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

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Solving initial value problem with delta function

Use the Laplace transform to solve the following initial value problem: $$y''+4y = 3\delta(t-\pi), \quad y(0)=0, y'(0)=0$$ I solve the equation based on what I learned in class, my answer is ...
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38 views

Show that these two differential equations have the same solution

Question: Show that the problems $ax'' + bx' + cx = f(t); x(0) = 0, x'(0) = v_0$ and $ax'' + bx' + cx = f(t) + av_0 \delta(t); x(0) = x'(0) = 0$ have the same solution for $t \gt 0$. Thus the effect ...
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24 views

Signal whose Laplace transform contains derived Dirac-deltas: How do I find the inverse transform?

I must reconstruct the input signal to a system, knowing the output signal and the system transfer function. At the end, I found that the Laplace-Transform of the input signal is something like: $$ ...
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60 views

Solving convolution problem with $\delta(x)$ function

Suppose we had the functions: $$g(t)=\theta(t)(e^{-t}+2e^{-2t})+2\delta(t)$$ and $$u(t)=2(\theta(t)-\theta(t-2))$$ Then we have ...
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15 views

Can this be expressed in terms of Laplace Transform?

I have an expression in the from: $$\mathbb{P}(H>\theta)=\int_0^{\infty}\exp(-m\theta I)f_I(i){\rm{d}}i$$ Here, $f_I(i)$ is the PDF of random variable $I$. Let, $\mathcal{L}_I(s)$ is the Laplace ...
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IVP solved with Laplace transformation - mistake?

I want to solve following IVP with Laplace transformation. \begin{align} x''(t) + 2x'(t) + x(t) &= \begin{cases} 0, & t < 0\\ 1, & t \in (0;2)\\ 3, & t > 2 \end{cases}\\ x(0) ...
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30 views

Find the Laplace transform f(s) for the figure showing below

Figure I tried to solve it and came up with this solution.
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18 views

Definition of Laplace

If a function is not of exponential order, then is it possible that function's Laplace transform exist? If yes, then how can we determine for a function that its Laplace transform exists or not?
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28 views

How to solve the laplace transform of $f_m(t_m)$ = $f_1(t)$ $\int_{0}^{\alpha} f_2(\tau) d\tau$ + $f_2(t)$ $\int_{0}^{\alpha} f_1(\tau) d\tau$.

Could you please help me to solve the following : if $t_m$ = min($t_1$,$t_2$) The probability density function $t_1$ is $f_1(t_1)$ and $t_2$ is$f_2(t_2)$ then $f_m(t_m)$ is the probability density ...
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19 views

Having some trouble with inverse Laplace tranform

How to solve this using inverse Laplace transform? 1/[($s$+1)($s$+2)$^4$] I though of this solution which is $A$/($s$+1) + $B$/($s$+2) + $C$/($s$+2)$^2$ + $D$/($s$+2)$^3$ + $E$/($s$+2)$^4$ Then ...
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33 views

Inverse Laplace Transform of an Infinite Sum

How to find the Inverse Laplace Transform of the following expression $$1+\frac{-Xs^{2/a}-Ys^{3/b}}{1!}+\frac{(-Xs^{2/a}-Ys^{3/b})^2}{2!}+\cdots$$ Any approximation is also okay... Here $a$ and $b$ ...
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47 views

Inverse Laplace Transform of Stretched Exponential

I have a Laplace tranform in the form given below $$\mathcal{L}(s)=\text{exp}(-As^{2/\alpha}-Bs^{3/\beta})$$ which is an multiplication of two stretched exponential decay function where $A,B >0$ ...
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47 views

Unilateral Laplace Transform vs Bilateral Fourier Transform

I would like to know why when we find the Laplace transform we use the one-sided (unilateral) version (all Laplace transform tables I can find are one-sided, like this one ...
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28 views

How to obtain a stabilization problem in linear system with controller?

The scheme of system: The equasion after Laplace transform: $$Y(p) = \frac{PID(p)\cdot H(p)}{1 + PID(p)\cdot H(p)} Y^d(p)$$ Now I want to make inverse Laplace transform and then plot $y(t)$, but ...
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49 views

How to determine if $z(x,y)=\ln(x^2 + y^2)$ is a harmonic function [closed]

So I know that a function is harmonic if it satisfies Laplace's equation: $\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = 0$ But I'm just not sure how I should put it in ...
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Find a solution to $xy'+y = x^k$ using the Laplace transform.

The part that is giving me trouble is: $$\mathcal{L}[xy']$$ I have never done this before so I ploughed through. $$\mathcal{L}[xy'] = \int^{\infty}_0ty'(t)e^{-pt}dt = ...
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18 views

Having trouble finding inverse Laplace Transform

I have this Laplace transform $$X(s)=\frac{1}{s\cdot(s^{2}+0.2s+1)}$$ I want to find its inverse transform. I did the following. First I decomposed that into partial fractions ...
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23 views

Laplace and unit step- multiplication vs convolution

Please be gentle if the question is stupid. When using the laplace transform, you often multiply the function of interest by a shifted unit step function to operate on the positive portion of the ...
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36 views

Prove this inequality using laplace transform

Let $n$ be a strictly positive integer, and $a_1,\cdots a_n,b_1,b_2 \cdots b_n$ strictly positive real numbers. Prove that $$\sum_{i=1}^n (\frac{a_i}{b_i})^2 \le 2\sum_{1\le j,i \le n} ...
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51 views

Pde using laplace transform

Could you help me to find a solution for this partial differntial equation by using laplace transform $$u_{t} - u_{xx} = xt$$ where $$u(0,t)=t, \quad u(1,t)=t^2, \quad u(x,0)= \sin \pi x$$ I tried ...
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59 views

Is the Laplace transform essentially a generalized version of the Fourier transform?

My current understanding of the two concepts is far from perfect, and I am essentially just a beginner. But it seems to me that while the Fourier tries to decompose functions as a composition of ...
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Find f(x) $\int_0^x f(u)du - f'(x) = x$

Find f(x) $$\int_0^x f(u)du - f'(x) = x$$ I was not given f(0) which makes it difficult for me to find f(x). This is what I have thus far: $$\frac{F(p)}{p}-pF(p)+f(0)=\frac{1}{p^2}$$ ...
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Laplace Transform: $g(x)=a\sin(x)+\int_0^x \sin(x-u)g(u) du$ [closed]

$$g(x)=a\sin(x)+\int_0^x \sin(x-u)g(u)du$$ I need to find $g(x)$ I believe I need to use Laplace Transform with this in mind (Convolution Thm): $$(f*g)(x)= \int_0^x f(x-t)g(t)dt$$ However I don't ...
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48 views

Two approaches to problem give different answers — which one is correct?

I approached this problem in two ways and arrived at different answers. Both ways seem logical to me. Are they both correct, or is one flawed? This is the original problem: $$\mathfrak L^{-1} ...
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62 views

Laplace transform of $\cos(at)/t$

If someone could help me solve for $$\mathcal{L}\left\{\frac{\cos(at)}{t}\right\}$$ it would be great. Step-by-step I have so far: $$\begin{align}\int_0^\infty \frac{\cos(at)\space ...
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35 views

Complex inversion of a function

I am trying to find the function whose laplace transform is below using the complex inversion formula: $$ f(s)= \frac{se^s}{(s-2)^3}$$ My attempt below seems to be giving me the wrong answer but I'm ...
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10 views

Existence/Uniquness/Solution of a countably infinite system of linear ODEs.

Consider the following system of ODE's. \begin{align*} \frac{\mathrm{d}^2}{\mathrm{d}t^2} x_0 &= F(t) + x_1\\ \frac{\mathrm{d}^2}{\mathrm{d}t^2} x_i &= x_{i+1}-2x_i+x_{i-1}\, \quad ...
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Laplace transformations on a homogeneous ODE

$$y^{\prime\prime} - 3y^{\prime} + 2y = 0$$ $y(0) = 14$, $y^{\prime}(0)=0$, and using the Laplace transformation I'm trying to solve this IVP
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Matched pole zero discretization

There are several techniques to discretize continuous-time transfer functions to discrete-time transfer functions. Some of them, such as, zero-order-hold, forward euler or Tustin, are well known. ...
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46 views

Solving Second-order non-linear ODE, with fractional expansions

I am solving a differential equation related to fluid mechanics, a rigid air bubble rising towards the surface of a liquid. Doing all of the maths, I have come to this differential equation, which I ...
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Proving that if $f \in \mathcal{E}$, then $f' \in \mathcal{E}$ (same for $\mathcal{E}_q$)

In the context of ordinary differential equations, I'm trying to prove that if some function $f$ is an element of $\mathcal{E}$, which is the function space of all exponential polynomials, then $f'$ ...
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27 views

Laplace transformation on an exponential

Using the definition of Laplace transformation (and without using a table), how to find the Laplace transformation of $$ g(t)= \begin{cases} 0,&\text{if }0\leq t\leq 4;\\ e^{3t}&\text{if }4\le ...
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inverse Laplace transform of gamma function

My problem is to get the inverse Laplace transform of the following equation. $$\hat{P}(s) = \frac{\Gamma(p+1+s T)}{p! N^{s T}}$$ $p$, $T$ and $N$ are positive constants. The denominator $N^{-s T}$ ...
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Partial fraction decomposition of $\frac{21}{s^{2}+4}$ for inverse-Laplace transform

So I have this number which I want to do inverse-Laplace transformation on, which is kind of complicated. So it would be easier to do some partial fraction decomposition first. I am trying to do the ...
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50 views

Solving for $x$ in a Laplace equation

So I have this Laplace equation: $$s^{2}x+4sx+5=\frac{s}{s-1}$$ And I want to solve for $x$. My result is the following: $$x = \frac{5-4s}{s^{3}+3s^{2}-4s}$$ Which is also the same answer that for ...
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Solving Bessel's equation by Laplace transform

I am learning Bessel function the solution of Bessel equation by book 'Advanced Engineering Mathematics' by Peter V.O'Neil and here i found its derivation by Laplace transform. In this derivation of ...
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Laplace transform identity $F(s) = \mathcal{L}(t^{-3/2} \mathrm{e}^{-1/t})$

I'm asked to prove the following result: If $F(s)$ is the Laplace transform of $f(t) = t^{-3/2} \mathrm{e}^{-1/t}$, show that $F'(s)=-s^{-1/2}F(s)$. I'm having a lot of troubles to prove this ...
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39 views

Laplace Transform of Dirac Delta function

I've seen everywhere that that the Laplace Transform of Dirac Delta function is: $$L[\delta(t-a)] = e^{-sa} \text{ when } a > 0$$ But they never explain what happens when $a < 0$. Can I assume ...
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Laplace Question $f(t) = e^{-t} \sin(t)$

I need help with this Laplace question. \begin{equation} f(t) = e^{-t} \sin(t) \end{equation} Answer should be $\dfrac{1}{s^2 + 2s + 2}$ What I'm currently doing is as follows: $u = \sin(t)\qquad$ ...
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Laplace transform problem with heaviside functions

Find the Laplace transform of (a) $[u(2pi/3)(t)]e^{-3t}cos(4t)$ (b) $[u(2pi/3)(t)]e^{-3t}(t)cos(4t)$ [Hint: Use the result from (a)] u is the heaviside function. For part a I got an answer of ...
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Tauberian theorems in queing theory

I'm trying to use Tauber's theorem below (Feller 1971, chapter XIII.5) "Let U be a measure with a Laplace transform $\omega(\lambda)$ defined $\forall \lambda >0$ and $t,\tau>0$ s.t. $t\tau=1$, ...
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51 views

Laplace Transform of Square Wave Function

I am given a problem in my textbook and I am left to determine the Laplace transform of a function given its graph (see the attached photo) - a square wave - using the theorem that $$F(s) = ...
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28 views

How to write a transfer function (in Laplace domain) from a set of linear differential equations?

Provided I have a system of linear differential equations (in time domain) such as: $$\begin{cases} \dot{x}(t)=Ax(t)+By(t)+Cz(t)\\ \dot{y}(t)=A'x(t)+B'y(t)+C'z(t)\\ \dot{r}(t)=B''y(t)\\ \end{cases}$$ ...
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23 views

Dynamic real-time system problem

I am struggling with a systems theory problem, the task is as follows: u(t) -> H(s) -> y(t) H(s) being the transfer function $$ H(s) = H(s) = \frac{s+1}{s(s+2)^{2}} $$ $$ u(t) = e^{-5t} $$ So ...
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Why M.G.F transform is injective a.s.?

We always use the theorem that If we know a random variable's MGF, we can determine its Pdf, which means the map from Pdf to Mgf is injective almost surely. And I just wanna know why this is ture.
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What topics should I study to understand Laplace transform?

If I'm a beginner to start understanding Laplace transform, from where should I start to understand Laplace Transform?
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117 views

How to evaluate integral $\int_0^{\infty} e^{-x^2} \frac{\sin(a x)}{\sin(b x)} dx$?

I came across the following integral: $$\int_0^{\infty} e^{-x^2} \frac{\sin(a x)}{\sin(b x)} dx$$ while trying to calculate the inverse Laplace transform $$ L_p^{-1} \left[ ...
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What is the laplace transform of $\delta(t-\pi /6)\sin (t)$

What is the laplace transform of $\delta(t-\pi /6)\sin (t)$ I know that $L\{\delta(t-\pi/6) \}=e^{-s\pi/6}$ I also know that $L\{\sin (t) \}=1/(s^2+1)$ I also know that ...
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47 views

Does $e^{1/t}$ have a Laplace Transform?

I'm having a little trouble understanding why some functions have a Laplace transformation and others don't. The definition I was given in class last week was "Given a suitable function $F(t)$ the ...
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29 views

Proving that the Laplace Transform is an isomorphism with convolution

My question is primarily more about the convolution integral/theorem than the proof in question, but I wanted to give some idea of why I'm asking. The Laplace transform of the convolution $$(f\star ...