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

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Solve integral (convolution) equation

Given a function: $u(t) = \exp\left( -\frac{At^2}{1+t}\right),$ $A>0, t>0,$ and an equation: $\frac{d u(t)}{dt} = \int^{t}_0 \phi(t-\tau) u(\tau) d \tau .$ How to find a closed expression for ...
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1answer
59 views

Transfer function of differential eqaution

I'm trying to find out the transfer function of simple differential equation: $$a_0\dot y + a_1y=b_0x+b_1$$ The problem is i have no idea what to do with $b_1$. If we apply the Laplace transform ...
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0answers
26 views

Maximising a net present value function

I am looking at an equation for profit derived from fishing operations. This is defined in terms of a bounded integral (with an upper bound of $+ \infty$), so it's a Laplace transform really. It gives ...
2
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2answers
60 views

Using the Laplace Transform solve $y''+6y'+5y=e^t$

The initial conditions are $y(0)=0$ and $y'(0)=1$. I began the process and ended up with $Y=1/(s-1)(s^2+6s+4)$. Since the second factor in the denominator does not factor so I have a feeling I messed ...
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1answer
44 views

Laplace, Correct Use of the Second Shift Theorem

I have invested some time now trying to understand how to use the Second Shift Theorem, mostly by doing the full integration first. What threw me off at first, I discovered, is that almost all books ...
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1answer
38 views

Inverse Laplace tranform via the table formulas

In my inverse Laplace table there is this inversion "formula": $(1) \frac{1}{s-a} \rightarrow e^{at}$ I understand that $\mathcal{L}^{-1}[\frac{1}{s+4}] = \frac{1}{2}\sin(2t)$ But why can I not do ...
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1answer
27 views

Laplace Transform

Let us assume that complex-valued differential equations as follows $\dot{z}(t)=-Az(t)+Bz(t-\tau)$, $z\in \mathbb{C}$ How to find the solution of the above equation by using Laplace transform.
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3answers
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Discrepancy with the book's solution and mine of Laplace transform of a piecewise defined function

Determine the Laplace transform of $f(t)$ below: $$ f(t)= \begin{cases} 0, & \text{if } t < 2 \\ (t-2)^2, & \text{if } t \geqslant 2 \end{cases} $$ So my answer is $$ 2e^{-2s}/s^3 $$ ...
2
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1answer
84 views

Transfer function for double cart system

System: Define X2 = Y2; I've described the system with the following diff equation: $$f_{tot} = m_1\ddot{x_1} + k(x_2-x_1)+m_2\ddot{x_2}+B(\dot{x_2}-\dot{x_1})$$ where m1, m2, k and B are Cart ...
0
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1answer
41 views

Exponential Order: $\forall t>M$ or $\forall t>0$?

The following comes from the discussion of Laplace transformation in ODE. Let $f(t)$ be piecewise continuous on $[0, \infty)$ and of exponential order. Prove that there exist constants $K$ and ...
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53 views

Laplace Transform for a difficult function

The Laplace Transform I'm having trouble with is: $$f(t) = 6te^{-9t}\sin(6t)$$ I'm not sure what the protocol is for multiplying t into it. The Laplace Transform for $f(t) = 6e^{-9t}\sin(6t)$ is ...
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1answer
134 views

Partial fraction expansion for non-rational functions

With regard to this answer to an inverse Laplace transform question, I derived the following result: $$\frac1{i 2 \pi} \int_{c-i \infty}^{c+i \infty} ds \, e^{s t} \Gamma(s)^2 = 2 K_0 \left ( 2 ...
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0answers
64 views

Solve for inverse Laplace transform using non-repeating complex partial fractions. (5.7-4)

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 ...
2
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0answers
37 views

Probability proof of inversion formula for Lapalce transform

Let $f:[0, \infty[\longrightarrow \mathbb{R}$ be bounded and continuous and define $L(\lambda)=\int_0^\infty e^{-\lambda x}f(x)dx$. Let $X_n$ be a sequence of independent random variables with ...
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2answers
27 views

Finding the Laplace transform of the solution of the given IVP

Find the the Laplace transform $Y(s)$ of the solution of the given initial value problem $$y''+y=\begin{cases}t & 0 < t < 1 \\ 0 & 1 < t < \infty \end{cases}$$ $$y(0)=0$$ ...
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1answer
172 views

Evaluate Integral with $e^{ut}\ \Gamma (u)^{2}$

I am trying to integrate this integral: $$f(x)=\frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty}x^{-s}\sigma ^{ms-m}\left [ \frac{\Gamma \left ( \frac{s}{\beta} \right )}{\Gamma \left ( \frac{1}{\beta} ...
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2answers
22 views

laplace transformation of a function using definition

I want to find the laplace transformation of $x^ne^{ax}$ using the definition. I'm stuck with the integral. How shall I proceed the integral and find the final answer with $n$?
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1answer
91 views

What is the laplace transform and how is it performed? (detailed explanation)

I am a high school student and I became interested after someone mentioned it. Although I am not quite at the level where I am taught this it just captured my attention. Could someone give me an ...
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0answers
27 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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1answer
78 views

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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0answers
26 views

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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2answers
47 views

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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25 views

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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1answer
60 views

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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82 views

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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1answer
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 ...
1
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0answers
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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1answer
258 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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2answers
70 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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38 views

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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1answer
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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1answer
54 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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1answer
208 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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2answers
47 views

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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2answers
28 views

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
306 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 ...
2
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1answer
15 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) = ...
0
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1answer
77 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 ...
1
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1answer
26 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 ...
1
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1answer
89 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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1answer
27 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: ...
0
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1answer
45 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
73 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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1answer
23 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 ...
3
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2answers
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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1answer
119 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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42 views

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 ...
0
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1answer
39 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?
1
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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 ...