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

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Inverse Laplace Transform Table, Absolution of Form

Do I need to ensure I don't stray from the transform in the table? $\frac{-2}{s-1}$ this looks like $-2*\frac{a}{s^2-a^2},$ for $a=1$ Does this yield $-2\sinh(t)$, or should it fit perfectly to ...
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23 views

Laplace Transform assistance

Find the inverse laplace transform of: $\frac{25}{(s-1)^2(s^2+4)}$ $\frac{25}{(s-1)^2(s^2+4)}=\frac{A}{s-1}+\frac{B}{(s-1)^2}+\frac{C}{s^2 + 4}$ $$25=A(s^2+4)(s-1)+B(s^2+4)+C(s-1)^2$$ ...
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Convolution Theorem to solve integral involving bessel functions [on hold]

The question is, using the convolution theorem (From Laplace Transforms), evaluage G(t), where Jn is the bessel function of order n. If anyone can at least give me a guide as to what I need to do ...
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39 views

What is the Laplace Transform for the next equation?

I have some doubts in the correct way to solve this part of a mathematical model using the Laplace transform: $8 y'(t) + 3 y'(t) - 6 y(t)$ = $4x'(t - 2) + 5x (t - 2)$ This is my solution: $Y(s) [8 ...
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24 views

Taylor series expansion and Laplace transform final value theorem

I cant figure out how some transformations are made in one article on physics. Here is expression in s-domain and they want to find its asymptotic value. $$ \xi(s) = \nu_1(s+1)=\frac{1}{(s+1)} ...
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14 views

Laplace transformation using second shifting theorem

can anyone tell me how to evaluate the solution? I really get stuck.
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10 views

Inverting weird Laplace transform

Solving a PDE gave me this expression: $U(x,s) = 2/((s+1)^2)+1) e^{-\sqrt{s}x} + sin x/(s+1)$ I suppose there's a trick involved since I can't find it in my table. How do I invert this thing?
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help in Laplace and partial fractions

Can any one teach me how to solve C2.(a) and (b) step by step? C2. (a) Resolve $\frac{1}{s^2(s^2+s+1)}$ into partial fractions of the form $\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+s+1}$. Hence, ...
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17 views

Using partial fraction for inverse Laplace transform of $1/[s(s+5)^2]$

my question is the last part $1/5(s+5)^2$, how is it become $-5te^{-5t}$ I thought is should be -$1/5 te^{-5t}$
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How to find the inverse Laplace transform of $s/(s^2+s+1)$? [closed]

How to find the inverse Laplace transform $\displaystyle L^{-1} \left\{ \frac s{s^2+s+1}\right\} $ ? Can someone explain this question I don't really understand it.
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25 views

Fourier Transform, Laplace Transform, but what about…

I have a question regarding the fourier and laplace transform. First, the Fourier transform essentially takes a function, divides it by a frequency (imaginary exponential), and then sees how much of ...
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27 views

the jump in $\ddot y$, Laplace transform

Given the following IVP: $$\ddot y+4y=\cos t-\cos t \cdot \theta(t-2\pi), y(0)=0, \dot y(0)=1$$ Check that $y(t)$ is continuous at $t=2\pi$. Find the jump in $\ddot y(t)$ at $t=2\pi$ i.e find $\lim ...
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38 views

Laplace transform of 1/(1+t)

In a table and also on WolframAlpha, I stumbled upon this http://www.wolframalpha.com/input/?i=laplace+transform+1%2F%281%2Bt%29 So the Laplace transform of $1/(1+t)$ is apparently $-e^s Ei(-s)$. ...
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33 views

linear time-constant causal system

I have a linear time-constant causal system with the transfer function: And I have the insignal How do I get the output signal? I thought of Laplace transform the insignal and then get Y and ...
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54 views

Laplace transform of $f'(t)/t$

A question regarding the computation of $\mathcal{L}_s[f'(t)/t]$, where $f(t)$ is a differentiable function, was asked few hours ago. Unfortunately, this question was voluntarily deleted by the OP. I ...
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31 views

Fourier transform of a Laplace transform

Is there an easy way to find the Fourier transform of a Laplace transform of function? $$ F[L[f(t)]_{s}] $$ Where my $f(t)$ is $\sqrt{t}$. However, Before finding the Fourier transform I do the ...
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25 views

Deriving Laplace Transform of Laguerre polynomial

I'm given this definition for the Laguerre polynomials: $$L_n(t)=\frac{e^t}{n!}\frac{d^n}{dt^n}\left[t^ne^{-t}\right],~\text{for }n=0,1,2...$$ and I have to show that the Laplace transform is ...
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1answer
19 views

Laplace transform of a product of two functions

I have read questions and answers about this topic and i am still confused, using this formula we can calculate the Laplace transform of a product of two functions: $$ L[a_{(t)} b_{(t)}]={{1}\over{2 ...
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29 views

PDE using Laplace transform

! Can anyone please explain how to solve this question?
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'Deriving' the Laplace Transform from the $z$ Transform: Missing a $\Delta t$

Textbooks normally give the following 'derivation' (or justification, if you prefer) of the z-Transform from the Laplace Transform. Let $x(t)$ be a signal defined on $t\geq 0$, and write a discretized ...
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Convolution and Total Response Differential Equations

Convolution with differential equations is extremely confusing to me. The two following questions were asked in class and we were asked to think about them. I want to work them out but I don't know ...
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31 views

ODE with Laplace transform: the jump of $\dot y$

I solved this eq. using the Laplace Transform: $\ddot y+4\dot y+13 y=\delta(t-2\pi)-\delta(t-7\pi)$ The sol. is: $y(t)=\frac{1}{3} e^{2 t} (-e^{14 \pi} \theta(t-7\pi) sin(3 t)+e^{4 \pi} \theta(t-2 ...
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Differential Equations with Discontinuous Forcing Functions

$$ y''+y'+1.25y = g(t), \quad t > 0, $$ $$y(0) = 0, \quad y'(0) = 0 $$ $$g(t) = \left\{ \begin{array}{ll} \sin{t} & 0 \le t < \pi \\ 0 & t \ge \pi \end{array}\right.$$ ...
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Inverse Laplace transform (using table) when denominator cannot be factored

Usually when performing inverse Laplace transforms, I decompose the function into partial fractions, and then look up standard transforms in a table. For example: $$Y(s) = ...
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28 views

Finding an inverse laplace transform for $\displaystyle\frac{a}{\left(s^2 + a^2\right)^2}$

I am asked to show that $x'' + w^2x = f\sin(wt)$ has a solution given by $x = \frac{f}{2w^2}(\sin(wt) - wt\cos(wt))$ where $w$ and $f$ are constants, by means of Laplace transforms. By taking a ...
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56 views

Initial values are lost (diff eq to Transfer function)?

I read eternal Julius O. Smith III and he says that $$x_{n-m} = z^{-m}X(z)$$ Particularly, difference relation $$y_{n} = y_{n-1} + x_{n}$$ is solved by by $$Y = z^{-1}Y + X = {X \over ...
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47 views

Elliptical Coordinates PDE, wave equation and separation of variables

I need some help with this problem. I know how to use the method of separation of variables and that the constant lambda should give you trig functions with solutions at some interval of pi, which ...
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2answers
95 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
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35 views

Why does s = z+1?

What exactly is Laplace transform? motivated me to ask why unit function is 1/s by Laplace transform and 1/(1-z) by Z-transform? Both seem to be integrals of delta-pulse and secondary integration ...
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The Laplace transform of the delta function

$f(t) = 1$ must equal to delta function in the Laplace domain since "constant in one domain is delta in the other domain". On the other hand, table says that it must be $1/s$ in the Laplace domain. ...
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32 views

Laplace transform of noncentral chi-square distribution

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function. Let me know whether it ...
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26 views

Determine the Laplace transform using the heaviside fuction

Kindly assist with the question below $$ \mathcal{L}\{ \mbox{Sinh}(3t) \, H(t-3t) \} $$ I tried using the Heaviside property to determine the laplace transform, but i got stock on the way. i used the ...
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Solving Differential equations with Laplace transform

$\displaystyle y''+4y'+3y=e^{-t}$, given $\displaystyle y(0)=y'(0)=1$ My Attempt: Taking Laplace transforms on both sides $\displaystyle $ $\displaystyle [s^2\bar y-sy(0)-y'(0)]+4[s\bar ...
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Solving simultaneous equations using Laplace transforms

$\displaystyle \frac{dx}{dt}+y=\sin t$ $\displaystyle \frac{dy}{dt}+x=\cos t$, given $\displaystyle x(0)=2, y(0)=0$ My Attempt: Taking Laplace transforms on both sides $\displaystyle $ ...
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Laplace transform - Heaviside algebra

I'm strugling with some algebra around a laplace transform with heaviside. The start function is $L(2tH(1-t)) + L(2H(t-1))$ so from this, I'm supposed to convert it to $L(2t) + L(2(1-t)H(t-1))$ ...
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Find the inverse laplace transform of $\displaystyle \frac{s}{a^2s^2+b^2}$.

Find the inverse laplace transform of $\displaystyle \frac{s}{a^2s^2+b^2}$ My Thoughts: Take $\displaystyle s^*=\frac{s}{a}$ and $b^*=\frac{b}{a}$ and divide numerator and denominator by $a^2$. ...
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39 views

How to find the Direct Discrete Laplace Transform of ${2n \choose n}$

Some time ago I developed a discrete version of the Laplace transform for the purpose of calculating sums and solve finite difference equations with constant coefficients. The notes below are a ...
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Heaviside Expansion Theorem with matrices

Is the Heaviside Expansion Theorem (HE) for the determination of inverse Laplace Transforms true for matrix expressions such that $\mathscr{L}^{-1}[\mathbf{P}(s)\mathbf{Q}^{-1}(s)] = \sum_i^n ...
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Determine tha Laplace transform using Heaviside fucntion

I want to determine the Laplace transform of the following function: $$f(t):t \mapsto \begin{cases}0, \quad t< 0 \\t, \quad 0\leq t \leq2\\2,\quad t>2\end{cases}$$ I have done it using standard ...
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Confused by a Laplace transform of $f(t)=t^ne^{at}$

Having looked at my lecture notes I was confused by the following part of a derivation of a Laplace transform for the function $\;f(t)=t^ne^{at} ,\quad n\ge0,\; a \in \mathbb{C}, \; f(t)=0 \;\forall ...
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1answer
14 views

Inverse Laplace transform, none factorable denominator

I am really stumpted on this problem and can't seem to figure out where to go from where I am. Can anyone give me some advice or hint where I should do next? Here is the problem: ...
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30 views

Using Laplace transforms to solve a convolution of two functions

Hi I have this problem where I need to take the convolution of functions and I am not sure if I got the right answer or something close so any advice or help would be very appreciated. So here is the ...
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Laplace transform of Differential Equation with a piecewise function

Hi I have this question and I am horribly stuck at one part and I cant seem to figure out if i did something wrong so any advice or help would be greatly apprecaited. Here is the question: ...
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60 views

$\lim_{s\to 0^+}\int_0^\infty a(t) e^{-st} dt $

$$\int_0^\infty a(t) e^{-st} dt = f(s)$$ What is the meaning of the limit of this integral as $s\to 0^+.$
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Origin of Laplace Transform

Is the Laplace transform the continuous version of the infinite power series? $$ \sum_{n=0}^\infty a_nx^n$$ becomes $$\int_0^\infty f(t)e^{-st}dt$$ I learned this by watching this video lecture: ...
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Inverse laplace Transform of

How to calculate the inverse laplace transform of $\frac{\omega }{\left ( s^{2}+\omega ^{2} \right )( s^{2}+\omega ^{2} )} $ ?
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Inverse Laplace Transform partial fraction

While solving a second order differential equation, I have reached at a stage where I have to calculate the inverse laplace transform of $\frac{\omega ^{2}}{\left ( s^{2}+\omega ^{2} \right )( ...
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31 views

Dirac Delta Function, Initial Value Problem

Hi I finished this IVP but I cant seem to get the right answer can someone give me some advice as to where I went wrong and point me in the right direction as to how to fix it. Here is the problem and ...
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Inverse Laplace Transform of a complicated function

Help with finding the inverse Laplace Transform of $$F(s) = \frac{1}{s\sqrt{M^{2}-s^{2}}}e^{\frac{\sqrt{N^{2}-s^{2}}}{s}}$$
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Maximum output value

I have the following transfer function: $\frac{sX_{2,r}(s)}{X_{2,r}^0} = \frac{s\omega_n^2R}{s^2 + 2\zeta \omega_n s + \omega_n^2}$ If I am correct, the signal $\dot{x}_{2,r}(t)$ is limited to the ...