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

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Can this definite integral of an inverse Laplace transform by simplified?

Can either of the below expressions involving an unknown analytic function $h(s,t)$ and the inverse Laplace transform $\mathcal{L}^{-1}$ be simplified? $$ \int\limits_{0}^1 \mathcal{L}^{-1} \left\{ ...
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Laplace diffrential equation

$$\frac{dx}{dt}=2x +3y$$ $$\frac{dy}{dt}=3x +2y$$ Find general solution. I know there is a solution through eigenvalues. But I want to solve it with Laplace transformation. I almost get the right ...
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The Laplace transform of $\frac{\ln(1+at)}{1+t}$

By expressing the square of the exponential integral as a double integral and then making a change of variables, one can show $$ \int_{0}^{\infty} e^{-2zt} \ \frac{\ln(1+2t)}{1+t} \, dt = \frac{e^{2z} ...
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Laplace Transforms

Solve the initial value problem for y(t) using Laplace Transforms. $$L\{y''+3y'\}=L\{f(t)\}$$ $$s^2Y-sy(0)-sy'(0)+3(sY-sy(0))=L\{t\}+L\{1\}-L\{u4(t)(t-4)\}-5L{u8(t)}$$ ...
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Laplace transform: $y"+6y'+25y=100\sin(10000t)$ [closed]

Can't figure out how to do the following problem: Find $y(t)$ of $y"+6y'+25y=100\sin(10000t)$ $y(0)=5$, $y'(0)=10$
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How to compare ZOH and tustin

I'm discretizing some continuous time systems. Now there (MATLAB) are of course different types of discrtization methods, among them tustin (bilinear), euler backwards, euler forward etc. Often one ...
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8 views

Unilateral Laplace transform calculation

I'd like to verify that $\mathcal{L}[e^{-at}]=\frac{1}{s+a}$, $t\ge 0$. So I calculate: $$\int_{0^+}^{+\infty} e^{-at} e^{-st} dt=\int_{0^+}^{+\infty} e^{-t(a+s)} dt = \frac{1}{-(a+s)} ...
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Inverse Laplace transform $\mathcal{L}^{-1}\left \{ \ln \left ( 1+\frac{w^{2}}{s^{2}}\right ) \right \}$

Where $s\in \mathbb{C}$. I assume that this would be pretty easily handled by solving it by definition, but I haven't taken courses in complex analysis yet. Also, I can't think of any nice property of ...
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49 views

transforming ordinary generating function into exponential generating function

I have seen a post here that says that you can convert an exponential generating function into an ordinary one with the aid of the Laplace transform. Is it possible to do the reverse transformation? ...
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The Laplace transform of $\exp(t^2)$

A naive attempt to calculate the Laplace transform of the function $f(t)=e^{t^2}$ results in integrals of the form $$\int_0^\infty e^{t^2-st}dt,$$ which obviously don't exist as the integrand grows ...
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Unilateral Laplace transform

I tried to do the same unilateral Laplace transform in two ways, but I got different results. I have to calculate: $\mathcal{L}[r(t-1)]$, where $r(t)$ is the ramp function, that is $r(t)=t, t\ge0$. ...
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22 views

Find the Laplace Transform

Could anyone enlighten me on how to find the Laplace Transform of $$\frac{1-\cos (t)}{t}$$
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39 views

Inverse Laplace Transform of $1/(s+1)$ without table

The pole is on the left half plane, so $\gamma =0$ $$\frac{1}{2i\pi}\int ^{i\infty}_{-i\infty}\frac {e^{st}}{s+1}ds$$ substituting $iu=s$ $$\frac{1}{2i\pi}\int ^{\infty}_{-\infty}\frac ...
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44 views

How do you find the inverse Laplace transform of $\frac{1}{\sqrt{s}(s-a)} $

When I use the convolution method, I can't avoid getting a divergent integral.
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Derivation of laplace tranform. [closed]

If $X$ is a random variable with probability density function $f$, how can i prove the following? The absolute value of the $n-$derivative of the Laplace transform $L(s)$ of $f$ is ...
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Laplace transform of f(x)

Let $\mathcal{L}$ be the Laplace transform and $$\mathcal{L}\{f(t)\} \cdot (s^3+s-1) = \frac 1 {s-1}$$ I am trying to find $f(t)$, it's complicated!
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Laplace of $\int_0^t \frac{sinx}{x}dx$

What is the Laplace transform of $\int_0^t \frac{\sin x}{x}dx$ I'm thinking about approaching it as a convolution but I am not sure how. Could I define it as the convolution of $1$ and $\frac{\sin ...
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1answer
32 views

System of differential equations using Laplace transform

Using Laplace transform, solve the system: $w'+y=\sin(x)$ $y'-z=e^x$ $z'+w+y=1$ where $w(0)=0$ and $z(0)=y(0)=1$.
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Integration Around Part of a Branch Cut

I am studying the integral, given by a Laplace transform, $$\int_0^\infty\!e^{-\alpha x}\sinh^{-2/3}x\left(1+\frac 12\sinh^2x\right)^{-1/6}\left(1-\beta\sinh^{4/3}x\right)^{1/2}\,\mathrm dx$$ From ...
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1answer
19 views

Relationship between Inverse Fourier and Inverse Laplace Transform?

Suppose we are given a fourier transform $$ F(\omega) = \frac{1}{\omega^2+4} $$ Can we use inverse laplace tranform by taking $i\omega = p$ to find the inverse fourier transform? I did this and got ...
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35 views

Laplace transform, Bochner integral

I have a quesition about linear operators on a Banach space. Let $B$ be a real Banach space. $(T_{t})_{t>0}$ is called strongly continuous contraction semigroup on $B$ if For all $t>0$, ...
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Inverse Laplace transform of s/s-1

Finding the inverse laplace transform: $$L^{-1}\left\{\frac{s}{s-1}\right\}$$ I wrote: $$L^{-1}\left\{\frac{s}{s-1}\right\}=L^{-1}\left\{\frac{1}{s-1}\right\} + L^{-1}\{1\}=L^{-1}\{1\} + e^{t}$$ And ...
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A proof which results in Gamma (or Erlang) distribution-From Karlin & Taylor's “A First Course in Stochastic Processes”

The random variables X and Y have the following properties: X is positive, i.e., $P\{X > 0\} = 1$, with continuous density function $f_X(x)$, and $Y\mid X$ has a uniform distribution on $\{0,X\}$. ...
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25 views

laplace step function $H(π-t)(\sin(t))^2$

How to calculate the laplace transformation of $H(π-t)(\sin(t))^2$ ? I know that I have to use $\sin^2(t)= 1/2(1-2\cos(2t))$ but i am stuck of how to proceed``
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Find the Laplace Transformation of $H(\pi-t)$.

I know how to find the Laplace Transformation of $H(t-\pi)$, but what about if the $t$ is negative. Any help is much appreciated.
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When does Fourier Transform be the same as Laplace's?

I have the TI nspire CX CAS... it can perform Laplace Transform but can't perform Fourier Transform. They are equal in some problems, but not all the time! So, when does both of them be equal so that ...
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27 views

Laplace transform (Simple factorization)

The question require me to find the inverse of Laplace transform. In the first line of solution, how does it go from LHS to RHS? Does it simply apply partial fractions?
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Invert laplace transforms

How do you find the invert the Laplace transform $\frac{6}{(s+2)^2+9}$? It seems I cannot use partial fraction. Do I need to expend the s+2 square?
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Solving initial value problem using Laplace transforms, one other method, and comparing results

So for my solution using characteristic equations I get (fixed a typo for first coefficient) $$\frac{11}{30} e^{-3t} - \frac{21}{20} e^{-2t} + \frac{21}{20} e^{2t} - \frac{11}{30} e^{3t}$$ For the ...
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Evaluating the inverse Laplace transform of $1/(s^2-\sum_{n=1}^\infty{n!s^{3-n}x^n})$

I want to evaluate at $t=1$ the inverse Laplace Transform $\mathcal{L}^{-1}\{F(s)\}\vert_{t=1}$ of $$ F(s) = \frac{1}{s^2-\sum\limits_{n=1}^\infty{n!s^{3-n}x^n}} $$ and find out the $x^n$ ...
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Are there functions that are not of exponential order for which you can define a Laplace transform?

I'am in a course of Introduction to Linear Differential Equations and teacher made us this question in class. we work in $\mathbb{R}$, and any help to answer this is welcome
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determine the time domain equation of the output responseusing inverse laplace transform, given a step input

I have this initial transfer function \begin{equation*} \frac{Y_s}{F_s}=\frac{1}{(1s^2+2s+2)} \end{equation*} unit step is $F_s=\frac{1}{s}$ so I then get \begin{equation*} \frac{1}{s(s^2+2s+2)} ...
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Find the roots of the corresponding characteristic equation

The equation is $${Y_s\over F_s}={1\over s^2+2s+2}$$ I have got to $$r^2+2r+2=0$$ what do i need to do next?
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Relation between Laplace and Fourier transform

I have a function that has the property $\tilde f(s) = \tilde{f}(abs(s))$. For this function, I need the inverse Fourier transform. I actually know the inverse Laplace transform of $\tilde f$ and I ...
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solving second order linear differential equation

Can somebody please show me how to solve the following differential equation: $$ a\ddot{x} + b\dot{x} = c $$ given these initial conditions $x(0) = 2$, $\dot{x}(0) = 0.5$ and $a = 4, b = 1.5$ First ...
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Laplace transform and “imaginary infinity”

I was recently studying Laplace transform for the first time, and I'd like to ask the following thing: there was an integral with limit of integration, something like that: a+j×infinity, j the ...
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Laplace transform involving the gamma function.

Does anyone know how to evaluate the following integral $$ \int_{0}^{\infty} \frac{e^{-qs}\alpha^{s}}{\Gamma(s)\Gamma(s)}\text{d}s $$ where $q,\,\alpha > 0$? I've done some digging in usual ...
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How can we make sure result of Laplace Transformation has no pole using lhopital's rule?

If there is $x(t) = rect(\frac{t}{2})$, then its L.T will be $X(s) = 1/s(e^s - e^{-s})$. right? and after that i tried to draw them on S-Plane to check if poles exist. In the L.T result, it looks like ...
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Solving IVP by Laplace transform

I'm trying to solve an IVP with non-constant coefficients $$ y'' + 3ty' - 6y = 1, \quad y(0) = 0, \; y'(0) = 0 $$ Taking the Laplace yields $$ s^2Y + 3(Y + sY') - 6Y = \frac{1}{s}$$ $$ Y' + ...
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1answer
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Inverse Laplace Transform with time delay and extra factor

I am attempting to solve a PDE $$y_{tt} = y_{xx}, -\infty < x < 0,\ t > 0$$ with boundary conditions $$ y_x(0,t) = k(t),\ y(x,t) \rightarrow 0\ \mbox{as}\ x \rightarrow -\infty,\ y(x,0) = 0,\ ...
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Find the solution for the spring-mass problem $y′′+9y=\cos(3t)$. Solve with initial conditions $y(0) = 0$, $y′ (0) = 0$. Using Laplace transform

I first took the Laplace transform of each part then getting $s^{2}Y+9Y=\frac{s}{s^{2}+9}$ then solving for Y, I got $Y=\frac{s}{(s^{2}+9)^{2}}$ but don't know how to simplify that to something that ...
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Help With Bromwich Inversion Formula Proof

To prove(copied from handwritten notes so possibly wrong): Bromwich Inversion Formula. Fix $x_0∈ℝ $. If $F$ is complex analytic on $\{z:\Re z > x_0\} $ and for every $x>x_0$, $y↦ F(x + iy )$ ...
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A difficult integral: Laplace transform of Gaussian*Erfi

$$\sqrt{\frac{\pi }{2}} e^{-\frac{t^2}{2}} \text{erfi}\left(\frac{t}{\sqrt{2}}\right) \rightarrow -\frac{1}{2} e^{\frac{s^2}{2}} \text{Ei}\left(-\frac{s^2}{2}\right)$$ or $$ ...
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y''+xy'+y=0, y(0)=1, y'(0)=-1

I have used laplace transform to get $Y'(s)-sY'(s)=-1+\frac{1}{s}$ $Y(s)=-e^\frac{s^2}{2}\int e^\frac{-s^2}{2}ds + e^\frac{s^2}{2}\int \frac{ e^\frac{-s^2}{2}}{s}ds +Ce^\frac{s^2}{2}$ what should ...
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How to prove that $\int_{a}^{+\infty}\int_{0}^{+\infty}e^{-xt}\sin t\,dx\,dt = \int_{0}^{+\infty}\frac{\cos a+x\sin a}{1+x^2}e^{-ax}\,dx$

I want to prove $\int_{a}^{+\infty}\int_{0}^{+\infty}e^{-xt}\sin t\,dx\,dt = \int_{0}^{+\infty}\frac{\cos a+x\sin a}{1+x^2}e^{-ax}\,dx$. Should the proof be done using some kinds of Laplace transform ...
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Fourier transform from Laplace transform

So what I did was Laplace transform $f(t)$ to $F(s)$ and then plug in $s=jw$. However, when I tried this for $cos(3t)$, $$F(jw)={jw\over9-w^2}$$ was the answer. I don't know if this is correct, and ...
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How to solve this Inverse Laplace Transform

How would I solve this Inverse Laplace transform? $$\mathscr{L}_s^{-1} \left\{ \frac{s}{s^2-s+\frac{17}{4}} \right\}$$ The solution is $$f(t) = (1/4 )e^{t/2} (\sin(2 t)+4 \cos(2 t))$$ I know I need ...
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Is the Laplace Transform of the convolution power the product of the Laplace Transformed convolution?

In statistics, the definition of $F^k$ is the k-fold convolution of $F$ with itself, where $F$ is some common distribution. I am wondering if the following holds, if: $$ L_{F^{k}(x)} = ...
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Solve pde using laplace?

I have to solve the following pde using Laplace transforms: $xw_x + w_t= xt$ i.c: w(x,0)= 0 Firstly, transforming the above wrt t, i get: $\bar{w_x} + s\bar{w}/x = 1/s^2$ But, in the textbook, the ...
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How to solve for the inverse Laplace Transform

How would one solve the following inverse Laplace transform? $$\mathscr{L}_s^{-1}\left\{\frac{2s}{\left(s-1\right)^2+7}\right\}$$ I know from WolframAlpha that the answer is: $$\frac{2 e^t ...