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

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Solving a non-linear integro-differential equation

I am trying to solve the following equation $$ f^2(x) - g^2(x) = \alpha\int_0^x f(u) (x-u)du $$ For $\alpha=0$ we get $f=g$. I would like to see how the solution moves away from $g$ when I increase ...
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s-plane and fourier transform, together in 3d space.

I dont understand how can varying the real part in the s-plane make the amplitude in the fourier plane go to infinity. Lets say the pole is at -3 + -j for example.. Then the laplace transform is the ...
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32 views

Inverse Laplace transform of $s^{k}$

How can I find the inverse Laplace transform of $s^{k}$ where $k$ is non-integer and negative? I know that $$\mathcal{L}^{-1}[s^k] = \frac{1}{2\pi i}\int e^{st} s^k ds$$ and since we have ...
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33 views

Laplace's Method Integration

Consider the integral \begin{equation} I_n(x)=\int^2_1 (\log_{e}t) e^{-x(t-1)^{n}} \, dt \end{equation} Use Laplace's Method to show that \begin{equation} I_n(x) \sim \frac{1}{nx^\frac{2}{n}} ...
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Laplace Transform of tsin(at) using only the definition

Hello I' am stuck on how to get the final result of the laplace transform of $f(t)=tsin(at)$using (a is a constant) only the definition of $$\int_0^{\infty}f(t)e^{-st}dt$$, I know $sin(at)= {1 \over ...
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Laplace transform of a majorated function

I have the following problem. I have an analytic function and I want to show that it is majorated by a convenient function. To do that, it is very helpful to solve the transformed equation. I have a ...
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38 views

Laplace's Method (Integration)

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} Use Laplace's Method to show that \begin{equation} I(x) \sim ...
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Solve second order differential equation with Heaviside function using Laplace transform

The equation is: $$y'' + 3y = u_4(t)\cos(5(t-4)), \quad y(0) = 0, \quad y'(0) = -2$$ Here $u_4$ is the Heaviside function with activation switch at $t=4$. I can get all the way to the partial ...
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Using Laplace Transform to solve a equation with piecewise function

Using Laplace Transform to solve$$y''+4y=f$$ Where $y(0)=0, y'(0)=-1,$ and:$$f(t)=\begin{cases}\cos(2t)&\text{if $0\le t \lt \pi$}\\0 &\text{otherwise}\\\end{cases} $$ Do I need to solve the ...
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27 views

Take the Laplace Transform

Take the Laplace transform of $$ \int_{0}^{t}x^2(x-t)^4 \cos(x)dx .$$ I'm not quite sure where to start...
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40 views

Deducing Laplace Formulas

I have to compute the followings integrals $\forall\; b\in \mathbb{C},\; \text{Re} \;b \gt0,p\gt 0$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x-ib}$$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x+ib}$$ ...
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Laplace transform using the definition

Find the Laplace of the given function using the definition $$f(t)=tsin(t)$$ I know what the answer is according to a sheet that I have of common transforms but I am not 100% on how to get there ...
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34 views

Laplace Transform of an integral

Find the Laplace transform of $$f(t)=t\int_0^{t} \tau e^{-\tau}$$ $L(f)(s)$= ?? My thought is that I can change the $\tau$ to $t$ by Transforming the integral to get $$t/s*L[t*e^{-t}]$$ But ...
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28 views

Laplace transform of integral equation

Use Laplace transforms to solve the integral equation $$y(t)-\frac{1}{2}\int_0^ty(t-v)~dv=1$$ First find the Laplace transform $Y(s)$ of $y(t)$
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Inverse Laplace Transform of Reciprocal Quadratic Function

Starting with the the equation: $$I(s)=\frac{6}{Ls^2 + Rs + \frac{1}{C}}$$ I need to find what $i(t)$ is by doing the inverse Laplace transform. I need to do some algebra to put it in a form that ...
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Laplace transform of $f(t)$ multiplied by $t^n$

How to prove that it is $(-1)^n\frac{d^n}{ds^n}F(s)$
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complex integral of z to the power alpha

I would like to perform an inverse laplace and at some point of the calculation I have to compute this integral $$\int_{\gamma-i\infty}^{\gamma+i\infty} z^{(1+n)\alpha-1}e^{z} \frac{dz}{2\pi i}$$ ...
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Which method to use when— higher order differential equations

Over the last unit in my class we have learned various methods of handling higher order diffeqs. However, I want to know how to decide which method to use to solve given diffeqs most efficiently. So ...
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53 views

Laplace transform.

This is a past exam question. I'm having a bit of trouble at finding the inverse laplace transform of the following function. Any help would be great. $$\frac{s^2+1}{(s^2+4s+5)^2}$$ Thanks for ...
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Existence of inverse Laplace tranform

I have two questions about inverse Laplace transform. Given a function $F(s)$, does its inverse Laplace tranform always exists ? If it's not, assume $F(s)$ has an inverse Laplace tranform, does the ...
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Find the solution of the IVP using Laplace transforms

The equation is as such: $y''+y=t\sin t$; $y(0)=1, y'(0)=2$ I took the Laplace transform of both sides to yield $F(S)(s^{2}+1)-(s+2)=\frac{-2s}{(s^{2}+1)^{2}}$, and then ...
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43 views

Laplace Transform of a rounded function (or an infinitely discontinuous function)

Working on an assignment today, I thought of a problem I haven't been able to solve and haven't been able to find any solutions for online. What would the Laplace Transform of a rounded function be? ...
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30 views

Help Understanding Inverse Laplace Transforms???

I've been looking over some examples regarding inverse laplace transforms, and my textbook doesn't really go into much depth regarding examples such as this: F(s) = (2(s-1)e^(-2s))/(s^2 -2s + 2) I ...
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Laplace of $e^{-s+3}$?

I understand that the $e^{-ks}$ will time shift the function in the time domain by $k$ and will result in a time function of $u(t-3)$, but what does it mean when you have $e^{-s-k}$? How will that ...
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Name this alternative to Laplace transforms

My professor taught a method in class to use for differential equations with step functions as the signal. Somehow he solved each piece of the piece wise function separately and then combined them. ...
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29 views

Help with basic Laplace Transform - unsure of procedure!!!

I am working on this Laplace Transform, and I've tried looking for a similar example off which to base my own work, but haven't been very successful. I'm confused by the formatting and don't know how ...
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37 views

Laplace transform of a piecewise function

I'd like to compute the Laplace transform of the following function: $$f(t) = \begin{cases} 0,& \mbox{if} \quad 0 \leq t \lt \pi \\ \sin(t), &\mbox{if} \quad t \geq \pi \end{cases}$$ Could ...
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28 views

properties of laplace transform

Obtain the transfer function for the following differential equation and check whether the input free solution is stable or not, $$\frac{dx}{dt} + 3x = f(t)$$ Please help, I don't even know where to ...
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41 views

Solving $y'-y=2\cos 5t$ using the Laplace Transform

Find the solution to the differential equation, using the Laplace Transform. $y'-y=2\cos 5t$, with initial condition $y(0)=0$. My attempt: First I take the Laplace Transform of each term. ...
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A question about Parseval's formula.

In operational calculus there is Parseval's theorem, which states that if $ f(t) \doteqdot F(p), \varphi(t) \doteqdot \Phi(p) $ and both $ F(p) $ and $ \Phi(p) $ are analytical in $ Re p \geq 0 $, ...
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Product of two Whittaker functions

According to 6.669.3 of Gradshteyn and Ryzhik the following identity $$ W_{a,b}(z_1)\,W_{a,b}(z_2) = \frac{2\sqrt{z_1z_2}}{\Gamma(1/2+b-a)\,\Gamma(1/2-b-a)}\int_0^\infty ...
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27 views

Is the Laplace transform additive? And why?

The first part is a simple question, but I cant find a clear answer. Does: $$\mathcal{L}(ax''(t)) = \mathcal{L}(a)\times\mathcal{L}(x''(t))$$ $a$ is a constant $x(t)$ is a variable that changes ...
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How can we take the LaPlace of a function raised to the power?

For example: $\mathcal{L}$((t-1)^1) Following simple linearity, we achieve the answer. However, following the power of theorem: (I'm not proficient enough in LaTex to write this...) I get the wrong ...
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How can we take the LaPlace transform of a piecewise function?

How can we take the LaPlace transform of a function, given piece-wise function notation? For example, $f(t)=\begin{cases} 0 &\mbox{for } 0<t<2\\ t&\mbox{ for } 2<t \end{cases}$ ...
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How to find the transfer function from the given differential equation

Question: find transfer function from differential eqn $y''(t)+2y'(t)+5=4x(t)$ I am confused about what happens to constant $5$ . will it be zero when we take laplace of whole eqn or not? Can ...
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Unit impulse / step response of a 1st order differential equation

You are given the equation $10v'(t) + 0.6 v(t) = f(t)$ $v(t)$ is the velocity of the object Determine the unit impulse response AND the unit step response. How would i approach this question? do i ...
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Laplace transform of gamma distribution

Gamma distribution has its pdf given by $f(t;k;\theta) = \frac{t^{k-1} e^{-t/\theta}}{\theta^k (k-1)!}$. Show that if the pdf's Laplace transform is $L_k (s)$, then $L_{k+1} (s) = \frac{L_k ...
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Are these Laplace transforms wrong in Stroud's Advanced Engineering Math Book?

I know that if you think a book is wrong, most probably it is your own mistake. However, I can't understand the following Laplace transforms in K. A. Stroud's "Advanced Engineering Mathematics". In ...
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Inverse Laplace Transform of $\frac{s^2+2s+2}{s+1}$

I want detailed steps of this if anyone can help.
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Zeros/poles at Laplace and at Fourier Transform

I recently started "relearning" the Laplace transform, and I noticed something. It seems to me that the intuitive idea of poles and zeros is different between these two transforms! For example, in ...
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63 views

Evaluating an integral with Laplace

We need to evaluate the following integral: $$\int_{0}^{\infty}\frac{\cos(tx)}{x^2+a^2}dx$$ There is the following note: "You may interchange taking the Laplace transform and integrating." I have ...
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54 views

Forced wave equation question?

I'm studying for my PDEs midterm and trying to do practice problems. I'm really not sure how to do this question - I've never seen anything like it. Thanks in advance for your help. Solve the ...
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How to show that the Laplace transform of $\exp(-t^2)$ is $\frac{\sqrt{\pi}}{2}\exp(\frac{s^2}{4})\rm erfc(\frac{s}{2})$

I obtained the answer from Maple. But still I want to know how it is derived.
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Undefined Laplace Transform

I'm in calculus II and our teacher gave us a problem as follows: Let f(t) be a function defined for all positive values of t. The Laplace Transform of f(t) is defined by: $$F(s) = \int_0^\infty ...
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Laplace Transform Damp Harmonic Motion

http://gyazo.com/19d18f085731c6dbc304fefdaece4f3c.png I'm currently on (a) where so far I have gotten; $ y'' + 2y' + 5y = f(t) $ Using Laplace transforms, I get; $ Y(s)$ = $ F(s) + s+2\over(s^2 ...
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The Laplace Transform of nonlinear terms (eg. cos(x(t)), x(t)^2)

I've been trying to solve a differential equation of the form $ax"+bx'+cx=d$, but I do not have a constant $c$, rather I have $\cos(c*x)$. (NB: I do NOT mean to find the LT of $\cos(a*t)$, this is ...
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69 views

Obtaining fundamental solution of the heat equation (1-d) through Laplace transform

A classic problem I'm having problems with (problem requires to use Laplace transform) $\frac{\partial ^2}{\partial x^2} u(x,t)=\frac{\partial}{\partial t} u(x,t) $ with conditions: ...
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Coupled mass spring system with damping and initial values

After researching through the web, I can't figure out how to express into a differential equation a coupled mass spring system with damping and initial values. Two masses and two springs, no external ...
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Inverse Laplace transform question help

I am having a hard time finding the inverse Laplace transform of $$\frac{1}{(s^2+1)^2} - \frac{1}{s^2(s^2+1)^2}$$ and would appreciate some guidance. I have tried breaking it down to partial fractions ...
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Coupled mass spring system with damping, I need help with the equation

I know that the equation $mx''+cx'+kx=f(t)$ is used for a normal mass spring system, but I don't know how to express the differential equation for a coupled mass spring system with damping. These are ...