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

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How to quickly compute the inverse laplace transform of $\dfrac{1}{(s^2+1)^2}$

I wish to find the inverse laplace transform of $\dfrac{1}{(s^2+1)^2}$ I tried using partial fraction but things do not seem to work out i.e. $\dfrac{1}{(s^2+1)^2} = \dfrac{A}{s^2+1} + ...
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Laplce transform solution to this system of Nonlinear ODEs

I want to solve this system of advective-diffusive-reactive equations analytically: $$\left(\alpha - k_0c_B\right)c_A+v\frac{dc_A}{dx}-D\frac{d^2c_A}{dx^2} = f_A $$ $$\left(\alpha - ...
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Inverse Laplace Transform of $(3s)/((s^2+2s+3)(s+1))$ [closed]

Use the partial-fraction expansion to calculate the time-response to a ramp input of $u(t)=t$ for a system with the following transfer function. $$G(s)=\frac{3s}{(s^2 + 3)(s + 1)}$$ enter image ...
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Laplace transform of $\frac{1}{\sqrt{t^2+1}}$

What is the Laplace transform of this function? $$\frac{1}{\sqrt{t^2+1}}$$
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Laplace Transform of uniformly convergent series

Let $\sum_{n=1}^{\infty} f_n(x)$ be a uniformly convergent series of functions each of which has a laplace transform defined for $s \geq \alpha$. Show that $f(x)=\sum_{n=1}^{\infty} f_n(x)$ has a ...
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44 views

Explain what the teacher did - system of ode, control theory.

There are a few things I'm not clear about in her solution and would appreciate a short explanation. We are given the system $\dot{x}=-ax+bu$. with an initial value $x(0)=x_0$. We want to find a ...
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44 views

Turn this integral into a Laplace transformation by Change of Variables

Question from Advanced Engineering Mathematics - Greenberg. Page 268, section 5.4 question 6. $C(T)$ = $\int_0^{\infty} e^{-0.0744v^2/T^2}p(v)dv$ is an approximate relation between frequency ...
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When does the Laplace Transform converge?

Consider the following function in the $s$-domain: $$F(s) = \frac{1}{(s^2 + 1)(2s-1)}$$ My book concludes that the ROC (Region of convergence) must be $\Re(s) > 1/2$ because a) The ROC can't ...
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39 views

Prove the laplace transform of $\sinh(at)$?

My problem today is as above. Here is what I have done: Use integral definition of laplace transform to get $$\int_0^\infty \sinh(at)\exp(-st)dt$$ $$= \lim_{b \to \infty}\int_0^b ...
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27 views

Find Laplace inverse

Let $${{{{x^{\ast}(s) = \left( \frac{1}{(s+\mu_1 + \mu_2) (s + \hat{\lambda}_2) (s + \lambda_1 +\lambda_2 )}\right)}}}}$$ be the laplace transform in question, where $\mu_1,\mu_2, \lambda_1,\lambda_2, ...
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Laplace Transform

Suppose that $F(s)=L[f(t)]$ and $G(s)=L[g(t)]$, where $L$ is the Laplace transformation $$F(s)=L[f(t)]=\int_0^{+\infty}e^{-st}f(t)dt.$$ I'm trying to prove that: $$\textrm{If}\ \ \lim_{t\to 0^+} ...
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130 views

Solving forced undamped vibration using Laplace transforms

I'm heaving trouble solving the following undamped forced vibration problem using Laplace transforms: $$\ddot{q}(t) + \omega_n^2 q(t) = \cos(\omega t).$$ I will show what I have done so far, and I'd ...
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86 views

Generalized Fresnel integral $\int_0^\infty \sin x^p \, {\rm d}x$

I am stuck at this question. Find a closed form (that may actually contain the Gamma function) of the integral $$\int_0^\infty \sin (x^p)\, {\rm d}x$$ I am interested in a Laplace approach, double ...
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48 views

Find the inverse laplace transform (step-function)

Find the inverse laplace transform for $$F(s) = \frac{e^{-2s}}{s^2+s-2}$$ I have narrowed it down to that it has something to do with step-functions but I can't see it, can I do something with the ...
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1answer
90 views

How do we solve the laplace transform of the Heat Kernel?

I am interested in the value of $$\int_0^\infty e^{-\alpha t}\frac{e^{-\frac{|x-y|^2}{2t}}}{\sqrt{2\pi t}}\, dt $$ this is the laplace transform of the Heat kernel (changing the time variable) This ...
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How to solve an integral with a fractional order.

How should I find a value of these integrals: $$ A:=\int_{0}^{\infty}\frac{\sin(x)}{(x+1)^{2-\nu}x^{\nu}}dx \quad\text{and}\quad B:=\int_{0}^{\infty}\frac{\sin(x)}{(x+1)^{1-\nu}x^{\nu}}dx, $$ where ...
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43 views

Why use complex numbers in Laplace transforms?

I'm starting to study System Dynamics and the textbook I have starts with a brief discussion of complex numbers, then goes on to explain the Laplace transform. The text very helpfully explained why ...
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42 views

closed form for the following integral which is similar to Laplace transform

I want to find a closed form for this integral: $\int\limits_{x=0}^{\infty} \exp(-\frac{1}{x})x^n\exp(-sx)dx$ I know that it has closed form for $n=0$ but what about $n\neq0$? Does anyone have any ...
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54 views

Inverse of the Watson's lemma

Watson's lemma basically says $$ f(t) \sim t^{\alpha} \,\,\,(\text{for small } t) \implies \int_0^{\infty} f(t) e^{-st} dt \sim \frac{\Gamma(\alpha + 1)}{s^{\alpha + 1}} \,\,\,(\text{for large } s). ...
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Integrable slowly varying function

We say a function $L$ is slowly varying if $$\lim_{t\to\infty} \frac{L(tx)}{L(t)} = 1$$ for every $x > 0$. Are there such $L$ that are integrable? Say $L$ is defined on $[0,\infty)$ and is ...
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Inverse Laplace transform with branch cut

For the purpose of my research on persistent random walks I need to compute the inverse Laplace transform of $$ F(s)=\frac{\mathrm e^{-b\sqrt{s^2-1}}}{s^2-1}.$$ I looked up in tables of integral ...
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Nonstandard analysis and integral transforms

Can integral transforms be evaluated without limits(i.e Laplace transform) such as in non standard analysis? Can the improper integral be bounded by a hyperreal number? I am not very familliar with ...
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How to find the inverse Laplace transform of $\frac{s}{(s+1)^2(s+2)}$? [closed]

I would need a little help in finding the inverse Laplace transform of the function: $$f(s)=\frac{s}{(s+1)^2(s+2)}.$$ Thanks in advance.
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Why won't this Laplace transform work?

Solving an IVP: $$y'' - 2y' +5y = 0 $$ $$ y(0) = 2, y'(0) = 4$$ Taking the Laplace transform of both sides $$\mathcal{L}{y''} - 2\mathcal{L}{y'} +5\mathcal{L}{y} = 0$$ $$[YS^2 - S(y0) - y'(0)] - ...
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Which method solves this integral equation? $\int_{-1}^{1}w(x)\,e^{tx}\,dx=6\left(\frac{\cosh(t)}{t^2}-\frac{\sinh(t)}{t^3}\right)$

Today I encountered this integral equation wrt. $w(x)$: $$\int_{-1}^1 w(x)\ e^{t x}dx = 6\left(\frac{\cosh(t)}{t^2}-\frac{\sinh(t)}{t^3}\right)$$ I never solved such equations, and when I tried to ...
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35 views

How to evaluate Integro-Differential Equation using Laplace convolution?

Can someone please explain how I begin to evaluate the following integro-differential equation? I know that it involves a convolution, but the $y(τ)$ within the integral is throwing me off. ...
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Help with a particular Ordinary Differential Equation

I did the following problem but I am coming up with the wrong answer. Problem: Use Laplace transforms to solve the following system. All unknowns are fuctions of $x$. \begin{eqnarray*} w'' + y + z ...
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building a laplace transform

F(s) is the laplace transform of the function (f(t)) Numerator is equal to 1 and the denominator is a 2nd degree polynomial with complex conjugate roots of my choice (both s and w are non zero in s ...
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Transfer Function, relation of numerator/denominator to output/input

simple transfer function: $ \frac{Y(z)}{U(z)} = \frac{B(z)}{A(z)} $ from my point of view the Y(z) is related to B(z) and U(z) is relater to A(z). The lecture notes says, it is the other way. the ...
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inverse laplace transform of a transfer function

So I'm working on this problem but the $$e^{-s}$$ term is throwing me off.. $$ G(s) = \frac{100(s+2)}{s(s^{2}+4)(s+1)}e^{-s} $$ Can someone help me out? I tried using partial fraction expansion to ...
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73 views

Cauchy's theorem and a contour integral

$$\frac1{2 \pi i} \int_{\gamma-i \infty}^{\gamma+i \infty} ds \, \frac{e^{-\sqrt{a s}}}{c s+s^{3/2}} \cos{\sqrt{b s}} \, e^{s t}$$ where $a, b$ and $c$ constant. To evaluate this, I used Cauchy's ...
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39 views

Finding inverse Laplace transform of a fraction of polynomials

I am trying to find the inverse Laplace transform of $$\frac{4s^3 + s}{s^2+1}$$ I tried polynomial long division and reduced it to the following expression: $$4s - \frac{3s}{s^2+1}$$ But I'm not ...
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Compute laplace transform of $\frac{\cos\sqrt t}{\sqrt t}$?

What is the Laplace transform of $\frac{\cos\sqrt t}{\sqrt t}$?
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Inverse Laplace Transform of $F(s) = \frac{3s+8}{(s^2+2s+20)^2}$

Having a little trouble solving this fractional inverse Laplace were the den. is a irreducible repeated factor $$F(s) = \frac{3s+8}{(s^2+2s+20)^2}$$ tried to apply partial fractions to it and i just ...
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1answer
61 views

Calculate the inverse Laplace transform of $\frac{1}{1-e^{-s}}$

During my signals and systems class i came across this and i have to find it's inverse Laplace transform. I don't know how. $$\mathcal{L}^{-1} \Big\{ \frac{1}{1-e^{-s}} \Big\} = \ ?$$ Any help ...
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Understanding the Bromwich Integral (Inverse Laplace Transform)

The formula for the Inverse Laplace Transform is (Bromwich Intergal): $$f_{(t)}=\frac{1}{2\pi i}\lim_{x\to\infty}\int_{\alpha-x i}^{\alpha+x i} \left(e^{st}\cdot F_{(s)}\right) \text{d}s$$ My ...
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Conditions of an inverse Laplace Transform

$$\mathcal{L}(c)=\int_{0}^{\infty} c\cdot e^{-st}\text{d}t=\frac{c}{s},s>0$$ $$\mathcal{L}^{-1}\left(\frac{c}{s}\right)=c$$ What are the conditions to the last inverse Laplace Transform?
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Conditions to have an inverse laplace transform?

If I had a function $\widehat f(s)$ how would I know if there exists a function $f(t)$ so that the laplace transform of $f$ is $\widehat f$? From looking at the formula for finding the laplace ...
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Transfer function from S-space to Z-space

simple question: I got: $$ G(s) = (\frac{s^2 + 1}{s - 1}) $$ I am supposed to get directly to Z-space with sampling period as parameter h. So far I have tried to divide the function into: $$ G(s) ...
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Solving system of 1st order ODE's using Laplace transforms - stuck on algebra

Here is the system: $$x'(t) + y'(t) + x(t) + y(t) = 1$$ $$y'(t) - x(t) + y(t) = -t$$ I simplified this to the following system of simultaneous equations, with ${\scr L}[x] = X$ and ${\scr L}[y] = Y$ ...
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How to calcute the inverse Laplace transform of $\hat{F}(z)=\sum_{i=0}^{\infty} \frac{A^{i}}{z^{i+1}}$

I am reading a paper which in part of that authors used to calculate the inverse Laplace transform in a way that I can not understand. actually suppose we have $ \hat{F}(z)=\sum_{i=0}^{\infty} ...
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Differentiation of Laplace Transform

It is known that The $s-$derivative rule states that $$ \mathcal{L} (t^{n} f) = (-1)^{n} F^{(n)} (s) $$ The proof for the laplace differentiation involves \begin{align*} F'(s) &= ...
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limitation of initial value theorem

I am student and stuck in this question , this question was asked to me on exam , what is the limitation of initial value theorem ,but i was not able to think of limitation Since the time was running ...
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ODE using Laplace transform

[ I got my Y(t) to be : $$12 \, e^{-4} \, e^{-2s} \, [\frac{1}{12(s+2)} + \frac{1}{4(s-2)} - \frac{1}{3(s-1)}] + \frac{1}{(s-2)} - \frac{1}{(s-1)}.$$ so i assume I need to use t shifting for the ...
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Choosing between SOV/Green's functions/Laplace transform for solving PDE - Guideline for choosing the most appropriate method?

Forgive me if this questions seems silly, but I have a question which is keeping me busy. I'm not really looking for a mathematical proof (but it is welcome), however I'm more looking for guided ...
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58 views

Still getting wrong answer after trying to solve $x''(t)+4x(t)=t^2$ where $x(0)=1$ and $x'(0)=2$

I am trying to solve this differential equation: $$x''(t)+4x(t)=t^2,x(0)=1,x'(0)=2$$ The answer should be: $$x(t)=\frac{1}{4}t^2-\frac{1}{8}+\frac{9}{8}\cos{2t}+\sin{2t}$$ Which is also verified ...
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50 views

Inverse Laplace transform of a product of integrals

I'm struggling in demonstrating that the relation in the first quoted equation. I've met this problem in finding the inverse Laplace Transform of an equation in the form $W=A\times B$, there $A$ and ...
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Solving general linear ODE $\sum_{k=0}^n y^{(k)}=0$

Is there a way to solve this general linear ODE: $$\sum_{k=0}^n y^{(k)}=0$$ For the first few $n$ here are the solutions: $$\begin{array}{c|c} n & y \\ \hline 0 & 0 \\ 1 & c_1 e^x \\ 2 ...
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1answer
36 views

Help with two functions - continuity, Laplace transform and Fourier series [closed]

I've been practicing for my exam lately, and there are two function that I've had a real trouble analyzing. 1.$f(x) = \sum_{n=1}^{\infty} \frac{\sin(nx)}{10^n \sin(x)}$, for $x \neq k\pi$ $f(x) = ...
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74 views

Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...