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

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How can I reduce $s^2 + 2\zeta\omega s + \omega^2$ into something like $(s+a)^2 + (\omega + b)^2$

I'm trying to solve an inverse laplace transform and I need to get this $s^2 + 2\zeta\omega s + \omega^2$ into something more workable. Can anyone help?
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1answer
24 views

Finding Laplace Transform of $(t-4)h(t-4)$

Find the Laplace Transform of $(t-4)h(t-4)$, where $$h(t) = \begin{cases} 1, & t > 0 \\ 0, & t < 0\text{.} \end{cases}$$ I am thinking about using the shifting theorem here. But coud ...
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18 views

Is definite integral of such function multiplication analytic?

If $f(x)$ is a general function (integrable) and $g(s,x)$ is an analytic function except for on its poles. Then, can some one judge about $$H(s)=\int_{a}^b f(x) g(s,x) dx $$ Is $H(s)$ analytic ...
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23 views

Need guidance with inverse laplace transform

I have the following formula, which I need to get the inverse laplace transform of: $\frac{2s}{s^2 + 6s + 13}$ I've managed to get $2\mathrm{e}^{-3t}\cos(2t)$, that's rather simple - but according ...
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25 views

Strange propery(possibly fake) of Laplace Transform

I was walking around by the CS hang out places at my school and saw the following equation on a white board. $\mathscr {L} (f) = f+f'+f"+...f^{n} =e^{-x} \int e^x f dx$ Where $n $ means $n $th ...
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39 views

Calculate the convolution of two constants. (5.6-1)

Request I am very new to this so please bear with me. I cannot duplicate the answer in the book. I believe I may be making a methodical error. Please correct it for me. Given: Find the convolution ...
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20 views

Derive Inverse Laplace transform by differentiating F(s) and Integrating f(t) (5.5-21)

Request: Please check my work. I cannot duplicate the answer in text although it is very close. I believe the problem lies in how I take the derivative. Is there a better way to calculate the inverse ...
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18 views

Calculate Laplace transform of the product of t and f(t) by differentiating f(t) (5.5-8)

Request: Please check my work. State where errors, if any, occurred and how to correct them. Is there a better way to calculate the transform other than the present method given? Given: Find the ...
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1answer
16 views

Calculate Laplace transform of the product of t and f(t) by differentiating f(t) (5.5-6)

Request: Please check my work. State where errors, if any, occurred and how to correct them. Is there a better way to calculate the transform other than the present method given? Given: Find the ...
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1answer
19 views

Calculate Laplace transform of the product of t and f(t) by differenitating f(t) (5.5-4)

Request: Please check my work. State where errors, if any, occurred and how to correct them. Is there a better way to calculate the transform other than the present method given? Given: Find the ...
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61 views

Inverse Laplace transform of $\exp(-1/\sqrt{s})$

I'm looking for the inverse Laplace transform of: $$F(s) = \exp(-1/\sqrt{s}).$$ Does the inverse Laplace transform exist? Do you have a reference in which this transform is given?
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22 views

Translated Laplace transform

Is there any way to rewrite the Laplace transform is such a way that that one can apply to an IVP not centred at zero, that is, at some $y^{(n)}(a_n) = b_n$ for $n\in\mathbb{N}$ and $a_n ...
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36 views

Laplace transform of solution to heat equation

in the book: Partial Differential Equations by Lawrence C. Evans (second edition) on page 203 the author uses the laplace transform to get an solution to the resolvent equation, which I do not really ...
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23 views

Using Laplace transform to solve the ODE $DP/dt = P + k(H(t-T)-1)$

A population of fish that's growing is harvest at a rate of $k$, from $t = 0$ to $t = T$ and follows the DE: $$DP/dt = P + k(H(t-T)-1), \qquad P>0$$ How do I solve the IVP with $P(0) = P_0> ...
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1answer
80 views

Evaluate $\int_{0}^{\infty} \cos(x^2)dx $ [duplicate]

Prove that the above integral is equal to $\frac{\sqrt{2\pi}}{2}$ I have already tried expanding using $\cos$ identity and also taking Laplace for it. I am getting nowhere with this.
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26 views

Fourier Transform - Laplace Transform - Which variable transform?

I need to know when do I have to transform $x$ and when $y$ in a PDE in Fourier Transform and Laplace Transform. In an exercise of Fourier Transform, I have to solve a Laplace Equation, where ...
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188 views

Inverse Laplace transform of exponentials and Incomplete gamma functions

I came to this final problem to be solved. I would like to understand a way to tackle this problem: Inverse Laplace transform of $$A(s)=\frac{1}{s}\exp{(s^{\beta}z)}\Gamma(0,s^{\beta}z)$$ ...
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49 views

Solve a very simple first order ODE in the form of an integral using Laplace Transforms. (5.4-5)

I cannot duplicate the answer in the text book. Please examine my work to find the error. Given: Solve the following differential equation for $i(t)$ by Laplace Transforms. One initial condition ...
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1answer
60 views

Solve a very simple second order ODE using Laplace Transforms. (5.4-9)

My question is where did I go wrong? I cannot seem to duplicate the answer given in text. The Problem: Solve the following differential equation using Laplace Transforms given that $y(0)=0$, ...
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31 views

How does the laplace transform diagonalize the derivative operator?

I was reading this post here and I got really confused at the part where the claim is that the laplace transform diagonalize the derivative operator ...
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19 views

Laplace–Stieltjes transform and renewal equation.

I'm wondering if somebody could check if I'm going about this the correct way. I have renewal equation, $$Z = z + H*Z$$ where $$Z(t) = \mathbb{P}(L > t),$$ $$z(t) = (1 - G(t))(1 - F(t)),$$ and ...
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64 views

Solve for the charge on a discharging capacitor in an RC circuit using Laplace Transforms. (5.3-61)

Please check my work. I need to solve the following problem but my answer varies from that of the book by a factor of $C$ for capacitance. A print screen of the problem is given below. Problem to ...
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48 views

Find inverse Laplace Transform using s-shifting and t-shifting. (5.3-57)

Please check my work. I need to find the inverse Laplace Transform for the following function. $$\mathcal{L}\{f(t)\}=\frac{e^{-s}}{s^2+\pi^2}$$ My solution: Recognizing the exponential factor as a ...
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1answer
39 views

Find inverse Laplace Transform having s-shifting and t-shifting. (5.3-56)

Please check my work. Did I calculate the following inverse Laplace Transform correctly? Our Laplace Transform is... $$\mathcal{L}\{f(t)\}=\frac{e^{-\pi s}}{s^2+2s+2}$$ My solution: Recognizing the ...
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92 views

Residue Theorem for Laplace Transform

I need to know what's the Residue Theorem for a Laplace Transform. Does anyone know the name or something, so I can search it? I couldn't find anything. For example, if I have this two equations: ...
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25 views

Find Laplace Transform of a windowed ramp function using unit step function and t-shifting. (5.3-44)

Please check my work. Did I calculate the following Laplace Transform correctly? Our function is $f(t)=t$ where $(1<t<4)$ and $f(t)=0$ everywhere else. My solution: First express function with ...
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24 views

Find Laplace Transform of exponential function using unit step function and t-shifting. (5.3-42)

Please check my work. Did I calculate the following Laplace Transform correctly? $$f(t)=e^{kt}u(t-a)$$ My solution: Use the following corollary from the second shifting theorem (t-shift)... ...
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46 views

Find Laplace Transform of trigonometric function using unit step function and t-shifting. (5.3-40)

Please check my work. Did I calculate the following Laplace Transform correctly? $$f(t)=sin(t)u(t-\frac{\pi}{2})$$ My solution: Use the following corollary from the second shifting theorem ...
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20 views

Laplace Transform of (t - 120) u(t - 120)

I am puzzled by what should be a simple problem. I am deriving the Laplace transform for $$ (t-120) u(t-120) $$ where $u(t)$ is the unit step signal. Using the second shift theorem, ie $$ ...
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Regarding the unilateral Laplace transform of LTI systems

Consider an LTI system described by the following differential equation, $$ \sum_{k=0}^{N}a_k\frac{d}{dt^k}y(t) = \sum_{k=0}^{M}b_k\frac{d}{dt^k}x(t) $$ With initial conditions, $$ y(t)|_{t=0}, ...
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laplace transform multiplication by power t

How to solve Laplace transform of $\displaystyle t^\frac{5}{2} e^{4t}$ . I know that this can be solve by multiplication by power of $t$ but how to differentiate $\frac{5}{2}$ part please reply
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1answer
29 views

Inverse Laplace of $\frac {s}{RCs+1} $

I was wondering how you would be able to solve the inverse laplace of $$\mathcal{L}^{-1}\left\{\frac{s}{RCs+1}\right\}\left(s\right)\tag{1}$$ where $R$ and $C$ are constants?
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How to solve this simple inverse Laplace transform

I haven't touched Laplace transforms in quite some time so I'm very rusty. I'm trying to reverse: $$ \frac{-(s+5)}{5(s+1)} = \frac{-s}{5(s+1)} + \frac{1}{(s+1)}$$ Obviously, the second term is $ ...
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28 views

Is it feasible to think of laplace transform and z transform as projections?

For Fourier transform, it has been ingrained in my head that all we are doing is projecting a function onto its Fourier basis, namely $(1, cos(t), sin(t),...cos(nt), sin(nt) ...)$ Can anyone comment ...
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21 views

Expressing a piecewise defined functions as a unit step function

I am trying to express the following function as a unit step function so that I can use Laplace: $ f(x) = \left\{ \begin{array}{lr} 0 & : t < 1\\ t^2-4t+5 & : 1\leq t ...
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Find Laplace Transform using unit step function and t-shifting. (5.3-38)

Please check my work. Did I calculate the following Laplace Transform correctly? $$\mathcal{L}\{t^2u(t-1)\}=\mathcal{L}\{(t-1)^2u(t-1)\}+\mathcal{L}\{u(t-1)\}$$ ...
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32 views

Find Laplace Transform using unit step function and t-shifting. (5.3-35, 5.3-36)

How do the Laplace Transforms vary between the two following functions? What I am really asking is if I calculated the following Laplace Transform correctly... ...
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118 views

Find Laplace Transform using unit step function given graph of a periodic impulse function. (5.3-33)

Please correct my work. The textbook answer which is expressed exactly like this $1/s(1+e^{-s})$ does not match my own. Find the Laplace Transform for one period of the perpetual periodic function ...
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23 views

Laplace Transform To Solve IVP

I need to use the Convolution theorem to solve $y'' +4y' +4y = g(t)$ with initial conditions $y(0)=2, y'(0)=-3$. This is what I have but it differs from the answer in the text so I'm wondering where I ...
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1answer
20 views

Laplace transform of x(t)^2*x'(t)

I got a result that doesn't seem correct, so I'm hoping someone can tell me if I went wrong somewhere (probably with the integration by parts or in the second to last line). $$ ...
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22 views

An Inverse Laplace Transform Problem

I am having problems solving this inverse Laplace transform: ℒ$^{-1}\Large [\frac{s-3}{s[(s-3)^2+9]}]$ I did partial fraction decomposition, but ended up with complex expressions in some ...
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65 views

The Laplace transform of the Heaviside function

I am studying complex analysis but, because I'm an engineer, I have a lot of doubts. I'm going to present my doubts and it would be nice if someone helps me to see things clearly. Let's start with ...
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1answer
26 views

Solve for a hyperbolic Laplace Transform by expressing as exponents and shiftig on s-axis (5.3-21)

I cannot get past a certain point on this problem as shall be shown. I need guidance in order to complete the problem. The exercise as stated in the text: Represent the hyperbolic function in terms ...
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54 views

Inverse Laplace Transform of $e^{c \cdot s^2}$

I am trying to find the Inverse Laplace Transform of the function $$ F(s)=e^{c \cdot s^2} $$ where $c > 0$.
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37 views

How do you find the state space representation of $G(s) = \frac {1}{s^2+s+1}$

Let $G(s) = \frac {1}{s^2+s+1}$ be the transfer function of the system Then $Y(s)(s^2+s+1) = U(s)$ Therefore $y'' + y' + y = u$ After this step, how should I set up my state transition variable $x$ ...
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42 views

Redundancy in the Laplace transform and Mellin's inverse formula

As I understand it, Mellin's inverse formula relates a sufficiently 'nice' function $f$ and its Laplace transform $F$ as follows: $$f(t)=\frac1{2\pi i}\lim_{T\to\infty}\int_{-T}^{T}e^{i\omega ...
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57 views

Example: How to find inverse Laplace Transform by integral of the function (5.2-29)

This is just a demonstration on how to solve the following type of problem. Find $\mathcal{L}^{-1}\{\frac{54}{s^3(s-3)}\}$ by the given method: $$\mathcal{L}\{ ...
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22 views

Solution of a heat transport PDE

Solve the system of partial differential equations: $$(1)\space\space \frac{\partial g}{\partial t} + v\frac{\partial g}{\partial x} = -k_1\left(g-h\right)$$ $$(2)\space\space \frac{\partial ...
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30 views

Laplace transform notation

I'm confused about the notation used, pretty much everywhere, to describe what a Laplace transform it. Wikipedia says something along the lines of "..Laplace transform of a function $f(t)$..", ...
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32 views

inverse laplace transform of F(s)

Let $f(x)$ be some arbitrary function , $F(s)$ is laplace transform of it I think inverse laplace transform of $\frac {F(s)}{s+r}$ where, r is constant may be $\int_0^t e^{-rt'}F(t') dt'$ ...