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

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Computing the Laplace transform of $\tan(pt)$

I've been thinking of using complex number approach , what's your view guys ?.
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Is it possible to say that $L(f^n)=s^nL(f)$ when the differential equation is not in the rest condition?

Question Use the Laplace transform to solve the following equation: $y'+2y=\cos(3t)$ ; where $y(0)=1$ In class our teacher wrote that "When in rest condition: $L(f^n)=s^nL(f)$", but I want to use ...
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Find the inverse Laplace transform of: $\frac{1}{(s^2+a^2)(s^2+b^2)}$

I'm having trouble doing this homework problem because I'm not sure how to deal with the $a$ and $b$. I did it the usual way we were taught - use partial fraction decomposition and then try to solve ...
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Solve the following initial value problem: $2y''+y'-y=e^{3t}$

$$ 2y''+y'-y=e^{3t}; \text{ with } y(0)=2,\ y'(0)=0 $$ I got to this point: $$ L(y)=\frac{1}{(s-3)^2}\cdot\frac{1}{(2s-1)(s+1)} $$ but now I'm not sure what to do with these polynomials. I know ...
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Inverse laplace transform with square completion

I need to find the inverse laplace of this : $$\frac{s+2}{s^2+2s+5}$$ I know that completing the square should help me to solve this so I get $$\frac{s+2}{(s+1)^2+4}$$ Then separating this equation ...
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About the causality of the signal whose frequency spectrum is not continuous as follows

Consider the signal in frequency domain: $$ \alpha(\omega) = \begin{cases} 1, & |\omega|<\omega_c \\ 0, & |\omega|\ge\omega_c \end{cases} $$ $$ =A(-j\omega)A(j\omega) $$ $$ =|A(j\omega)|^2 ...
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About Laplace transform, how to solve it [on hold]

I have a challenge from my prof, so please help me answer it, only one for exam. Laplace transformation: $$y" - 10y' + 9y = 5t , \quad y(0) = -1, \quad y'(0) = 2$$
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Laplace-Stieltjes :Functions of independent random variables

I am reading a book about stochastic modelling and I came across something and I couldn't really figure it out. First question would be are Probabilty Generating Functions (PGF) only for discrete ...
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Laplace Transform of Kelvin functions

What is the value of the Laplace transform, in terms of the G-function, \begin{align} \int_{0}^{\infty} e^{-st} \, t^{m} \, \left(ber_{\nu}^{2}(t) + bei_{\nu}^{2}(t)\right) \, dt \hspace{5mm} ? ...
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What does s=jω actually mean in terms of the complex plane and Laplace transforms?

I was trying to solve a problem on RC circuits. The current source was of the form $\cos (\omega t)$ which transforms in the manner of Laplace to $\frac{s}{s^2+\omega^2}$. I thought I’d use the ...
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Prove that $\mathcal{L}\left( \int_{0}^t f(u)du \right)=\frac{1}{s}\mathcal{L}(f)$

Prove that $$\mathcal{L}\left( \int_{0}^t f(u)du \right)=\frac{1}{s}\mathcal{L}(f)$$ I started out with the following identity: $$ \frac{1}{s}\mathcal{L}(f)=\frac{1}{s}\int_{0}^\infty e^{-st}f(t)dt ...
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Laplace Inverse

I want to find the laplace inverse of $$s^{-3/2}$$ the steps given in the solution manual are as follows: $$\frac{2}{\sqrt\pi}\frac{\sqrt\pi}{2s^{3/2}}=2\sqrt{\frac t\pi}$$ I know the first part ...
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27 views

Evaluate $L\{t(\sin^3 t-\cos^3 t)\}?$ [closed]

How to find $L\{t(\sin^3 t-\cos^3 t)\}?$ I am absolutely stuck. Please help me.
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Inverse Laplace Transform using Hetnarski's Algorithm

I'm trying to find the velocity component of an MHD flow using Laplace transforms. R.B. Hetnarski's algorithm for inverting the laplace transforms of some exponential functions was recommended to me ...
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26 views

How do I find the Laplace Transform of $ \delta(t-2\pi)\cos(t) $?

How do I find the Laplace Transform of $$ \delta(t-2\pi)\cos(t) $$ where $\delta(t) $ is the Dirac Delta Function. I know that it boils down to the following integral $$ \int_{0}^\infty ...
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Inverse Laplace transform of $\operatorname{arccot}(s)$, $\arctan(s)$

How would one find inverse Laplace transforms of $\operatorname{arccot}(s)$ or of $\arctan(s)$ without knowing in advance that this is related to $\dfrac{\sin x}{x}$?
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18 views

Inverse Laplace Transform and the Unit Step Function

I need to find the inverse Laplace transform of the following function: $$ F(s) = \frac{(s-2)e^{-s}}{s^2-4s+3} $$ I completed the square on the bottom and got the following: $$ F(s) = (e^{-s}) ...
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Laplace transform and IVP at $\infty$

Solving the following differential equation $$ty^{''}\left ( t \right )+\left ( t-1 \right )y^{'}\left ( t \right )-y\left ( t \right )=0$$ with initial values $$y\left ( 0 \right )=5, y\left ( ...
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39 views

Laplace transform of convolution integral

If $f(t)$ an $g(t)$ are piecewise continuous functions on $[\ 0, \infty)$ then the convolution integral of $f(t)$ and $g(t)$ is, $$(f*g)(t) = \int_{0}^{t}f(t-\tau)g(\tau) \text{d} \tau.$$ The text ...
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Laplace Transforms of Step Functions

The problem asks to find the Laplace transform of the given function: $$ f(t) = \begin{cases} 0, & t<2 \\ (t-2)^2, & t \ge 2 \end{cases} $$ Here's how I worked out the solution: ...
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40 views

Finding the eigenvalues and eigenfunction (tricky)

I'm given $$X"- vX' +X \lambda=0$$ (v is a constant) I have worked x' to be: X'(x) = $$\frac{1}{2} B v e^{\frac{v x}{2}} \sin \left(\frac{1}{2} x \sqrt{v^2-4 \beta ^2}\right)+\frac{1}{2} B ...
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Bessel equation of half-order (asymptotic)

Not really optimistic about getting a reply for a question tagged under "Bessel function" but here goes, I have $$J_{\frac{1}{2}} = (a_1 \cos(z) + a_2 \sin(z))Z^{-\frac{1}{2}} $$ and ...
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Solving a differential equation by using Laplace transform

I need to solve this equations by using laplace-transform. I tried to solve it but when I reach to the point that it's needed to use partial fraction expansion in order to transform the laplace ...
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Do we have a inverse Laplace transform of $\frac{1}{\arctan s}$

Do we have a closed form of this seemingly very simple inverse transformation? If no closed form, what about its asymptotic form? Does this satisfies the criterion to have its inverse ...
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$F^{(n)} (p)$ do you first differentiate and afterward apply the Laplace?

If you have a Laplace transform: $F^{(n)} (p)$, do you first differentiate and afterwards apply the Laplace? $F(p)$ meaning $L[f(t)](p)$
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28 views

Fourier transform of PDE on finite and infinite bound simultaneously.

Consider $$u_{xx} + u_{yy} = 0 $$ on the bounds: $$o < x < L$$ and $$-\infty<y<\infty$$ The initial condition is: $$u(0,y) = f(y)$$ and $$u(L,y)=g(y)$$ I've tried performing fourier ...
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40 views

The inverse Laplace transformation of $e^s$

I am solving the differential equation: $$y'' + 3xy' -6y = 1, \ y(0) = y'(0) = 0$$ Using Laplace transformations. I arrived at: $$L(y)(s) = \frac{c}{s^3} e^{s^2 / 6} + \frac1{s^3}$$ Where $c$ is ...
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32 views

Applying Fourier transform to heat equation with source

I haven't had any experience with applying of FT to heat equation with source. But this popped up in an exercise. Any help in the right direction would be great. Consider: $$\frac{\partial ...
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Fourier transform of a piecewise

How should I go about seeking the Fourier transform for the piecewise function: $$f(x) = \left\{\begin{matrix} 0 ,&|x|>a \\ 1 ,&|x|<a \end{matrix}\right.$$ Is this the correct ...
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Reference for an identity from Abramowiz and Stegun

I am curious as to where this identity was originally obtained. Any suggestions? $$ \frac{1}{\mathop{\Gamma}\nolimits\!\left(1+2\mu\right)2\pi i}\int_{-\infty}^% ...
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Integral of Absolute Value of $\sin(x)$

For the Integral: $\int |\sin (ax)|$, it is fairly simple to take the Laplace transform of the absolute value of sine, treating it as a periodic function. $$\mathcal L(|\sin (ax)|) = ...
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The Dirichlet problem for the Laplace equation: classical solutions versus weak solution

Let $B_R$ a ball in $\mathbb{R}^n$. Consider $u^{\star} \in H^{1}(B_R) $ and $f \in H^{1}(B_R) \cap C(\overline{B_R})$. Suppose that $u^{\star}$ minimizes $$\int_{B_R} |\nabla u|^2, u \in \{ v \in ...
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Why does the imaginary part of $s$ have no effect in analyzing region of convergence for Laplace Transform?

The tutorial that brought this assertion to me was: http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node2.html "As the imaginary part $\omega=Im[s]$ of the complex variable ...
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Laplace Transform of derivative squared

I'm trying to solve a problem while I'm studying Control Theory and I came up with a difficult question. $ \mathcal{L}\left[y'(t)^2 \right] $ Basically I need to find the Laplace Transform of this ...
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“eigenfunction” of a transformation

Fourier transform of a gaussian is another gaussian. Fourier/Laplace transforms of $\frac{1}{\sqrt t}$ is something like $\frac{1}{\sqrt \omega}$. I realize that we can't call these eigenfunctions ...
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What is the inverse laplace transform of $\large{\frac{ s^3 - a^2s }{(s^2 + a^2)^2}}$

I tried convolution and partial fractions but both turned out to be too much work. Is there any easy work around??
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Laplace transfer function and quasi-sinusoidal input

Let's suppose we have an LTI system whose Laplace domain transfer function is: $$ F(s)=\frac{1}{s^2 + \frac{\omega_y}{Q_y}s + \omega_y^2} $$ Its input is the Coriolis force. Such force is experienced ...
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Instantaneous DC output using Laplace transform

I am testing a feedback system in which a feedback signal from sensor is applied to a correction network and then compared with threshold values. I need some clue how to achieve this, I will be ...
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Convert an equation in Laplace “s” space to Discrete “z” space using a table

I'm trying to discreteize an equation. I have the equation in laplace form, but I do not have the original differential equation. The equation is: $$\frac{\theta(s)}{V(s)} = \frac{a}{s(s+b)(s+c)}$$ ...
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Evaluate $\int_{0}^{\infty}\frac{\sin^{2}\left ( t \right )}{t^{2}}dt$ with help of Laplace transform

Using the following identity $$\int_{0}^{\infty}\frac{f\left ( t \right )}{t}dt= \int_{0}^{\infty}\mathcal{L}\left \{ f\left ( t \right ) \right \}\left ( u \right )du$$ I rewrote ...
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Transform function in terms of Heaviside functions.

I am working on the red-shaded problem below however I am unsure how to use my answer from part b to answer the question. Any help is appreciated.
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Initial Value Problem involving Laplace Transforms

I am trying to solve the initial value problem: $y' + 5y = $ 0 if $t \in [0,3]$ 9 if $3 \in [3,6)$ 0 if $t \in [6, \infty)$. with $y(0)=9$. I am asked to take the Laplace Transform of both sides ...
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Laplace Transform of $e^{2t-12}u(t-6)$.

I am trying to find the Laplace Transform of $e^{2t-12}u(t-6)$. All I know is that $\mathcal{L}\{e^{-at}\} = \dfrac{1}{s+a}$ and that $\mathcal{L}\{u(t-a)\} = \dfrac{e^{-as}}{s}$. I also know that ...
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Using the Heaviside function to represent a given graph

The question is the following: The graph is zero between 0 and 2, is a straight line from the point (2,0) to (5,5), a straight line down from (5,5) to (4,0) and zero everywhere else. So far, my ...
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28 views

How to explain this simplification here?

I can't understand this simplification the book says without explanation. Could someone help me? It is the calculatation/development of the transfer function of a digital system composed by a dac ...
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An initial-value problem and a corresponding Laplace Transform

I am to solve the 1st-order ODE \begin{align*}y^\prime + 3y = 45t, \qquad y(0) = 6 \end{align*} using a Laplace Transform for a problem set. So far, by taking the Laplace Transform of both sides, I ...
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Inverse Laplace Transform of $\dfrac{6s -19}{s^2 - 6s + 13}$.

I am trying to figure out the inverse laplace transform of $\dfrac{6s -19}{s^2 - 6s + 13}$. Looking at my table of Laplace Transforms in my textbook, it seems that either I must break up this fraction ...
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What's the logic in this equation? [laplace and z transform] [closed]

Why/How the z^{-l} goes there in 3.25? Also I can't understand the 3.26 statement.
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48 views

Initial Value Problem using Laplace Transforms

Solve using Laplace Transform: $$y''(t)+2y'(t)+5y(t)=xf(t), \\ y(0)=1,y'(0)=1$$ where $x$ is a constant. Once the solution is found, evaluate the limit as $t \to\infty$. Progress: If I have ...
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38 views

How to solve an intro-differential equation with integrals.

$y'(t)-y(t)-3\cdot\int[e^{x-t}\cdot y(x),x,0,t]=16\cdot t, y(0)=16$ I am having a difficult time figuring out how to evaluate the integral to solve with the rest of the problem.