The Laplace transform is a widely used integral transform, similar to the Fourier transform.

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Laplace transformation of $u(t-1)e^{2-2t}$

I actually need to refer to the table of transforms but I cannot recognize anything. Instead I directly applied the definition of the Laplace transformation and got $$F(s)=\frac{e^{-s}}{s+2}$$ Is ...
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25 views

Finding the Inverse laplace transform .

Can anyone give me some hint how can I find the inverse Laplace Transform of : $$\textrm{log}\bigg\{\frac{s+1}{s-1}\bigg\}$$
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Closed form of certain integral.

I am solving the following problem in heat transfer using the Laplace transform $$\rho\,c{\frac {\partial }{\partial t}}T \left( x,t \right) =k{\frac { \partial ^{2}}{\partial {x}^{2}}}T \left( x,t ...
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Find the leading order asymptotic behaviour of the integral

$$I(x) = \int_0^{\infty}e^{-t-\frac{x}{t^2}}dt \mbox{ as } x \mbox{ tends to infinity} $$ I know this has a moveable maximum so you need to make a substitution which transforms it into the integral: ...
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Inverse Laplace Transform of e$^{-c \sqrt{s}}/(\sqrt{s}(a - s))$

I am trying to find the Inverse Laplace of the following function: $$ F(s) = \frac{\mathrm{e}^{-x b \sqrt{s}}}{ b (a - s)\sqrt{s}} $$ I really don't know where to start on this one as I have only ...
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Laplace transformation of 1/(s-1)^4

"Hi, I have a question about Laplace transformation.The question is:" $y ′′(t) − 2 y ′ (t) + y(t) = t e^ t , y(0) = y ′ (0) = 0$ "I know" $Y(s)(s^2-2s+1)=1/(s-1)^2$ $Y(s)=1/(s-1)^4$ "and I know we ...
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38 views

Weak convergence with the condition of Laplace transform

Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of probability measures on $\mathbb{N}$, such that the Laplace transform $\phi_n(\lambda)=\int e^{-\lambda x}\mu_n(dx)$ converges pointwise to a limit ...
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Laplace transform, Brownian semigroup

I have a question about the Laplace transform of a measure and a semigroup. Let $(\eta_{t})_{t>0}$ be a family of measures on $(]0,\infty[,\mathcal{B}(]0,\infty[))$. The Laplace transform ...
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35 views

We have the ode $y''−x^{−1}y'+x^{−2}y=0$, if $t=x^{−1}$, then how to get the new ode? [duplicate]

We have the ode $y''−x^{−1}y'+x^{−2}y=0$, if $t=x^{−1}$, then how to get the new ode? I just want to know the name of this method.
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Introductory treatment of N-/Multidimensional Laplace on Gaussian

A reasonable treatment of derivatives of two dimensional Laplace on Gaussian (LoG) is provided here: normalized Laplacian of Gaussian I am looking for an intermediate level (in terms of math) ...
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27 views

Existence of the laplace transform $sin(8t^{3})$

Apparently the laplace transform of $sin(8t^3)$ doesn't exist. No program lets me calculate it. I was asked if it existed in an exam and I said yes because it happens to meet all the criteria. Can ...
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1answer
24 views

Solve system of diff equations using laplace transform and evaluate x(1)

I keep getting the wrong answer, and wolphram seems to back me up. Here's the system of equations The answer I get for $x(1)$ is 10492.1... The supposedly right answer is -1426.16 Can anyone try ...
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39 views

Find the Laplace transform of this function $f(t)$

I'm pretty confused trying to find this Laplace transform: $$f(t)=\begin{cases}2, & 0\leq t < 1 \\ 2+t, & 1\leq t < 2 \\ 0, & 2\leq t < \infty \end{cases}$$ I know that I can ...
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59 views

Laplace Transform to solve differential equation (with IVP given at a point different from $0$)

how could I use Laplace Transform to solve the following differential equation: $$y''+2y'+y=0; \;\;\;\;\;y'(0)=2\;\;\;\;\;\mbox{and}\;\;\;\;\;\boldsymbol{y(1)=2}.$$ The solution may involve the ...
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25 views

Solving a system of equations using Laplace transforms

Problem: I want to use Laplace transforms to solve the following: $$\def\b{\begin{pmatrix}}\def\e{\end{pmatrix}}\def\d{\dot}$$ $$ \b\d y_1 \\ \d y_2\e = \b 2y_1 + y_2 \\ -y_1 \e$$ where $y_1(0)=0$ ...
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75 views

Does really Laplace transform convert cartesian coordinates into polar form?

Are we really trying to change a function $f(t)$ of cartesian (rectangular) coordinates into a function $F(s)$ by applying cartesian to polar conversion. $\mathcal{L} (f(t)$=$\int_0^\infty e^{-st} ...
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29 views

basics needed for studying Laplace Transform

Could anyone list out the basic concepts needed to study Laplace Transform or from where should I start.I was studying Z transform but I knew that Z transform is the finite version of Laplace ...
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Laplace transform - exponential rise

I have a hard time understanding how the exponential rise can be represented by $1/1+sT$ where T is the time constant. The table gives $1/s-a$ for the same case. Please help me understand this.
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56 views

Inverse Laplace Transform with cosh -fick's law diffusion

I'm trying to demonstrate highly quoted result of the Fick's second law of diffusion for a composite sphere of radius R2 (whose inner core, 0 ≤ r < R1, and outer shell, R1 ≤ r < R2, have ...
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32 views

Inverse Laplace transform of $\frac{1}{s} \frac{\sqrt{s}-1}{\sqrt{s}+1}$ [duplicate]

I have been desperately trying to find the inverse laplace transform using the complex inversion formula for this question. $\frac{1}{s} \frac{\sqrt{s}-1}{\sqrt{s}+1}$ I have found it extremely ...
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1answer
30 views

Find inverse Laplace transform of $H(s)=\frac8{s^4+4}$

How can we find the inverse Laplace transform of the function $$H(s)=\frac8{s^4+4}?$$
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27 views

ODE - Laplace transform

I have an ODE $\psi^{'}(s)_{3 \times 3}=(A+Bs)_{3 \times 3}\psi(s)_{3 \times 3} \tag1$ where A,B are constant skew symmetric matrices with zero determinant. $\psi(s)$ is a rotation matrix. It implies ...
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How to find inverse laplace transform

$$ F(s) = \dfrac{6s+9}{s^2-10s+29} $$ How do you solve the inverse Laplace transform of this above equation?
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31 views

How to solve Laplace initial value problem

$$ y''+36y = f(t) $$ $$ f(t) = \begin{cases} 1, & \text{0 ≤ t < 8} \\ 0, & \text{8 ≤ t < ∞} \end{cases} $$ $$ y(0) = 0 $$ $$ y'(0) = 1 $$
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inverse laplace transform by using complex integral

given function $$f(s)=\frac{1}{s}\frac{\sqrt{s}-1}{\sqrt{s}+1}$$ and $$\int_{0}^{\infty}{\frac{e^{-xt}}{\sqrt{x}(x+1)}dx=\pi e^t {erfc}(\sqrt{t})}$$ my steps: ...
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18 views

tail limit of Laplace transform of a bounded random variable

Suppose that $X$ is a variable such that $0<X<m$. I would like to know some information on the behavior of the function $$\phi(p)=\frac{1}p \log E e^{pX} $$ when $p\to\infty$. Here are some ...
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27 views

Inverse Laplace transformation of (s^2-4s-2)/((s^2+2)^2)

I approached this problem as follow: $1.$ rewrote $(s^2-4s-2)$ into $(s-2)^2-6$ $2.$ Now break the function into 2 parts: $\frac{(s-2)^2}{(s^2+2)^2} + \frac{6}{(s^2+2)^2}$ the Laplace inverse ...
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How injective is the Laplace transform?

Denote the Laplace transform by $\mathcal{L}$, and assume $\mathcal{L}[f]$ and $\mathcal{L}[g]$ exist for some functions $f$ and $g$. Then we know that $\mathcal{L}[f*g]=\mathcal{L}[f]\mathcal{L}[g]$. ...
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All possible inverse laplace transform of $ (s+1)^2/(s^2+2s+1)$

Well this was the question given in our previous week's test: Find all possible Inverse Laplace Tranforms of $(s+1)^2/(s^2+2s+1)$ . I just expanded the numerator and so the function reduced to 1.And ...
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Rewriting solution in terms of hyperbolic trigs

I have to find the inverse laplace transform of: $\mathcal{L}^{-1}(\frac{s}{-8+2s+s^2})$ I found it was $\frac{2}{3}e^{-4t}+\frac{1}{3}e^{2t}$ But the question I'm asked is, determine $A,B,C,D$ ...
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43 views

Can anyone tell me the name of this laplace transform?

I need to apply this rule to solve a Laplace transform: $\mathcal{L(\frac{f(t)}{t})}=\int_s^\infty F(u) du$ I've been given a table on laplace transform "rules" but I don't know how to use this one. ...
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final value theorem in the presence of white noise

I apply the final value theorem to get the steady-state error with the presence of white noise and I just keep getting zero. To me, it seems wrong to have zero steady-state error when there is noise ...
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47 views

Inverse Laplace of $\frac{\sinh{x\sqrt{s}}}{s^2\sinh{\sqrt{s}}}$

What is the inverse Laplace of $\frac{\sinh{x\sqrt{s}}}{s^2\sinh{\sqrt{s}}}$? Using the residues, I can calculate the residues at $s_n=2n\pi i$, but I have problem in calculating residue at $s=0$. ...
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Inverse laplace transform in a physics problem.

This came up during a physics problem, where we need to find the inverse laplace transform of $$X(s) = \left( 1+ \frac{k}{ms^{3/2}}\right)^{-1} \left( \frac{c_1}{s^2} + \frac{c_2}{s} \right)$$ to ...
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43 views

Calculating improper integral

Does anyone know how to solve the following integral: $$I =\int_{0}^\infty \cos(t \mathrm{log}( x))\,\mathrm{e}^{-ax}\, \mathrm{d}x,$$ where $t$ and $a$ are real. Please show some intermediate ...
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Find the impulse response to a RLC filter

I have a serial RLC-filter for which I should first determine the transfer function and then the impulse response. I figured out that the transfer function is: $H(s)=V(s)/U(s)$ And my circuit has the ...
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46 views

Integration and Laplace-Stieltjes of a multiplied Weibull and Exponential distribution Function

I have a trouble for integrating a multiplied weibull and exponential distribution. The expression is as follows: $$ Y(t) = \int_0^t e^{-\lambda T}e^{-(T/\mu)^z}dT\,. $$ Then, I need to take ...
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80 views

Laplace Transform and Convolution of Three Functions

So I'm trying to solve this differential equation: $y'' - 4y' + 3y = f(t)\:\:\:$ with initial conditions $y(0) = y'(0) = 0$ using laplace transform. After taking the Laplace transform and doing ...
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1answer
23 views

Laplace transform of $ \frac{1}{C}\int_0^1i(t)dt$

I have the integral: $$ \frac{1}{C}\int_0^1i(t)dt$$ which I should transform with Laplace. There is a rule saying that $$ \int_0^ti(t)dt$$ has the transform $$ s^{-1}F(s) $$ can I use this to ...
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Show $\mathcal{L}\left\{\frac{1}{t}f(t)\right\} = \int_{s}^{\infty}F(u)du$ [duplicate]

Show for $\mathcal{L}$, the Laplace transform, that $$\mathcal{L}\left\{\frac{1}{t}f(t)\right\} = \int_{s}^{\infty}F(u)du.$$ I know that $\mathcal{L}\left\{ t^n f(t) \right \} = (-1)^n ...
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How to solve this Laplace transform? $f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$

Find the laplace transform of $$f(t)=e^{-2t}\cos^2 3t - 3t^2 e^{3t}$$ The answer is $$\frac{1}{2(s+2)}+ \frac{1}{2} \frac{s+2}{s^2 + 4s + 40} - \frac{6}{(s-3)^3}.$$ This took me about an hour to ...
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Using calculus to find inverse functions

High schooler here. Last summer I taught myself a little bit of calculus, and I have been doodling about it. So I began writing some problems for myself, and one of them was this: Find the inverse ...
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Laplace transform the expression $\int_0^t(t-u)y(u)du$

I laplace transformed the expression $\int_0^t(t-u)y(u)du$ in Wolfram and it seems like the answer is just $\frac{Y(s)}{s^2}$. If I change the expression to this: $$ \int_0^t(t-u)y(u)du = t\int_0^t ...
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57 views

Evaluate the $I=\frac{1}{\pi}\int_0^{\infty}\frac{e^{-xt}\sin (a\sqrt{x})}{x}\,\mathrm dx$

I want to evaluate $$I=\frac{1}{\pi}\int_0^{\infty}\frac{e^{-xt}\sin (a\sqrt{x})}{x}\,\mathrm dx$$ It seems that the solution should be in the form of the error function and also it involves contour ...
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What will happen after Laplace Transform?

Consider the Laplace transform $\int_{0}^{\infty} e^{-px}f(x)\,dx$ Assume $f(x)=1$ , then the Laplace transform is $\frac {1}{p}$. Assume $f(x)=x$ , then the Laplace transform is $\frac {1}{p^2}$. ...
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31 views

Inverse of Mellin transform

I would like to invert the following Mellin transform $M(s)$ of a function $f(x)$ defined on $[0,a]$ with $a>0$ (or get the $x\rightarrow 0$ asymptotics): $$ M(s) = \frac{2a^s}{s-2(1-a^s)} $$ We ...
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70 views

asymptotics from Laplace transform

Suppose I know that a non-negative random variable with density $f$ has the following Laplace transform: $$\hat{f}(s)=\int_0^{\infty}e^{-st}f(t)dt=\frac{1}{\cosh(\sqrt{2s}x)}$$ where $s>0$ and ...
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59 views

The existence of the laplace transform

I don't understand why the laplace transform of some function, say f(t), has to be "piecewise continuous" and not "continuous". Is "piecewise continuous" sort of like the minimum requirement? This ...
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inverse Laplace transform by integral

I've seen this formula for the inverse Laplace transform in several books: $$f(t)=\mathcal{L}^{-1}\{F\}(t)=\frac{1}{2\pi i}\lim_{T\to\infty}\int_{\alpha -iT}^{\alpha +iT}e^{st}F(s)ds$$ where $f$ is ...