+2}{(s+1)^2+4}$$ Then separating this equation ...
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About the causality of the signal whose frequency spectrum is not continuous as follows

Consider the signal in frequency domain: $$ \alpha(\omega) = \begin{cases} 1, & |\omega|<\omega_c \\ 0, & |\omega|\ge\omega_c \end{cases} $$ $$ =A(-j\omega)A(j\omega) $$ $$ =|A(j\omega)|^2 ...
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Laplace-Stieltjes :Functions of independent random variables

I am reading a book about stochastic modelling and I came across something and I couldn't really figure it out. First question would be are Probabilty Generating Functions (PGF) only for discrete ...
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Laplace Transform of Kelvin functions

What is the value of the Laplace transform, in terms of the G-function, \begin{align} \int_{0}^{\infty} e^{-st} \, t^{m} \, \left(ber_{\nu}^{2}(t) + bei_{\nu}^{2}(t)\right) \, dt \hspace{5mm} ? ...