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

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Laplace transform involving two functions of t

I need to solve the following $$ \int_0^{\infty} f(t)g(t)e^{rt} dt$$ Where $$g(t)=t^n$$ Letting r=-s we have the definition of $$ \mathcal{L} [ f(t)g(t) ]$$ and am unsure how to continue.
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How to find Laplace transform of a differential equation?

$y′′ + 3y′ + 2y = f$ , $y(0) = 0$ , $y′(0) = 1$ where $f$ is given by $f(t) = \sum_{n=1}^\infty \delta(t−n)$; find a 1-periodic function $y_*$ with $\lim_{t\rightarrow \infty} |y(t)−y_*(t)| = 0$. I ...
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How to re-write one fraction as two others.

I have the two following fractions. $$ \dfrac{A}{Bx^{\alpha+1}}$$ and $$ \dfrac{C}{Dx^{\alpha+\beta}}$$ The form i want $$ \dfrac{E}{Fx^{\alpha+\beta+1}}$$ I was thinking to do partial fractions or ...
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55 views

Trying to show $\int_0^1 e^{-xt}sin(t) dt \sim \frac{1}{x^2}$

I am using Laplace's Method and I am trying to show $$I =\int_0^1 e^{-xt}sin(t) dt \sim \frac{1}{x^2}$$ $h(t) = -t$ has a maximum at $0$ and as it is a simple function there is no need to expand it. ...
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Laplace transform of a differential equation?

$y′′ + 2y′ + 2y = δ(t − \pi) + aδ(t − T)$ , $y(0) = y′(0) = 0$ $a$ and $T$ are positive numbers and $T > \pi$. I need to find values for $a$ and $T$ such that $y(t) = 0$ for all $t \ge T$? I just ...
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44 views

Solve initial value problem with unspecified right-hand side $g(t)$

Consider the initial value problem $$y''-6y'+9y=g(t),\quad y(0)=1,\ y'(0)=3.$$ 1) Use the Convolution Theorem to find the solution to the IVP for any piecewise continuous function $g(t)$ that is of ...
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74 views

Laplace transform via complex analysis

Let $Y(s) = \frac{2e^{-s}}{s(s^2 + 3s + 2)}$. Then the inverse Laplace transform is \begin{align} y(t) &= \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{2e^{s(t - 1)}}{s(s^2 + 3s + ...
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Solution of 1d wave equation by Laplace transform

This is a homework problem that I can almost finish. I just can't invert the Laplace transform at the end. $$u_{xx}=u_{tt}, u(t=0)=u_t(t=0)=0, u(x=0)=\sin\omega t, u(x=2)=0.$$ Taking the Laplace ...
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Laplace transform for IVP not at zero in system of differential equations

Suppose we have a system $\boldsymbol X'=\boldsymbol A\boldsymbol X$. Let's denote the laplace transform of a vector $\boldsymbol Y$ as $\mathscr L\{\boldsymbol Y(t)\}(s)=\boldsymbol y(s)$. If we ...
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66 views

Understanding block diagrams

If I have block diagram with input $X(s)$ that goes to a block with $\frac{1}{s + 2}$ in it and then by way of $w(s)$ to a block with $s$ in it, and finally to the output $Y(s)$, how do I find the ...
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36 views

Partial fractions in Laplace Transform

Solve: $$y''+y'+\frac{5}{4}=U_\frac{\pi}{2}(t)f(t-\frac{\pi}{2})$$ becomes: $$[s^2+s+\frac{5}{4}]Y(s)=\frac{1-e^\frac{-\pi*s}{2}}{s^2}$$ becomes: ...
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116 views

Why the Laplace transform of u(-t) is 1/s?

Yesterday I had my first contact with the Laplace transform, in an Electric Circuits class. $\mathcal{L} \left\{ u \left( -t \right) \right\}$ showed up. Our teacher said it was equal to ...
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61 views

Matlab calculate output

I'm trying to write a matlab function that takes in a transfer function and the input so it can calculate the output. So far, based on this information under I have the following piece of matlab ...
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42 views

Calculate the laplace transform…

Calculate the laplace transform of $$t^2u(t-2)$$ I don't know how to manipulate t^2 in order for it to meet the form of the product between a function and a heaviside function. Number (27) on ...
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How to find $\mathcal{L}^{-1}\left\{1\right\}$?

This is probably a really simple question, but I cannot figure it out and it's not mentioned in my books. How do I find \begin{align} \mathcal{L}^{-1}\left\{1\right\}?\tag{1} \end{align} It seems like ...
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42 views

Help with inverse Laplace transform

Please help me to find the inverse Laplace transform of : $$ \frac{1}{S(aS^2 + bS + c)} \left( 1 - \exp (-TS) \right) $$ where a,b,c,T are constants thanks
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Laplace transform of a differential equation??

Find unique solution of $y′′ + y = f$ using $y(0) = y′(0) = 0$ and periodic function $f(t) = t$ if $0 \leq t < 2\pi$ Attempted work: $L[y'' + y ] = L[f(t)]$ $L[y''] + L[y] = L[f(t)]$ $s^2 L[y] ...
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What is the Laplace transform of this random variable?

Define a random variable that takes only one value for example $$X=c$$ where c is a positive constant. What does the Laplace of it evaluate to i.e the following $$\mathcal{L}_X(s)= ...
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231 views

Solving PDE by Laplace Transform

Use Laplace transforms to solve the boundary value problem $$Y_{xx}(x,t)-2Y_{tx}(x,t)+Y_{tt}(x,t)=0, \quad 0<x<1, t>0$$ $$Y(x,0)=Y_t(x,0)=0, \quad 0<x<1$$ $$Y(0,t)=0, \ Y(t,1)=F(t), ...
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34 views

Laplace transform of a differential equation?

Find the unique solution of $y''+ y = f$, $y(0) = y'(0) = 0$ with the $2\pi$ periodic function given by $f(t)=2\pi \sin(t)$. I am having trouble setting up and starting the the question. I would be ...
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87 views

Using complex analysis to find the Inverse Laplace transform

I have been reviewing for my comprehensive graduation exam where I have been solving the Inverse Laplace transform via complex analysis. Consider $$ H(s) = \frac{s^2 - s + 1}{(s + 1)^2} $$ Then we ...
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Laplace transform of $f(t)=\left|\sin\frac{t}{2}\right|$?

If you are given a rectified sine wave, $$f(t)=\left|\sin\frac{t}{2}\right|$$ how do you find the Laplace transform of this? I tried using the equation $$L\{ f(t)\} = \frac{1}{1-e^{-sT}} \int_0^T ...
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253 views

Laplace transform of a sawtooth wave

Find the Laplace transform of the periodic function such that $f(t) = t$ if $0\leq t < 2\pi$ I am having trouble setting up this question. Am I on the right path? $$ \mathcal{L}\{f(t)\} = ...
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56 views

How to determine $2\pi$ periodic function?

Let $f(t) = 2\pi \sin t$, and determine a $2\pi$-periodic function $y^∗$ with the property that $\lim_{t\to+\infty} |y(t) − y^∗(t)| = 0$ for every solution $y$ of $y′ + y = f$. I am having trouble ...
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1answer
54 views

Heaviside function in the function whose Laplace transformation is $e^{-(\gamma+s)}/[(s+\gamma)^2+b^2]$

This is from a homework question 13.22 part (c) from "Mathematical Methods for Physic and Engineering" by Riley et. al on p. 464 I don't understand why the heaviside function is in the solution to ...
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25 views

Laplace transform of $tf'(t)$

I know that $\mathcal{L}(tf'(t)) = -\frac{d}{ds}\mathcal{L}(f'(t))$ and that this $= -\frac{d}{ds}(sF(s) - f(0))$ but the solution says that this becomes $-F(s) - F'(s)$ and I can't figure out why ...
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41 views

IVP with Laplace Transform

My attempt: Y = Laplace $$s^2Y -sy(0) - y'(0) - 3Y = ??$$ How do I set up $$h(t)$$ in the form of laplace?
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Sign problem, Laplace transform of sin(at)

I have a problem in my integration by parts but I can't find it: $$L(\sin(\alpha t)) = \int_0^{\infty}\sin(\alpha t)e^{-st}dt$$ $$= -\frac{1}{\alpha}\left[\cos(\alpha ...
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38 views

Difference between the Rectangular “Window” Function and the Rectangle Function

I'm getting ahead in my differential equations textbook (Fundamentals of Differential Equations by Nagle et. al) and in the chapter of Laplace Transforms it states that the rectangular window function ...
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Partial Fraction Decomposition — Inverse Laplace Transforms

I apologize if this is a rather lame question, but I've always been a little touchy with my partial fraction decompositions and I'm hoping to get better at them. Could you verify (or correct?) my ...
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Find $\mathcal{L}\left\{\cos^3\left(t\right)\right\}$

I began by breaking the problem up as follows: \begin{align} \mathcal{L}\left\{\cos^3\left(t\right)\right\}=\int_0^\infty e^{-st}\cos^3\left(t\right)\:dt & = \int_0^\infty ...
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218 views

Black's formula and feedback system stability

Consider a hypothetical system with open-loop transfer function $G(s)$. Place it in positive feedback with unit gain. (That is, take its output and directly add it to its input.) The closed-loop ...
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31 views

Laplace Transformations and Piecewise Functions

I am trying to understand why it is that Laplace transformations can simply be "added together" when finding the transform of a piecewise function. My professor has quite extensively talked about the ...
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1answer
20 views

Correct partial fraction construction?

Is the below the correct partial fraction decomposition? $$\frac{s^2 - 6s + 9}{(s-2)^3}=\frac{A}{s-2}+\frac{B}{(s-2)^2}+\frac{C}{(s-2)^3}$$ I can see that the numerator doesn't have a factor of ...
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51 views

Laplace transform with the Heaviside unit step function

I want to find the laplace transform for the function: $$f(t) = \left\{\begin{array}tt,\quad t\lt 2 \\ t^2 , \quad t\geq 2 \end{array} \right.$$ So I thought that the proper setup was: ...
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34 views

Find $\mathcal{L}\left\{t e^{2t}\cos\left(5t\right)\right\}$

This is what I have so far: \begin{align} \mathcal{L}\left\{t e^{2t}\cos\left(5t\right)\right\}=\int_0^\infty e^{-st}t e^{2t}\cos\left(5t\right)\:dt,\tag{1} \end{align} but notice that if ...
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170 views

Shortcut methods for Partial fraction decomposition in IVPs solved by Laplace transformation?

I have an IVP I'm trying to solve with Laplace transformations: $$y''-4y'+4y=te^{2t}$$ Given that: $y(0)=1$ and $y'(0)=0$ I've gotten to the part where I isolate $Y(s)$: ...
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37 views

Laplace Transform solution verification: $\ddot{y} + 2y = 2e^t\implies \frac13\cos(\sqrt{2}t)-\frac{2}{3\sqrt{2}}\sin(\sqrt{2}t)+\frac23e^t\,\text{?}$

Does $$\ddot{y} + 2y = 2e^t\quad y(0)=1,\dot{y}(0)=0$$ Give $$\frac13\cos(\sqrt{2}t)-\frac{2}{3\sqrt{2}}\sin(\sqrt{2}t)+\frac23e^t\,\text{?}$$ This is what I have got, and it seems to go back and ...
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Is this the correct setup for partial fractions? $\frac{1-e^{-s} + se^{-s} + s^3}{s^2(s^2+2)}=\frac{A}{s}+\frac{B}{s^2}+\frac{Cs+D}{s^2+2}$

I am trying to inverse laplace transform the following: $$F(s)=\frac{1-e^{-s} + se^{-s} + s^3}{s^2(s^2+2)}$$ and I believe what I do is take: $$\frac{1-e^{-s} + se^{-s} + ...
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41 views

Inverse Laplace transform for $\frac{1-e^{-\pi s}}{s(s^2 + 16)}$

I want to find the inverse Laplace transform for the following:$$\frac{1-e^{-\pi s}}{s(s^2 + 16)}$$ This was obtained from a piecewise function and required the heaviside step function to simplify. ...
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Computing transfer function then static gain

I am working on a problem for a flight controls class. I have an equation related to pure yaw. My goal is to get the transfer function associated with it, and then obtain the system static gain. The ...
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1answer
36 views

How to justify this complex substitution using contour integration

I tried to solve the laplace transform of $\cos(at)$ and $\sin(at)$ using Euler's formula. That is, $$\int^\infty_0e^{-(s-ia)t}dt\color{red}{=}\frac{1}{s-ia}\int^\infty_0e^{-t}dt=\frac{1}{s-ia}$$ ...
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Laplace Transform…

Find the Laplace transform of $t^2e^{at}cos(bt)$ My attempt: $\large\mathit{L}\{t^2e^{at}\cos(bt)\}$ = $\large\mathit{L}\{\frac12t^2e^{at}(e^{ibt}+e^{-ibt})\}$ ... ... ...
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LaPlace transform of $t^2\sin(at)$

$$\int^{\infty}_{0} e^{-st}t^2\sin(at)dt$$ I keep running into a problem when using: $$u=e^{-st}t^2$$ $$du=2te^{-st}-st^2e^{-st}$$ $$v=\frac{1}{a} \cos(at)$$ $$dv=\sin(at)$$ Anyone have any ...
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22 views

What is the inverse Laplace of a complete square?

How do I find the inverse Laplace for something like this ? $${8 \over ( s^2 + 16 )^2 }$$ I tried using partial fraction but it didn't help any ideas on how to do it using differentiation or ...
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177 views

Evaluate $\int_{0}^{\infty}\frac{1-e^{-t}}{t}e^{-st}\;dt$

This is laplace transform of $\dfrac{1-e^{-t}}{t}$ and the integral exists according to wolfram Do i get any help/hints about how to work this ? I have been trying integration by parts with different ...
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Additive (causal + anticausal) decomposition of a transfer function

Define the causal/anticausal decomposition of a function $F(s)$ as follows. Let $f(t)$ any function such that $$F(s) = \int_{t=-\infty}^{+\infty} f(t)e^{-st} dt.$$ Then the causal part of $F(s)$ is ...
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23 views

Laplace Transform of $\cosh(at)/(at)$

Can someone give me a clue on how to compute this Laplace transform? $$\mathcal{L}\left[ \frac{\cosh(at)}{at} \right]$$
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Why does the Laplace transform of $t^2 \exp(at)$ exist?

My book states a theorem : "Let $f(t)$ be a function piecewise continuous on $[0, A]$ for $ A > 0$ and have an exponential order at infinity with $|f(t)| \leq M \exp(at)$. Then the Laplace ...
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417 views

Using laplace transforms to solve a piecewise defined function initial value problem

I want to use laplace transforms to solve the following: $$\frac{d^2 y}{dt^2}+16 y = f(t) = \left\{\begin{array} 1 1&t\lt\pi\\0&t\geq \pi\end{array}\right.\text{ with } y(0)=0 \text{ and } ...