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

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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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Solve 2nd order ordinary differential equation with unit-step driving function by Laplace transforms and convolution theorem. (5.6-42)

Synopsis: Please check my work. I do not have a text "answers to odd problems" for reference as this is an "even" numbered problem. The following documents in good detail the steps taken to solve for ...
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Inverse Laplace transform of $ \frac{7s-6}{s^2-s-6}$

IT is asked to find the inverse Laplace transformation of $$\frac{7s-6}{s^2-s-6}$$ Writing it with partial fractions $$\frac{7s-6}{s^2-s-6} =\frac{4}{s+2}+\frac{3}{s-3}$$ Ive found that the ...
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Partial Fraction Decomposition for Laplace Transform

As part of trying to solve a differential equation using Laplace transforms, I have the fraction $\frac{-10s}{(s^2+2)(s^2+1)}$ which I am trying to perform partial fraction decomposition on so that I ...
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How to find Bilateral Laplace Transform of $e^{at}$ Using Changing of the Time Horizon

Ok, this has me a bit stumped. In my class the teacher "showed" us how to find the bilateral Laplace transform of x(t)=$e^{at}$ where $-\infty<t<\infty$. Breaking them into the two parts ...
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Discrete Laplace transform. Analogy to change of basis

Assume $$f=\sum_{k=0}^{N-1} c_k\cdot E^{k}$$ where the vector $E^k$ is $$E^k = (e^{2 \pi i k\cdot M}(0),e^{2 \pi i k\cdot M}(1),\cdots,e^{2 \pi i k\cdot M}(N-1))$$ (M is a constant and e represents ...
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Wave Equation with outgoing wave boundary conditions

I need some help with this problem: I have a to solve the wave equation with two initial conditions and with outgoing wave boundary conditions; i.e., $$\begin{cases} u_{tt}-u_{xx} & =0\\ u(x,0) ...
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47 views

Solve 2nd order ordinary differential equation by Laplace transforms and convolution of their inverse functions. (5.6-40)

Synopsis: Please check my work. I do not have a text "answers to odd problems" for reference as this is an "even" numbered problem. The following documents in good detail the steps taken to solve for ...
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35 views

Existence of Inverse Laplace Transform

Let $F(s)$ the Laplace transform of a function $f(t)$ . Under which conditions on $f(t)$ there exist a unique $g(t)$ such that $g(t) = \mathcal{L}^{-1}\{e^{- F(s)}\}(t) \quad $ ? ...
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220 views

Solve 2nd order ordinary differential equation by Laplace transforms and convolution of their inverse functions. (5.6-39)

Synopsis: I cannot duplicate the answer in my text although it does appear very similar. This tells me that my method is correct but I am making another kind of error -- perhaps in my integration? ...
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69 views

How do you solve a homogeneous ODE using Laplace Transform?

Suppose I want to solve $$y'' - 4y = 0$$ all I.C. zero Taking the laplace transform I get $$(s^2 -4)Y(s) = 0$$ So y(t) can be anything. Why am I running into this problem and how can I get around ...
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Uniqueness of Laplace Transform

The theorem for Uniqueness of Laplace transform is as below: Suppose $f$ and $g$ are continuous functions. If $Lf(s)$ = $Lg(s)$ then $f(t)$ = $g(t)$. I am following the proof given in the notes in ...
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22 views

Method for taking this inverse laplace transform

i am having trouble taking the answer to this problem that i found a book called "Differential equations with applications and historical notes": $$ e^{-x} = y(x) + 2 \int_0^x\cos(x-t)y(t) dt $$ ...
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1answer
29 views

Using a Laplace transform to solve piecewise functions that are also an infinite sums

Consider $y'' + 2y = f(t), y(0)=y(L)$ $$f(t) = \begin{cases}0 \quad 0<t<L/4 \\ 1 \quad L/4<t<3L/4 \\ 0 \quad 3L/4 < t < L \end{cases}$$ Suppose that we can wrtie $f(t)$ and $y(t)$ ...
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52 views

Are all function transforms special cases of Gelfand's transform?

Reading about Gelfand-Naimark theorem I've seen that the Fourier transform is a special case of Gelfand transform for the space $L^1(\mathbb{R})$ with the convolution product. In a related question on ...
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188 views

Calculating Inverse Laplace Transform of stretched exponential

I am trying to solve a Laplace transform problem that has gotten way over my head in terms of complex analysis knowledge. I would like to solve the Inverse Laplace Transform $(s\rightarrow t)$ of ...
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Find the inverse laplace transform of $F(s) = \frac{a(s^{2}-2a^{2})}{s^{4}+4a^{4}}$

Find the inverse laplace transform of $F(s) = \frac{a(s^{2}-2a^{2})}{s^{4}+4a^{4}}$ I factored denominator by completing square but how do i proceed next . Thanks
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Find the inverse laplace transform of $F(s) = \frac{2s}{s^{4}+s^{2}+1}$

I need to find the inverse Laplace transform of $F(s) = \frac{2s}{s^{4}+s^{2}+1}.$ I realise that I need to do something with the denominator so I can convert to partial fractions, but I am not ...
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1answer
212 views

Calculate the convolution of the product of two simple functions. (5.6-12)

Synopsis: I cannot duplicate the answer given in a very reputable online symbolic integral calculator as shown in this link ($x$ is $\tau$) although my answer does appear very similar. This tells me ...
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1answer
14 views

How to determine the bounds of integration for an inverse Laplace transform?

I don't completely understand how to find the original time signal when I'm given a Laplace transform and its region of convergence. For instance, if I'm given the Laplace transform: $$ X(s) = ...
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1answer
71 views

Calculate the inverse Laplace transform by convolution. (5.6-26)

Synopsis: I cannot duplicate the answer in my text although I do get somewhat close. This tells me that my method is correct but I am making another kind of error -- perhaps in my integration? The ...
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1answer
21 views

Calculate the inverse Laplace transform by convolution. (5.6-25)

Synopsis: I cannot duplicate the answer in my text although I do get somewhat close. This tells me that my method is correct but I am making another kind of error -- perhaps in my integration? The ...
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1answer
87 views

Calculate the convolution of the product of a unit step function and t. (5.6-14)

Request Please check my work. I am not certain how to calculate the convolution of the unit step function. Given: Find the convolution of $f(t)=t$ and $g(t)=u(t-1)$. $$h(t)=(f*g)(t)=\int_0^t ...
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25 views

Calculate the convolution of the product of two sine functions. (5.6-13)

Request: I cannot duplicate the answer in the book although I do get very close. This tells me that my method is correct but I am making another kind of error -- perhaps in my integration? Given: ...
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1answer
39 views

Calculate the convolution of the product of two identical sine functions. (5.6-7)

Request I am very new to this so please bear with me. I cannot duplicate the answer in the book although I do get very close. This tells me that my method is correct but I am making another kind of ...
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1answer
71 views

What is the Laplace transform transfer function of affine expression $\dot x = bu + c$?

For the one dimensional case, with $a, b, c$ being real constants, $u$ being the system input, $x$ the state, what is the Laplace transfer function of: $$\dot x = bu + c$$ Ideally I'm looking for ...
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22 views

Is there a shortcut for performing this integration (Laplace definition)

My assignment for a linear systems course is to "Use the definition of the unilateral Laplace transform and an integral table to verify the following Laplace transforms, and specify their regions of ...
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61 views

Inverse Laplace Transform of $\frac{s}{(s-a)^{3/2}}$

Find the inverse laplace of: $\frac{s}{(s-a)^{3/2}}$ I tried working through this using partial fractions and convolution but I can't seem to get a requitible answer. How would I go about solving ...
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101 views

Difference between transform and transformation.

I was told that there is a difference between a transform and a transformation. Can anyone point out clearly. For example : Is Laplace Transform not a transformation ?
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Find initial value, given the Laplace transform

Find the initial value , \begin{equation} F(s)=\frac{s+1}{(s+1)^2+10} \end{equation} this is the question, I can implement the inverse laplace method but I do not know what to do at the last stage ...
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38 views

How can I reduce $s^2 + 2\zeta\omega s + \omega^2$ into something like $(s+a)^2 + (\omega + b)^2$

I'm trying to solve an inverse laplace transform and I need to get this $s^2 + 2\zeta\omega s + \omega^2$ into something more workable. Can anyone help?
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1answer
26 views

Finding Laplace Transform of $(t-4)h(t-4)$

Find the Laplace Transform of $(t-4)h(t-4)$, where $$h(t) = \begin{cases} 1, & t > 0 \\ 0, & t < 0\text{.} \end{cases}$$ I am thinking about using the shifting theorem here. But coud ...
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Is definite integral of such function multiplication analytic?

If $f(x)$ is a general function (integrable) and $g(s,x)$ is an analytic function except for on its poles. Then, can some one judge about $$H(s)=\int_{a}^b f(x) g(s,x) dx $$ Is $H(s)$ analytic ...
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25 views

Need guidance with inverse laplace transform

I have the following formula, which I need to get the inverse laplace transform of: $\frac{2s}{s^2 + 6s + 13}$ I've managed to get $2\mathrm{e}^{-3t}\cos(2t)$, that's rather simple - but according ...
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Strange propery(possibly fake) of Laplace Transform

I was walking around by the CS hang out places at my school and saw the following equation on a white board. $\mathscr {L} (f) = f+f'+f"+...f^{n} =e^{-x} \int e^x f dx$ Where $n $ means $n $th ...
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1answer
52 views

Calculate the convolution of two constants. (5.6-1)

Request I am very new to this so please bear with me. I cannot duplicate the answer in the book. I believe I may be making a methodical error. Please correct it for me. Given: Find the convolution ...
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1answer
24 views

Derive Inverse Laplace transform by differentiating F(s) and Integrating f(t) (5.5-21)

Request: Please check my work. I cannot duplicate the answer in text although it is very close. I believe the problem lies in how I take the derivative. Is there a better way to calculate the inverse ...
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19 views

Calculate Laplace transform of the product of t and f(t) by differentiating f(t) (5.5-8)

Request: Please check my work. State where errors, if any, occurred and how to correct them. Is there a better way to calculate the transform other than the present method given? Given: Find the ...
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1answer
17 views

Calculate Laplace transform of the product of t and f(t) by differentiating f(t) (5.5-6)

Request: Please check my work. State where errors, if any, occurred and how to correct them. Is there a better way to calculate the transform other than the present method given? Given: Find the ...
2
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1answer
19 views

Calculate Laplace transform of the product of t and f(t) by differenitating f(t) (5.5-4)

Request: Please check my work. State where errors, if any, occurred and how to correct them. Is there a better way to calculate the transform other than the present method given? Given: Find the ...
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1answer
63 views

Inverse Laplace transform of $\exp(-1/\sqrt{s})$

I'm looking for the inverse Laplace transform of: $$F(s) = \exp(-1/\sqrt{s}).$$ Does the inverse Laplace transform exist? Do you have a reference in which this transform is given?
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Translated Laplace transform

Is there any way to rewrite the Laplace transform is such a way that that one can apply to an IVP not centred at zero, that is, at some $y^{(n)}(a_n) = b_n$ for $n\in\mathbb{N}$ and $a_n ...
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Using Laplace transform to solve the ODE $DP/dt = P + k(H(t-T)-1)$

A population of fish that's growing is harvest at a rate of $k$, from $t = 0$ to $t = T$ and follows the DE: $$DP/dt = P + k(H(t-T)-1), \qquad P>0$$ How do I solve the IVP with $P(0) = P_0> ...
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84 views

Evaluate $\int_{0}^{\infty} \cos(x^2)dx $ [duplicate]

Prove that the above integral is equal to $\frac{\sqrt{2\pi}}{2}$ I have already tried expanding using $\cos$ identity and also taking Laplace for it. I am getting nowhere with this.
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43 views

Fourier Transform - Laplace Transform - Which variable transform?

I need to know when do I have to transform $x$ and when $y$ in a PDE in Fourier Transform and Laplace Transform. In an exercise of Fourier Transform, I have to solve a Laplace Equation, where ...
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Inverse Laplace transform of exponentials and Incomplete gamma functions

I came to this final problem to be solved. I would like to understand a way to tackle this problem: Inverse Laplace transform of $$A(s)=\frac{1}{s}\exp{(s^{\beta}z)}\Gamma(0,s^{\beta}z)$$ ...
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49 views

Solve a very simple first order ODE in the form of an integral using Laplace Transforms. (5.4-5)

I cannot duplicate the answer in the text book. Please examine my work to find the error. Given: Solve the following differential equation for $i(t)$ by Laplace Transforms. One initial condition ...
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70 views

Solve a very simple second order ODE using Laplace Transforms. (5.4-9)

My question is where did I go wrong? I cannot seem to duplicate the answer given in text. The Problem: Solve the following differential equation using Laplace Transforms given that $y(0)=0$, ...
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1answer
52 views

How does the laplace transform diagonalize the derivative operator?

I was reading this post here and I got really confused at the part where the claim is that the laplace transform diagonalize the derivative operator ...
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23 views

Laplace–Stieltjes transform and renewal equation.

I'm wondering if somebody could check if I'm going about this the correct way. I have renewal equation, $$Z = z + H*Z$$ where $$Z(t) = \mathbb{P}(L > t),$$ $$z(t) = (1 - G(t))(1 - F(t)),$$ and ...