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

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Contour integral: different answers with different contours

Good day to everyone. I have a following contour integral problem. I have to find a solution for the integral $$\underset{\gamma_r }{\oint }\frac{e^{\lambda s} }{(1-s) s^{a-b} \left(s-\theta ...
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22 views

Laplace transform and Power series

As it can be read here, Discrete to Continuous Representations of Functions via Laplace Transforms? the Laplace transform is a continuous analog of a power series in which the discrete parameter n is ...
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58 views

Inverse Laplace Transformation of an exponential function

How one could find the inverse Laplace transformation of $\exp(-(b/(b+s))^k)$? Where both $b$ and $k$ are positive.
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2answers
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Prove that some function is the solution of some equation

Show that $$x(t)=\sum_{n=0}^{\infty}\frac{(-1)^n(t/2)^{2n}}{(n!)^2}$$ is the solution of $$x*x=\int_{0}^t x(u)x(t-u)du=\sin t$$ My approach: I suppose that I have to use the Laplace transform. I ...
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1answer
52 views

Laplace Transform of $\frac{a}{2\sqrt{\pi}}t^{-3/2}\exp(-a^2/(4t))$

I'm trying to prove that the Laplace Transform of $$\frac{a} {2\sqrt{\pi}}t^{-3/2}\exp(-a^2/(4t)) $$ is $$\exp(-a\sqrt{s}); $$ from the definition of Laplace Transform we should compute ...
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1answer
34 views

Laplace transform non exponential order.

There's an existance/uniqueness theorem which states: if function f is of exponential order, then the Laplace transform of f exsists and is unique. Since there's no theorem in my book which states ...
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59 views

Laplace transformation $y''+2y'+2y=3\sin x+\cos x$

Given$$y''+2y'+2y=3\sin x+\cos x$$ Transform to image region $$Y(s)(s^2+2s+2)=\frac{3}{s^2+1}+\frac{s}{s^2+1}-s-2$$ $$Y(s)((s^2+2s+1)+1)=\frac{3}{s^2+1}+\frac{s}{s^2+1}-s-2$$ ...
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48 views

Inverse Mellin transform of simple function

How should the following inverse Mellin transform integral be evaluated: $ f(x) = \displaystyle \frac{\alpha}{2\pi i}\int_{c-i\infty}^{c+i\infty}(\alpha ...
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44 views

Transfer Function non-zero initial conditions Laplace Transform

okay I know how to get transfer functions using the Laplace Transform assuming zero initial conditions but I would like to know how to deal with non-zero initial conditions. One of my maths books ...
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1answer
45 views

Explanation on how is simplified expression $\frac{s^2+3s+3}{2s^2+7s+7}$

This is done in the solution of exercise in order to make it possible to do inverse Laplace transform. Though I am not sure how is that done, so here it is: ...
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15 views

Confusion regarding bilinear transform.

I was reading my book where the z-transform of a signal is derived to be ${1-e^{-2bT}z^{-1}}$ . Then it goes on to say that by applying the bilinear transform we can get ...
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81 views

What is Wolfram Alpha returning when I ask it to calculate $\mathcal{L}[e^{x^2}](s)$?

As my instructor mentioned more times than could ever seem necessary, arguments of the Laplace transform have to have growth of exponential or less. I typed this "nonexample" into W|A and noticed ...
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1answer
38 views

Laplace Transform with sin and cos

Hi I am having trouble figuring out the solution of this Laplace transform: $$L_t{(u(t- \pi)(2\cos(t)-3\sin(3t))}$$ Where I am stuck if I am even on the right track is: $$L_t{(u(t- ...
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1answer
62 views

Laplace Transform, Unit Step Function

Hi I have been trying to do this Laplace Transform and cant seem to figure it out and was wondering if someone could point me in the right direction; here it is: $$L_t{(u(t-2)(2t^2-6t+5)})$$ What I ...
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1answer
78 views

Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function. Note that I have the Laplace transform of : ...
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1answer
42 views

Laplace transform of a differential equation

Given the Laplace transform \begin{align} \mathcal{L}\{g(r)\} = f(t) = \int_{0}^{\infty} e^{-tr} g(r) \ dr \end{align} can it be shown that the transform of the differential equation \begin{align} ...
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39 views

Solve Laplace Integral (3 factors) [closed]

Please provide steps to solve this integral: $$3\int^t_0{\sin{u}(t-u)e^{-(t-u)}du}.$$
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60 views

Solving an ODE using Laplace Transforms

$$y′′′′ + 2y′′ + y = \sin x$$ $$y(0) = y′(0) = y′′(0) = y′′′(0)= 0$$ After solving I got $y(s)=\dfrac1{(s^2 + 1)^3}$ for which I am unable to find the inverse Laplace transform. Please let me know ...
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1answer
34 views

Find Inverse Laplace Transform

I'm trying to calculate $$L_s^{-1}\left({\cfrac{2s+12}{s^2+9}}\right)=2L_s^{-1}\left({\cfrac{s+6}{s^2+9}}\right)$$ But I do not know how to go from here. I have noticed that the bottom does look ...
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40 views

How to find the inverse Laplace transform of this function?

$F(s) = \dfrac{s}{s+3}$ I can't find this term in any Laplace lists around
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Heat Equation derivative in terms of Laplace

If the heat equation is $ \frac{\partial u}{\partial t} - \alpha \nabla^2 u=0$ Is the second derivative of u w.r.t t is the laplacian of the lapacian?
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Taking Laplace-Stieltjes transform to find virtual idle time in G/M/1 queue

I am reviewing some queueing problems from Gross and Harris, and had a question on problem 5.40 part b. The problem is stated as follows: Part B: Show that the stationary output of a $ G/M/1 $ queue ...
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23 views

analytical solution for linear 1st order PDE using laplace and seperation of variables

I am looking for the solution of the following pde: $\frac{\partial y(x,t)}{\partial t} = a* \frac{\partial y(z,t)}{\partial x} + b* y(x,t) + c$ and need help with the boundary and initial ...
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Inverse Laplace with $\ln$

How can I compute the inverse Laplace of 1) $\ln\left(\dfrac{s+1}{s-1}\right)$ 2) $\ln\left(\dfrac{s-1}{s}\right)$. Can someone please hep me to do these two problems
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29 views

Inverse Laplace transform of a rational function

Could you please help me the following Inverse Laplace problem $$\frac{2s^2+4s+3}{s(s^2+s+0.5)}$$ $F(t)$ is required.
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2answers
58 views

Laplace transform of $L({1-e^{-t}\over t})$

I have to find the Laplace transform of $$\mathcal{L}\left[\dfrac{1-e^{-t}}t\right],$$ then this is equivalent to $$\mathcal{L}\left[\dfrac{1}t\right]-\mathcal{L}\left[\dfrac{e^{-t}}t\right]$$ But ...
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34 views

Laplace transform of $f(t)/t$

For a function $f(t)$ Laplace transform is defined as $F(s)=\int_0^{\infty} f(t)e^{-st}dt$. I have to show the property that the Laplace transform of $f(t)\over t$ is $\int _s^\infty F(s')ds'$. ...
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22 views

How to compute transfer function from Laplace Transform

My system of interest has the following EOM (V is my input variable): $\ddot{x} = g - k_{1}V(t) + \dot{x}k_2$ Taking the Laplace with initial conditions of zero, I get: $s^2X(s) = \frac{g}{s} - ...
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30 views

Inverting Laplace transform

I am trying to solve the integral-differential equation: $$x'(t) + \int_0 ^{t} (t-s)x(s) ds = t + \frac{1}{2}t^2 + \frac{1}{24}t^4$$ With $x(0) = 1$ Taking the Laplace transform of this and using the ...
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What are origins of the Laplace variable $s$

When I learnt the Laplace Transform I was just told the very standard formula that: $F(s) = \int_{-\infty}^{\infty}f(t) e^{-st} dt$. From this we went on to the table of transforms at its properties ...
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inverse laplace transform of $$p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$$

I used the Residue theorem to solve this problem. But, I could not obtain the solution given by $$\mathscr{L}^{-1}\left( { p^{-3/2}e^{-\sqrt{pa}}\over{2\sqrt{2}}} [\cos(\sqrt{ap})+\sin(\sqrt{ap})] ...
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math Laplace transform transfer function

What is the transfer function of $A\sin(\theta) = B\theta''$ (2nd order). I mainly want to find the Laplace transform of $\sin(\theta)$, where $\theta$ is a function of $t$.
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188 views

Prove that the Laplace trasform is a Linear trasformation

Could you help me prove that the Laplace Trasform is a Linear trasformation? Thank you.
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54 views

Inverse Laplace Transform,

I have been stuck on this problem for quite a bit, have tried to look at similar answers on website but no help... The original questions is, Solve the IVP $\ y''+y=\sin(t);y(0)=1;y'(0)=0$ I ...
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Laplace transform of $\sin(x)$

I am confused with Laplace transform of $\sin(\theta)$. For example, what is the LT of $A \sin(x(t))=Bx''(t)$ ($x$ is second order), $A,B$ are constants.
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Laplace transform $y''''+37y''+36y=g(t)$

Hey this problem is making me insane so have at it and let me know what I keep screwing up. Express the solution of the initial value problem in terms of a convolution integral: ...
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38 views

Laplace Transform of $\sin(t-3)$

I wanted to compute the Laplace Transform of $\sin(x-3)$ using the shift rule: $\mathcal{L}(f(t-a)) = e^{-as}\mathcal{L}\left(f(t)\right) \Rightarrow \mathcal{L}(\sin(t-3)) = e^{-3s}\mathcal{L}(\sin ...
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83 views

Evaluate $\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$

Prove or disprove that$$\int_{0}^{\infty}\sqrt{\frac{\sqrt{(a^2-y^2)^2+4y^2}+a^2-y^2}{(a^2-y^2)^2+4y^2}}dy=\sqrt{2}\pi$$ for any $a>1$. I came across with this integral evaluating inverse ...
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Laplace Transform: Basis

I tend to think of the Fourier Transform (FT) as projecting a function onto a basis of cosines and sines. The Laplace Transform (LT) has a similar form to the FT, except it has been generalised. ...
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1answer
37 views

Inverse Laplace Transform of the following complicated form

What would be the inverse laplace transform of the following: I mean I want to solve this: $$ \large \mathcal {L^{-1} [ \mathcal {L}[{sin(at+b)}] . \mathcal{L} [{e^{xt}}] . e^{cs}}] = ? $$
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Laplacian transform of division by square root of t?

In this formula: $$f(t)=e^{-3t}t^{\frac{-1}2}$$ I saw examples on $t^n$ where $n>0$. But in above example $n<0$. I don't know how to deal with the $t^{\frac{-1}2}$. I know that ...
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29 views

Laplace transform of function's derivative

As stated in wikipedia: $$\mathcal{L}\{f(t)\}=\int_{0^-}^{\infty}e^{-st}f(t)dt$$ ...
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52 views

solving differential equations with function coefficients using Laplace Transform

Does there exits a method to solve an $n$-th order liner differential equation with "function coefficients" using Laplace transform. It is well known that the identity $$L\left\{ {{t^n}f\left( t ...
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42 views

Solving PDE using Laplace transform

Use the Laplace transform to solve $U_t=kU_{xx} $ in$ (0,l)$ with $U_{x}(0,t)=0$, $U_{x}(l,t)=0$ and $U(x,0)=1+cos({2 \pi x \over l})$ .The answer I get is $U_{x,t}=cos({2 \pi x \over l}) ...
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Rolling $n$ times with an $m$-sided dice. Closed, finite formula for the distribution of the sum? [duplicate]

My current idea is the following: practically we want to get the distribution of the sum of $n$-times of a discrete uniform distribution between $1,...,m$ . It is practically the discrete convolution ...
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43 views

Inverse Laplace transfer second order equation

How should I inverse Laplace transform this function? $$\frac{(\omega_n)^2}{s^2+bs+c}$$ I don't even know were to start and how should I transform it? A little help would be helpful here.
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37 views

Convolution in frequency domain

I have a time domain function$[f(t)=\cos(wt).e^{-t^{2}}]$ I want to find out the laplace transform of the above function. The convolution property says that a product in time domain can be obtained ...
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23 views

Unit step function value

I have understand question c)(i) But i do not understand question c)(ii)
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35 views

Don't know how to solve Unit Step Function

This is the solution that i found in the solution sheet but i can't seem to know how to get to the step that is in the red rectangle.
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47 views

Solving and graphing an IVP involing unit step function

Im trying to solve this ODE and find a simplified expression for $x(t)$. $$\ddot x+4x=-2\sum_{n=1}^{4} e^{in\pi}u(t-n\pi);\space x(0)=0=\dot x(0),i=\sqrt{-1}$$ First i found the the laplace ...