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

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Laplace transform of $\sin(\sqrt t)$

How can I use this differential equation $$4tf''(t) +2 f'(t) + a^2 f(t)=0$$ to show that $$L(\sin(\sqrt{t}))=\frac{1}{2}\sqrt{\pi}\,\frac{1}{s^{\frac{3}{2}}}\,e^{\frac{-1}{4s}}$$
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Difficult Inverse Laplace Transform

I've had this question in my exam, which most of my batch mates couldn't solve it.The question by the way is the Laplace Transform inverse of $$\frac{\ln s}{(s+1)^2}$$ A Hint was also given, which ...
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40 views

Stability and characteristic roots of difference equations

I often hear "For a process to be stable its characteristic roots or poles must be outside the unit circle (for casual process)". All right, consider the recurrence relation: $$y_t-5y_{t-1}+6y_{t-2}...
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23 views

Parseval like theorem for Laplace transform?

I was wondering if there any parseval like theorem for Laplce transform? Some statement such as $\int f(x)g(x) dx=\int F(s)G(s) ds$ where $F(s)=\mathcal{L}\{f(t)\}$ and $G(s)=\mathcal{L}\{g(t)\}$?
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31 views

Complicated exponential function inverse laplace transform

The problem is from wave equations for describing flexural vibration of Euler-bernoulli beam. The equations are listed in the following pics. I. equations and 16 elements to be analyzed II.cF:wave ...
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31 views

Inverse Laplace transform of exp(-as)/s via an infinite series

It is known that: $$\mathscr L^{-1}[F_1(s)\,+\,F_2(s)\,+\,F_3(s)\, ...] = \mathscr L^{-1}[F_1(s)]\,+\,\mathscr L^{-1}[F_2(s)]\,+\,\mathscr L^{-1}[F_3(s)]\,+\;...$$ where $\mathscr L^{-1}$ is the ...
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15 views

Laplace transform of this problem

I'm a bit confused in finding the laplace transform of the following: $f(t) = e^{2t} u(t) $ I know how to solve it if it we're only $e^{2t}$ but the $u(t)$ is confusing.
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What does this “proof” tell us about the Laplace transformation?

An exercise in the book stated "Find $2$ functions $f_1 \not =f_2$ such that $L\{f_1\}=L\{f_2\}$. Now I think I know the answer ($f_1=f_2$ except at any number of discrete points) but previously I was ...
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37 views

Why is the Laplace Transformation defined the way it is?

I can see the many uses of Laplace Transformation, but why is it defined the way it is? What is its physical significance?
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Conditional expectation of a hitting time of a Brownian motion and Laplace transform

I am trying to solve the following problem: Suppose B is a 1-dim Brownian motion, let $\mathcal{T}_a = inf\{t: B_t = a\}, \mathcal{T}_{a,b}=min\{\mathcal{T}_a,\mathcal{T}_b\}$. For $a < 0 < b$ ...
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23 views

Inverse Laplace of a piecewise function

$$ F(s)=\frac{e^{-3s}-e^{-6s}}{s^8} $$ I've tried convolution, partial fractions, and guess and check. I know $t^7$ is involved somehow. Help!
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25 views

How do I find the Laplace transform of the function $f(t)=3t$ if $t\le 6$ and $f(t)=18$ if $t>6$?

I found the step function: $u(6-t)\times 3(t-6)+18$ where $u(t)$ is the step function, but I do not know how to find the inverse laplace of this. Do I need to change the form of the step function?
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42 views

Show that $\int_{0}^{\infty} \int_{0}^{\infty} e^{-s(x+y)}f(x)g(y) \ dx \ dy = \int_{0}^{\infty} \int_{0}^{t} e^{-st}f(t-u)g(u) \ du \ dt$.

While learning how to compute the product of two Laplace transform and the inverse transform, I faced with this equality : $$\int_{0}^{\infty} \int_{0}^{\infty} e^{-s(x+y)}f(x)g(y) \ dx \ dy = \int_{0}...
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1answer
17 views

Finding the inverse Laplace of this function

Finding the inverse Laplace transform of $L^{-1}\left( \dfrac {1}{\left( x+1\right) ^{2}}\right)$ I understnd that the inverse laplace of $L^{-1}\left(\dfrac {1}{\left( x+1\right) }\right)$ Is ...
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29 views

What is the quickest method of finding the inverse laplace transform of $ 7e^{-6}/(s^2+6)^4 $

Solving for a differential equation, I found this which I need to obtain the inverse laplace of. What method should I use to solve this as quick as possible? Partial fractions and convolution seem to ...
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27 views

Solve integral equation using Laplace transform and convolution

Solve $$y(x)=e^x(1+\int_{0}^{x}e^{-t}y(t) dt)$$ Here's what I did: take derivative on both sides, $y'(x)=e^x+e^x \int_{0}^{x}e^{-t}y(t)dt +e^xe^{-x}y(x) \Rightarrow y'(x)=e^x+y(x)-e^x+y(x) \...
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On the relation between Laplace transform of a function and its derivative

Suppose that $s>s_0$. Through integration by parts we can write $$\int_{0}^{\infty}e^{-st} f'(t)~\mathrm dt= \Big[ e^{-st} f(t)\Big]_0^\infty + s \int_{0}^{\infty}e^{-st} f(t)~\mathrm dt. $$ Book ...
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38 views

Convolution of Gaussian and parabolic function.

What is convolution of $\exp(-x^2)$ i.e Gaussian function and $2x^2$? I don't have any idea,as I have found that Laplace transform of Gaussian function involves complementary error function,so inverse ...
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Laplace transform

Hello how can I solve this problem $ty''-y'=t^2$, $y (0)=0 $ I though about several ways like $y''-y'/t =t$ but I did not know how to solve the integral of $( y'/t) $. Then I thought about ...
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1answer
32 views

Inverse Laplace transform

Using partial fractions, find the inverse laplace transform of the function $f(s)$ = $\frac{s^2+2}{s^2(s^2+3)}$ I have tried to seperate the whole function using a partial fraction via $\frac{s^2+2}...
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51 views

Contradiction between Fourier and Laplace transforms?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function that has both Fourier and Laplace transforms. Also let $f(t)=0$ for all $t<0$. The Fourier transform of $f$ is $$\mathcal{F}(\omega)=\mathcal{F}...
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Inverse Laplace transform with an arbitrary parameter

I am trying to find this: \begin{equation} \mathcal{L}^{-1}(s^nF(s))=?, \end{equation} where the parameter $n$ can be an arbitrary value. I know when $n$ is a positive integer, it can be written as a ...
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24 views

Prove that L{t*y'(t)}(s) = Y(s) - s*Y'(s) and show that L{t*y(t)}(s) = -d/ds(Y'(s))

I believe that both these questions should be solved using integration by parts and definitions of the Laplace transform. Prove that$$\displaystyle{\mathcal{L} \{ t*y'(t) \}(s) = Y(s) - s*Y'(s)}$$ ...
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How do I find the laplace transform of a product? [closed]

How do I find the laplace transform of a product? Specifically $e^{5t}\cos{t}$?
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20 views

Differentiating Laplace transform of random variable

Let $Y$ be a random variable and $A$ an event, such that $g(q):= E(e^{-qY} ; A)$ exists for all $q \geq 0$. (Here $E(X ; A) := \int_{A} X \hspace{3pt} dP$ for a random variable $X$). I want to check ...
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30 views

Use Laplace Transformations to solve $y''+2y'+5y=3e^{-x}sin(x)$, with $y(0)=0$, $y'(0)=3$

I've gotten this far and I cannot proceed: $L[y]=\frac{L[3e^{-x}sin(x)]+3}{p^2+2p+5}= \frac{3}{((p+1)^2+1)(p^2+2p+5)}+\frac{3}{p^2+2p+5}$ I'm finding it impossible to find the inverse to solve for $...
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What am I doing wrong when I try to deduce the Laplace transform formula?

The Laplace transform of a function $f(t)$ is the projection of $f(t)$ vector (indexed with $t$) onto the linearly independent set of vectors $e^{st}$. The projection of a vector $\vec{v}$ onto ...
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Solve the PDE $\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=x$ using Laplace transform in $t$

Using Laplace transform in $t$, or otherwise, solve the equation for $u$: $$\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=x$$ in the region: $x$ > 0, $t$ > 0, subject to the boundary ...
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44 views

laplace transform of $t^nf(t)$

I have: $$\mathcal{L}(t^nf(t)) = \int_0^\infty t^nf(t)e^{-st}\ dt = \left(-\dfrac{d}{ds}\right)^n \int_0^\infty f(t) e^{-st}$$ I don't understand where the derivative came from
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Finding a function given as a part of a convolution integral

I am trying to solve the following equation for the function $f$. $$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 \alpha} \right)} = \int_0^t \frac{f\left(x, s\right)}{t - s}ds$$ where $\alpha$ and $\...
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94 views

(Laplace Method) $y'' - 4y' = 6e^{3t} - 3e^{-t}$

For this problem $y(0) = 1$ and $y'(0) = -1$ I need to solve this problem using this: \begin{align*} y(t) &\longrightarrow Y(s)\\ y'(t) &\longrightarrow sY(s) - y(0)\\ y''(t) &\...
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Which rules are used to make function like one in Laplace Transformations table?

I have function like this: $$\frac{s^2+3s+3}{(2s^2+7s+7)} $$ It needs to be brought to the level of Laplace Transformations from table, like these two: $$\frac{a}{(s-b)^2 + a^2} $$ $$\frac{s-b}{(s-b)...
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82 views

Inverse Laplace Transform With Dead Time

Here is the transfer function that needs to be transformed back into the time domain: $$Y(s)=\frac{K_{2}e^{-\theta s}}{s(\tau_{1} s + 1)(\tau_{2} s + 1)}$$ Then would the response be: $$y(t) = K_{2}(...
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69 views

Non-Linear Regression for Parameter Estimation

I have a second order system, it's response to a step change can be expressed in the s-space as: $$Y(s)=\frac{K_{2}e^{-\theta s}}{s(\tau_{1} s + 1)(\tau_{2} s + 1)}$$ Which can be inverse ...
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23 views

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 $(3/...
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40 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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1answer
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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: $$ s^...
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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 $$u*g=\int_{-\infty}^{\infty}g(\tau)u(t-\tau)d\tau=2\int_{t-2}^{t}(e^{-\...
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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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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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1answer
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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34 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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1answer
48 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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1answer
61 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 http://people.seas.harvard....
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30 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 $y^...
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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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28 views

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 = -\frac{d}{dp}\int^{\infty}_0y'(...