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

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Don't know how to solve Unit Step Function

This is the solution that i found in the solution sheet but i can't seem to know how to get to the step that is in the red rectangle.
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Solving and graphing an IVP involing unit step function

Im trying to solve this ODE and find a simplified expression for $x(t)$. $$\ddot x+4x=-2\sum_{n=1}^{4} e^{in\pi}u(t-n\pi);\space x(0)=0=\dot x(0),i=\sqrt{-1}$$ First i found the the laplace ...
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406 views

deriving second order transfer function from spring mass damper system..

I am having a hard time understanding how a differential equation based on a spring mass damper system $$ m\ddot{x} + b\dot{x} + kx = 0$$ can be described as an second order transfer function for an ...
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23 views

Converting to a partial fraction.

I'm trying to do an inverse Laplace operation on $I(s)$ shown below but I'm struggling on finding what $A$ & $C$ are on the partial fraction and how to do it. I calculated what $B$ equals by ...
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17 views

problem in $z$ transform

Im having a hard time trying to solve this... if $ (x_{k})_{k=0}^\infty $ is a causal succession such as $$ Z(x_{k}) = X(z) \parallel z \parallel > R $$ prove that $$ Z( {{x_{k+2}}}) ...
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55 views

Unit Step Function Finding slope

How do i find the using $y=mx+c$ for the slope above from $1$ to $3$
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45 views

Find the poles and residues in an awkward Laplace inversion

Assume that part c) has been proved and ignore parts c) & d). To invert the Laplace transform we would do $\displaystyle u(x,t)=\frac{1}{2\pi ...
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107 views

Contradiction in inverse Laplace transform problem with Mellin's inverse formula?

Let say we have to solve a given differential equation $$ty''+y'+ty=0$$ $$y(0)=1,\ y'(0)=0$$ (which is Bessel equation with the solution $y=J_0 (t)$, of course) with the Laplace transform. Then we ...
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Unit Step Function

Question: What is $\mathcal{L}\{u(t-1)u(t-2)\}$? My calculations $e^{-2}s \mathcal{L}\{u(t+2)-1\}$ $e^{-2}s \mathcal{L}\{t+1\}$ $e^{-2}s (s^{-2}+\frac{1}{2})$ I'm confused, I gotten the wrong ...
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21 views

Inverse Laplace Transform of Polynomial

I'm trying to find the inverse laplace transform of the following function: The resulting inverse laplace is in the form: Not sure where the derivatives came from, or what the inverse laplace of ...
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40 views

Laplace transform and Laguerre Polynomials

let be the Laplace transform of a function $$ F(s)= \int_{0}^{\infty}dtf(t)e^{-st} $$ then if the function satisfies that for every integer 'n' $ D^{n}f(x)=0 $ then $$ s^{n}F(s)= ...
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laplace non homogeneous parts [closed]

Consider the following initial value problem: $$y'' + {81} y= \begin{cases} 4 t, & 0 \leq t < 4 \cr 0, & t \geq 4 \end{cases} \hspace{0.5in} y(0)=0, ...
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39 views

Convolution, indicator function

I need to calculate $(f*f)(x)$ of $f(x) = 1_{[0,1]}(x)$, which is the indicator function defined with Calculating the integral $(f*f)(x) = \int_{0,}^{x}1_{[0,1]}(t) \cdot1_{[0,1]}(x-t) dt$ gives ...
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42 views

Solution of a system of differential equations for a continuous time Markov chain.

The equations arise as the Laplace transforms of the forward equations of a continuous time Markov chain for a three-state system, with the following transition rates: Transition , rate $0 ...
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32 views

What good is the final value theorem?

I'm having trouble understanding the utility of the Final Value Theorem* in Laplace Transforms. My book said that the if $x(t)=0$ for $t<0$ and if $x(t)$ contains no impulses at the origin, the ...
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122 views

inverse Laplace transfor by using maple or matlab

I want to use inverse Laplace transform to F function by using maple or matlab. However I cannot get any answer. I know the answer from table but I want to use one of softwares. from table: ...
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44 views

Cant solve this differential equation using laplace transforms

$$\left\{\begin{array}{ccc} y''(t) &=& x(t) - 2y(t)\\ x''(t) &=& - 2x(t) + y(t) \end{array}\right.$$ $y(0)=1,\;x(0)=1,\; y'(0) = \sqrt3,\; x'(0) = -\sqrt3$. I am trying to solve ...
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60 views

Laplace transform of unit step function

Im given a graph of $f(t)$ and i need to find the Laplace transform of $f(t)$. From looking at the graph i have $$f(t) = \begin{cases} t, & \text{$0 \le t \le 1 $} \\ 0, & \text{$1 \lt t \lt ...
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42 views

How do I take inverse Laplace transform of $\frac{-2s+3}{s^2-2s+2}$?

How do I take inverse Laplace transform of $\frac{-2s+3}{s^2-2s+2}$? I have checked my transform table and there is not a suitable case for this expression.
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Complete y(t) solution for Laplace Transform? [duplicate]

Trying to figure out how to use Laplace Transform to find $y(t)$: The problem is $$y''+4y'+4y=f(t)$$ where $f(t) = \cos(\omega t)$ if $0 <= t < \pi$ and $f(t)=0$ if $t >= \pi$? Initial ...
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Laplace Trouble to find solution

Trying to figure out how to use Laplace Transform to find $y(t)$: The problem is $$y''+4y'+4y=f(t)$$ where $f(t) = \cos(\omega t)$ if $0 < t < \pi$ and $f(t)=0$ if $t > \pi$? Initial ...
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46 views

Laplace transform of initial value problem, stuck on partial fractions.

The problem im given is: Use Laplace transforms to solve the initial value problem. $$\ddot x +x=\sin(2t)$$ $$x(0)=0=\dot x(0)$$ I first do the following Laplace transforms: $$\mathcal{L}\{\ ...
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105 views

Boundary integral method to solve Poisson equation

Suggest how to solve Poisson equation \begin{equation} σ ∇^2 V = - I δ(x-x_s) δ(y-y_s) δ(z-z_s) \nonumber \end{equation} by using the boundary integration method to calculate the potential ...
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From position to velocity?

I have an transfer function which tells what the angular displacement of an DC motor. This transfer function is in the S-domain, and normally when you differentiate (*s) the angular position you would ...
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51 views

Inverse Laplace transform of a given function

1) The Laplace transform of f(t) is $\overline{f}(p)=\frac{1}{p}$ when $f(t)=1$ 2) The Laplace transform of $f(at)$ is $\frac{1}{a}\overline{f}(\frac{p}{a})$ 3) The Laplace transform of the ...
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35 views

Using Laplace Transform to solve a 3 by 3 system of differential equations

I have been trying to solve this system of equations using Laplace transforms for a while. It is very easy to solve it using eigenvalues and eigenvectors, but when I tried to do it using Laplace I ...
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41 views

Laplace transform of $t\,e^{-t}$

Can someone explain why the answer is $\frac{1}{(s+1)^2}$? My understanding was to multiply the Laplace transforms of $t$ and $e^{-t}$. So $\left(\frac{1}{s^2}\right)\left(\frac{1}{1+s}\right)$? ...
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54 views

Z - transform of a transfer function

I have to apply a z-transformation to my transfer function which looks like this: $$\frac{K}{s} - \frac{K\cdot T}{T\cdot s}+1$$ I have tried it and this is my result: $$K \cdot \frac{z}{z-1} - K ...
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39 views

Manually trying to calculate output of an transfer function.

I am trying to calculate the output of an transfer function due to the input of an step, But some weird reason, I am only getting the inverse output, what Matlab says it should be. My transfer ...
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27 views

Differential Question about Laplace/Delta/Convolution

I need help understanding a part of this question. Let $a.) y''+4y = \delta (x)$, $y(0)=y'(0)=0$. and $b.) y'' + 4y = f(x)$, $y(0)=y'(0)=0$ where $f(x)$ is some continuous function of finite ...
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53 views

Inverse LaPlace Transform of the square root of Rational, Monic 1st Degree Polynomials

I tried to find this in Churchill's Operational Mathematics which has a good variety of transform pairs, but no matches for what appears a simple expression. Does anyone have a solution for the ...
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48 views

Solve equation taking Laplace transforms

I'm solving this equation using Laplace transform $$ Y''(t) + (t+1)Y'(t) + tY(t) = 0 $$ and I know that $Y(0)=1$ and $Y`(0)=-1$, so I start solving it taking Laplace transform $$ ...
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Solving a differential equation using the laplace transform involving convolution

The problem is the following The thing that puzzles me here is the integral on the right hand side, so: How to take the laplace transform on the right hand side? Any help to get me going would be ...
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63 views

Solving an integral using Laplace transform and inverse Laplace transform

I want to solve this integral equation using Laplace: $$ Y(t) + 3{\int\limits_0^t Y(t)}\operatorname d\!t = 2cos(2t)$$ if $$ \mathcal{L}\{Y(t)\} = f(s)$$ then, $$ f(s) + 3 \frac{f(s)}{s} = ...
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83 views

Fourier-Laplace Transform of Heaviside Step function multiplied to Sine

In a Advanced Solid State lecture I encountered the following assertion- Fourier Transform of $\Theta(t)\sin(\omega_0 t)$ is ...
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Demonstration with Laplace

How can I demonstrate this? If $F(t)$ is a periodic function with a period of $T>0$, then $$ \mathcal{L}\{F(t)\} = \frac{\int\limits_0^T e^{-st} F(t)\operatorname d\!t}{1-e^{-sT}}\operatorname ...
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70 views

inverse laplace using partial fractions and completing square

what is the inverse Laplace transform of this equation $$\frac{1}{(s+1)(s^2+s+1)}$$ I know that completing the square for the quadratic term is required to avoid complex roots and then I need to use ...
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Integral equation solve using Laplace transform

How can I solve this integral equation using Laplace transform? $${\int\limits_0^{\infty}\ }\frac{e^{-t}(1-\cos t)}{t}\operatorname d\!t$$ Knowing that $$ \mathcal{L}\{\cos t\} = \frac{s}{s^2+1} $$ ...
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Laplace transforms: Convolution

Find $$1*1*1*\cdots*1\quad n\,\,\text{ factors}$$ that is, a function $f(t)=1$ convolution with itself for a total of $n$ factors. Would anyone mind helping me? I have no idea what I should do. ...
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Inverse Laplace Transform of $ \left(\frac{1-s^{1/2}}{s^2}\right)^2$

I found this question in my N.P Bali's Engineering Mathematics 7th Edition. I could not find any solved questions related to this. How can I find the Inverse Laplace Transform of : ...
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60 views

Finding the inverse Laplace transform of $\frac{s^2-4s-4}{s^4+8s^2+16}$

$$F(s) = \frac{s^2-4s-4}{s^4+8s^2+16}$$ My work is as follows, $$\frac{s^2-4s-4}{(s^2+4)^2}=\frac{s^2+4}{(s^2+4)^2}-\frac{8}{(s^2+4)^2}-\frac{4s}{(s^2+4)^2}$$ The inverse laplace of the first term ...
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integro-differential equation with application in quantum mechanics

I am trying to solve for the time dynamics for a simple quantum system (two-site system with sinusoidal coupling and a decay parameter on one site) and the math is looking not so simple. Here is the ...
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557 views

Laplace transformation: second shifting theorem

I know the answer is $1/(s^2) +e^-6s (2/s^3 -14/s -1/s^2 )$, but can anyone tell me how to evaluate the solution? I really get stuck.
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laplace transform with heaviside step function

There are some similar questions around but they didn't help me much. I've got a graph that I'm supposed to perform a laplace transform on. From $0<t<2$, it's a ramp function from $f(0)=0 to ...
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80 views

Solving a differential equation using Laplace transform

The problem has two parts: 1. Solve the initial value problem: $$ y''+y=\sum_{j=0}^\infty \delta_{2j\pi}(t) $$ with the initial conditions: $y(0)=y'(0)=0$ 2.Show that if $2n\pi<t<2(n+1)\pi$ ...
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Why Laplace transfrom uses Exponential function

I probably don't have enough background to post this question, but I am very curious about it. The way I think about the Laplace transform now, the Laplace transform multiplies a given signal by ...
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1answer
64 views

Laplace Transform Piecewise Function

I've never seen these types of bounds on a piecewise function of a Laplace transform before, can someone help explain how to solve this problem, particularly the Laplace transform of g(t)? Thanks in ...
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35 views

What Laplace transformation is used for this?

I cannot see the transition here. I was unable to find what Laplace transformation was used here.
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51 views

laplace transform of a sine function

I'm a little confused about how to find Laplace transforms of a sine function when it is a function of time. As in, suppose the function is $x(t)=\sin(at)$ , then I can proceed to get ...
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How to draw a diagram for the following PDE

Subject: Partial Differential Equations. Here are the details of the question: $$ \frac{\partial ^2u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} = 0 $$ for $0 < x < 3, 0 < y < 1$ ...