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

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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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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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53 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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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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38 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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The inverse laplace transform of $p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$ can be written in Fresnel integrals?

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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204 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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68 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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55 views

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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56 views

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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61 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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109 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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67 views

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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50 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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40 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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108 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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82 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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47 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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68 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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35 views

Unit step function value

I have understand question c)(i) But i do not understand question c)(ii)
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81 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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98 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 ...
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deriving second order transfer function from spring mass damper system..

I am having a hard time understanding how a differential equation based on a spring mass damper system $$ m\ddot{x} + b\dot{x} + kx = 0$$ can be described as an second order transfer function for an ...
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Converting to a partial fraction.

I'm trying to do an inverse Laplace operation on $I(s)$ shown below but I'm struggling on finding what $A$ & $C$ are on the partial fraction and how to do it. I calculated what $B$ equals by ...
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21 views

problem in $z$ transform

Im having a hard time trying to solve this... if $ (x_{k})_{k=0}^\infty $ is a causal succession such as $$ Z(x_{k}) = X(z) \parallel z \parallel > R $$ prove that $$ Z( {{x_{k+2}}}) ...
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130 views

Unit Step Function Finding slope

How do i find the using $y=mx+c$ for the slope above from $1$ to $3$
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53 views

Find the poles and residues in an awkward Laplace inversion

Assume that part c) has been proved and ignore parts c) & d). To invert the Laplace transform we would do $\displaystyle u(x,t)=\frac{1}{2\pi ...
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Contradiction in inverse Laplace transform problem with Mellin's inverse formula?

Let say we have to solve a given differential equation $$ty''+y'+ty=0$$ $$y(0)=1,\ y'(0)=0$$ (which is Bessel equation with the solution $y=J_0 (t)$, of course) with the Laplace transform. Then we ...
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Unit Step Function

Question: What is $\mathcal{L}\{u(t-1)u(t-2)\}$? My calculations $e^{-2}s \mathcal{L}\{u(t+2)-1\}$ $e^{-2}s \mathcal{L}\{t+1\}$ $e^{-2}s (s^{-2}+\frac{1}{2})$ I'm confused, I gotten the wrong ...
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Inverse Laplace Transform of Polynomial

I'm trying to find the inverse laplace transform of the following function: The resulting inverse laplace is in the form: Not sure where the derivatives came from, or what the inverse laplace of ...
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151 views

Convolution, indicator function

I need to calculate $(f*f)(x)$ of $f(x) = 1_{[0,1]}(x)$, which is the indicator function defined with Calculating the integral $(f*f)(x) = \int_{0,}^{x}1_{[0,1]}(t) \cdot1_{[0,1]}(x-t) dt$ gives ...
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Solution of a system of differential equations for a continuous time Markov chain.

The equations arise as the Laplace transforms of the forward equations of a continuous time Markov chain for a three-state system, with the following transition rates: Transition , rate $0 ...
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220 views

inverse Laplace transfor by using maple or matlab

I want to use inverse Laplace transform to F function by using maple or matlab. However I cannot get any answer. I know the answer from table but I want to use one of softwares. from table: ...
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Cant solve this differential equation using laplace transforms

$$\left\{\begin{array}{ccc} y''(t) &=& x(t) - 2y(t)\\ x''(t) &=& - 2x(t) + y(t) \end{array}\right.$$ $y(0)=1,\;x(0)=1,\; y'(0) = \sqrt3,\; x'(0) = -\sqrt3$. I am trying to solve ...
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113 views

Laplace transform of unit step function

Im given a graph of $f(t)$ and i need to find the Laplace transform of $f(t)$. From looking at the graph i have $$f(t) = \begin{cases} t, & \text{$0 \le t \le 1 $} \\ 0, & \text{$1 \lt t \lt ...
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How do I take inverse Laplace transform of $\frac{-2s+3}{s^2-2s+2}$?

How do I take inverse Laplace transform of $\frac{-2s+3}{s^2-2s+2}$? I have checked my transform table and there is not a suitable case for this expression.
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Complete y(t) solution for Laplace Transform? [duplicate]

Trying to figure out how to use Laplace Transform to find $y(t)$: The problem is $$y''+4y'+4y=f(t)$$ where $f(t) = \cos(\omega t)$ if $0 <= t < \pi$ and $f(t)=0$ if $t >= \pi$? Initial ...
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1answer
83 views

Laplace Trouble to find solution

Trying to figure out how to use Laplace Transform to find $y(t)$: The problem is $$y''+4y'+4y=f(t)$$ where $f(t) = \cos(\omega t)$ if $0 < t < \pi$ and $f(t)=0$ if $t > \pi$? Initial ...
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Laplace transform of initial value problem, stuck on partial fractions.

The problem im given is: Use Laplace transforms to solve the initial value problem. $$\ddot x +x=\sin(2t)$$ $$x(0)=0=\dot x(0)$$ I first do the following Laplace transforms: $$\mathcal{L}\{\ ...
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Boundary integral method to solve Poisson equation

Suggest how to solve Poisson equation \begin{equation} σ ∇^2 V = - I δ(x-x_s) δ(y-y_s) δ(z-z_s) \nonumber \end{equation} by using the boundary integration method to calculate the potential ...
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Inverse Laplace transform of a given function

1) The Laplace transform of f(t) is $\overline{f}(p)=\frac{1}{p}$ when $f(t)=1$ 2) The Laplace transform of $f(at)$ is $\frac{1}{a}\overline{f}(\frac{p}{a})$ 3) The Laplace transform of the ...
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45 views

Using Laplace Transform to solve a 3 by 3 system of differential equations

I have been trying to solve this system of equations using Laplace transforms for a while. It is very easy to solve it using eigenvalues and eigenvectors, but when I tried to do it using Laplace I ...
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43 views

Laplace transform of $t\,e^{-t}$

Can someone explain why the answer is $\frac{1}{(s+1)^2}$? My understanding was to multiply the Laplace transforms of $t$ and $e^{-t}$. So $\left(\frac{1}{s^2}\right)\left(\frac{1}{1+s}\right)$? ...
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Z - transform of a transfer function

I have to apply a z-transformation to my transfer function which looks like this: $$\frac{K}{s} - \frac{K\cdot T}{T\cdot s}+1$$ I have tried it and this is my result: $$K \cdot \frac{z}{z-1} - K ...
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Manually trying to calculate output of an transfer function.

I am trying to calculate the output of an transfer function due to the input of an step, But some weird reason, I am only getting the inverse output, what Matlab says it should be. My transfer ...
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35 views

Differential Question about Laplace/Delta/Convolution

I need help understanding a part of this question. Let $a.) y''+4y = \delta (x)$, $y(0)=y'(0)=0$. and $b.) y'' + 4y = f(x)$, $y(0)=y'(0)=0$ where $f(x)$ is some continuous function of finite ...
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Inverse LaPlace Transform of the square root of Rational, Monic 1st Degree Polynomials

I tried to find this in Churchill's Operational Mathematics which has a good variety of transform pairs, but no matches for what appears a simple expression. Does anyone have a solution for the ...