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

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Inverse Laplace by phase shifting

Can anyone help me out here? I have to find the inverse laplace of $$ \frac{s (1-e^{-s/2})}{s^2+\pi^2}$$ Sorry it looks bad, I just don't know how to format. Here's the wolframalpha link: ...
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74 views

Triple Laplace transform of a first order partial differential equation

How to find $L_x L_y L_t \{(\frac{\partial}{\partial x}- \frac{\partial}{\partial y}- \frac{\partial}{\partial t}) f(x,y,t)\}$? I have obtained ...
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1answer
40 views

Laplace Transform with translation theorem?

How do I solve for something like L{te^(t-5)U(t-5)} ?
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1answer
87 views

Laplace transform with 3 functions

I'm supposed to evaluate: $L\{t^{2}e^{7t}\sinh(3t)\}$. I know that this can be broken using $(-1)^n \frac{d^n}{ds^n}F(s)$ so I end up with $(-1)^2 \frac{d^2}{ds^2}L\{e^{7t}\sinh(3t)\}$. I'm not sure ...
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3k views

Finding the Laplace Transform of sin(t)/t

I'm in a Differential Equations class, and I'm having trouble solving a Laplace Transformation problem. This is the problem: Consider the function $$f(t) = \{\begin{align}&\frac{\sin(t)}{t} ...
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1answer
117 views

Functional form of a series of a product of Bessels

This question arises from my answer to an inverse Laplace transform question. The result I got took the form $$ f(t)= e^{-r_0 t/2} H(t-a) \left [ J_0\left(\frac{1}{2} a r_0\right) ...
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1answer
333 views

find the inverse Laplace transform of complex function

It would be appreciate if someone help me to obtain the inverse Laplace transformation of the complex function $F(s)=\frac{e^{-\frac lc\sqrt {s(s+r_0)}}}{\frac lc\sqrt {s(s+r_0)}}$ where $r_0,l,c$ are ...
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1answer
106 views

Inverse Laplace transformation of a complex function

Consider the complex function $\displaystyle f(s)=\frac{1}{\frac lc\sqrt{(s(s+r_0)}}$ where $r_0, l, c$ are positive real number and s is a complex variable. How I can obtain the inverse Laplace ...
2
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2answers
340 views

Tandem queue - response time distribution

In tandem queue with two queuing system, each server has exp(mu0) and exp(mu1) service time distribution and arrival rate is poisson(lambda). Scheduling policy is FCFS. What would be the response time ...
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2answers
287 views

Complicated Inverse Laplace Transform

I don't know how to show that the inverse Laplace transform of: $$k\cdot s^{-1/2} e^{-2\sqrt{s}}$$ is $$K\cdot t^{-1/2}e^{-1/t}$$ where $k$ and $K$ are constants. In the standard tables of the ...
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347 views

Laplace Transform of $tf(t)$

Q. prove that $\mathfrak{L}\{tf(x)\}=-\frac{d\mathfrak{L}\{f(x)\}}{ds}$ where the notation used is standard one. Attempt I tried what would seem obvious way to start: ...
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1answer
45 views

How to distinguish between the different frequency domains?

Sometimes the terms 'Fourier domain', 'complex frequency domain', 'Frequency domain' and 's domain' are used interchangeably. Take those answers here for example: ...
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1answer
83 views

Laplace Transform and Uniqueness of Solutions of ODE's

Without using Picard's theorem for existence and uniqueness of solutions of ordinary differential equations, if we solve a differential equation with the method of the Laplace transform, do we get ...
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2answers
127 views

Inverse Laplace Transform (with parameter?)

How does one find the Inverse Laplace transform of $$\frac{6s^2 + 4s + 9}{(s^2 - 12s + 52)(s^2 + 36)}$$ where $s > 6$?
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1answer
26 views

$L\{\cos{at}\}$: What happens to $i^2$?

Below I'm showing all the steps (which they don't in my book) to show you the whole process so you can let me know where I went wrong. In my book they go from Line 1 to line 5 and then line 7 (which ...
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1answer
100 views

Proving that $L\{e^{at}\} = \frac{1}{s-a} s>a$

The question states: if $f(t)=e^{at}$ find $L\{f(t)\}$. Solution $$\begin{align} L\{e^{at}\} &= \int^\infty_0e^{-st}\cdot e^{at}\\ &=\lim_{B\rightarrow\infty}\int^B_0e^{-t(s-a)}dt\\ ...
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1answer
237 views

Calculation of the Inverse Laplace Transform of $\frac{1}{p}$ by contour integration.

I am always told in my lessons of control engineering that the inverse Laplace Transform of $\frac{1}{p}$ is the Heaviside step function $\theta(t)$. But I have a problem when I calculate the inverse ...
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2answers
4k views

Compare Fourier and Laplace transform

I would like to clarify main difference between Fourier and Laplace transforms and also understand if exponential factor is main difference between this two method. So Fourier transform is ...
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2answers
46 views

Finding the Inverse Laplace Transform

I am having trouble finding the inverse Laplace transform of: $$\frac{1}{s^2-9s+20}$$ I tried writing it in a different way: ...
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2answers
1k views

Inverse Laplace Transform of zero

How to prove that the inverse Laplace Transform of zero is zero itself? $$\mathscr{L}^{-1}\{0\}=0$$ I know that the inverse Laplace Transform of a constant is Dirac's Delta. But I think that that ...
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1answer
337 views

Laplace Transform Convolution Theorem Applied to Functions without Transforms

My differentials prof taught us the convolution theorem and applied it to a differential equation $ay'' + by' + cy = e^{t^{2}}$ Then he transformed it and left the transform as $L(e^{t^{2}}) = ...
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361 views

Laplace transform of a periodic function

Knowing that $$L[f(t)]=\frac1{1-e^{-sp}}\int_0^{p} e^{-st}f(t)dt$$ $p$ indicates the period of the function If $f$ is a continuous function by segments in $[0,\infty)$ and $F(s)=L[f(t)]$ exists for ...
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1answer
5k views

Inverse Laplace Transform of s/(s+1)

What is the inverse laplace transform of $\frac{s}{s+1}$? My work was: $$ X(s)=\frac{s}{s+1}\\ X(s)=s\frac{1}{s+1}\\ x(t)=\frac{d}{dt}e^{-t}=-e^{-t} $$ My only issue is that when I check my answer ...
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2answers
154 views

Laplace of two functions multiplied together?

I'm trying to figure out how to do the Laplace transform of $\delta(t-\pi) \sin{t}$. I know the Laplace transform of each of these. But how would I find it when they are multiplied together?
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About evaluating $\mathcal{L}^{-1}_{s\to x}\left\{\dfrac{F(s)}{s}\right\}$ by considering contour integration with different entire functions $F(s)$

Detailedly compare the difficulties of different entire functions $F(s)$ where $F(0)\neq0$ when evaluating $\mathcal{L}^{-1}_{s\to x}\left\{\dfrac{F(s)}{s}\right\}$ by considering contour integration, ...
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29 views

The Square of the Laplace Transform

I have been looking at the Laplace transform $$\mathcal{L}f(s)=\int_0^{\infty}f(t)e^{-st}dt$$ and I'm trying to find The norm of $\mathcal{L}^2$ The nullspace of The norm of $\mathcal{L}^2$ So ...
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1answer
52 views

Help with partial fraction decomposition

So I was working on problem 16 in Elementary Differential Equations 9th edition by DiPrima and I get to the point where I'm using partial fraction's to separate : $\displaystyle {1 \over ...
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206 views

About the inverse laplace transform of sinc function

How to calculate $\mathcal{L}^{-1}_{s\to x}\{\text{sinc}(s)\}$ ? Note: $\text{sinc}(s)=\dfrac{\sin s}{s}$ when $s\neq0$ . Also note that $\lim\limits_{s\to\pm\infty}\dfrac{\sin s}{s}=0$ .
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1answer
89 views

Laplace inverse of $\frac{e^{-s}(3s^2-s+2)}{(s-1)(s^2+1)}$

I thought maybe you could fist solve $\frac{(3s^2-s+2)}{(s-1)(s^2+1)}$ using partial fractions and later solve the $e^{-s}$ separately as it is a $d(t-1)$ (Dirac delta function). As you solve the ...
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1answer
57 views

Please tell me if i did this inverse laplace correctly. Thanks

The question is : find the inverse laplace transformation of $$\frac{13s^2+3s+6}{(s-2)(s^2+9)}.$$ Please tell me if i did this correctly Here is my work: Using partial factions: \begin{align} Y(s) ...
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Laplace transform of $\cos(at)$

I need to find the Laplace transform of $\cos(at)$ I know that $L\{\cos(at)\}= \int_{0}^{\infty} e^{-st} \cos (at) dt$ but I am having trouble finding the integral Thank you
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1answer
147 views

Laplace Transform of a Brownian motion

If $v(\omega,t) : \Omega \times [0,\infty) \to \mathbb{R}$ is a Standard Brownian motion, then for what values of $s,\omega$ does the Laplace transform $l(\omega,s) = \int_0^\infty e^{-st} v(\omega,t) ...
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1answer
772 views

Finding the Laplace Transformation of $|t-1|$ (absolute value)

I am unsure how to find the Laplace Transformation to this. I tried breaking the limit from $0$-infinity up, but it did not help.
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Continuity of the inverse Laplace Transform

If I know $Y(s)$, can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions? For example; I'm solving an ODE with the Laplace ...
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52 views

How would I solve this Laplace transform for $Y(s)$ and $y(t)$

How would I solve this Laplace transformation for $Y(s)$ and $y(t)$? I'm lost and I don't know what to do. Take the Laplace transform of the following initial value problem and solve for ...
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40 views

Find $N$ independent solutions through Laplace Transform

The Laplace Transform method gives us one solution of an Ordinary Differential Equation. How can we use the same procedure to get $N$ independent solutions, being $N$ the order of the ODE? Where can ...
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1answer
222 views

Laplace inverse transform of the complex expression

I have an expression below which I need to do the laplace transform Any help is highly appreicated. The expression is : $$ ...
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3answers
222 views

Help in proof of theorem about Riemann-Liouville Fractional Calculus

Theorem: Let, $$\left[D^{nv}+a_{1}D^{\left(n-1\right)v}+\dots+a_{n}D^{0}\right]\left(y\right)=0$$ be a fractional differential equation of order $\left(n,q\right)$, where $v=\frac{n}{q}$, and let ...
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2answers
325 views

Find the inverse Laplace transform of $F(s)=\frac{e^{−6s}}{s^2+0s−16}$

Find the inverse Laplace transform of $$F(s)=\frac{e^{−6s}}{s^2+0s−16}$$ Here is my work: $$F(s)=\frac{e^{−6s}}{s^2+0s−16}$$ $$s^2+0s−16 = (s+4)(s-4)$$ $$\frac{1}{(s+4)(s-4)} = \frac{A}{s+4} + ...
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1answer
122 views

Find the inverse Laplace transform of $F(s)=\dfrac{5e^{−6s}}{s^2+4}$

Find the inverse Laplace transform of $F(s)=\dfrac{5e^{−6s}}{s^2+4}$ $f(t)=$ __________? Here is my work: $L{(5/2) \sin(2t)} = 5/(s^2 + 4)$, we have by the shifting theorem $f(t) = (5/2) \sin(2(t - ...
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1answer
112 views

Find the Laplace transform of $f(t) = \begin{cases} 0, & \text{if $t<5$} \\ t^2−10t+31, & \text{if $t\ge 5$} \\ \end{cases} $

Find the Laplace transform of $$f(t) = \begin{cases} 0, & \text{if $t<5$} \\ t^2−10t+31, & \text{if $t\ge 5$} \\ \end{cases} $$ $F(s)=$ __________? Here is my work. I went wrong ...
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1answer
60 views

Find the Laplace transform of $f(t)=4u_2(t)−2u_4(t)−5u_6(t)$ [closed]

Please Find the Laplace transform of $$f(t)=4u_2(t)−2u_4(t)−5u_6(t)$$ Appreciate the help. I don't understand this or where to start.
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247 views

Laplace transform of a product of functions

While trying to compute the Laplace transform of a certain product, part of the calculation leaves me with a Bromwich integral which has the form: ...
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1answer
804 views

Inverse Laplace transform using contour integration

I want to show by contour integration that $\displaystyle\mathcal{L}^{-1} \{\text{arccot}(s) \}(t)= \frac{\sin t\ }{t}$. In other words, I want to evaluate $\displaystyle \frac{1}{2 \pi i} \int_{a - ...
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1answer
981 views

Compute the inverse Laplace transform of $e^{-\sqrt{z}}$

I want to compute the inverse Laplace transform of a function $$ F(z) = e^{-\sqrt{z}}. $$ This problem seems very nontrivial to me. Here one can find the answer: the inverse Laplace transform of ...
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2answers
166 views

Taking Laplace transformation for $y′′−4y′−45y=\sin(3t)y(0)=5$, $y′(0)=3$

Consider the following initial value problem: $y′′−4y′−45y=\sin(3t)y(0)=5$, $y′(0)=3$. How can we find the equation obtained by taking the Laplace transformation in terms of $Y(s)$? Here is where i ...
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1answer
75 views

Finding the inverse laplace transfromation. differential equations.

Im trying to find the inverse laplace transformation of $\frac{(2s^2 + 9s + 5)}{((s^2 - 16s + 73)(s^2 + 25))}$. when $s \gt 8$. Here is my work: $$\begin{align} &\frac{(2s^2 + 9s + 5)}{((s^2 - ...
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1answer
333 views

Solving a recurrence relation using Z transform

I'm trying to solve the following recurrence using Z transforms: For $n\in \mathbb{N}^{*}$ $T(n)=1\ for\ n< 4$ $T(n)=T(\lfloor \frac{n}{4} \rfloor)+T(\lfloor \frac{3n}{4} \rfloor)+n\ for\ n\geq ...
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3answers
119 views

Inverse Laplace Transform of a polynomial fraction

How do I find the inverse Laplace transform of $\;\;\large\frac{4s}{(s^2+4)^2}\;\;$?
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1answer
50 views

Laplace Transform of Steps

So I'm trying to do the laplace transform of unit step functions. From the laplace table in my book it says : $ \mathcal{L}(u_c(t)f(t-c)) = e^{-cs}F(s) $ So my problem asks for the laplace transform ...