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 of $\cosh(bt)$

So, one of my homework assignments is to take the Laplace transform of a function such that $f(t)=\cosh{bt}$. I figured it would be equivalent to: ...
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27 views

What is the General procedure for graphing heavidside functions?

I was given an example of a second order differential equation with U1(t)-U(3t) as the forcing function. I was asked to graph the forcing function and the answer is that the function is 1 when t is ...
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33 views

Inverse Laplace transform and convolution

Suppose we have two functions of $t$, $f(t)$ and $g(t)$. Letting $\mathcal{L}\{f(t)\} = F(s)$ and $\mathcal{L}\{g(t)\} = G(s)$, I know that: $\mathcal{L}\{f(t) \star g(t)\} = F(s) \cdot G(s)$, but ...
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36 views

Finding the Laplace Transform Inverse

Solve by Laplace Transforms. So I'm stuck on how to find this $\mathcal{L}^{-1}$ $( \frac{\frac{5s}{4} + \frac{13}{4}}{s^2+5s+8} ) $ I'm not sure what t odo. I was thinking I need to use the ...
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1answer
179 views

A bessel function integral

$$\int_{-\infty}^{\infty} dy \frac{J_1 \left ( \pi\sqrt{x^2+y^2} \right )}{\sqrt{x^2+y^2}} = \frac{2 \sin{\pi x}}{\pi x} $$ How do I show this?
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Solving second order nonhomogeneous differential equation with non-constant coefficients using Laplace Transform

$ty''(t) + y'(t) -ty(t)= tf(t)$ How to solve the problem using Laplace Transform? Using Laplace transform I got $$Y(s)= C(s^2-a^2)^{-1/2} + (s^2-a^2)^{-1/2}\int (s^2-a^2)^{-1/2}F(s)\,ds$$ where ...
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45 views

Final value theorem for closed system

We have a system with output given by $\frac{Y(s)}{R(s)} = \frac{F(s)G(s)}{1+F(s)G(s)}$ where $F(s)G(s) = K\frac{s+1}{s^2+s+1}$. Let $K=4$ and $R(s) = 10/s$. Using the final value theorem, ...
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solve $y''-4y=\delta(t-1)$ with initial conditions $y(0)=0, \; y'(0)=1$ using Laplace transforms

I took the Laplace transform and solved for $Y$ which resulted in $Y=\frac{1+e^{-s}}{s^2-4}$. I began to break up the problem separating the result into two equations but the fact that there is a $1$ ...
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1answer
34 views

Checking region of validity for a standard Laplace transform

In these notes, http://www.math.psu.edu/papikian/Kreh.pdf, Theorem 2.14 it states that $$\mathcal{L}[J_0](s)=\frac{1}{\sqrt{1+s^2}} $$ which I suppose is equivalent to $$ ...
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36 views

Laplace Transform to solve $R\frac{dQ}{dt}+\frac{Q(t)}{C}=V(t)$

I have the differential equation $R\frac{dQ}{dt}+\frac{Q(t)}{C}=V(t)$ where $R,C\in\mathbb R$ and $Q,V$ are functions of $t$. If I take the laplace transform of the differential equations I get: ...
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1answer
39 views

Final value of 1/(( s+2 )² * (s² - s + 1)) in the time domain

The original question is given as $$\frac {d^3y}{dt^3}+y=u=(1-t)e^{-2t}$$ The initial value y(0) = 0 and the same for all derivatives of y. Determine Y(s) What happens to u(t) and y(t) when ...
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1answer
49 views

Solve ordinary differential equation using Laplace transform

I have trouble to solve the differential equation. I can write derivatives of Laplace transforms but I can't do anything $$ \ddot y(t)+3y(t)=\sin(t)\text{ with } y(0)=1,\,\dot y(0)=2 $$
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1answer
56 views

Computing the Laplace transform of $\frac {f(x)}{x}$

I am having trouble computing the following Laplace transform: $\frac {f(x)}{x}$. From Wikipedia it should be equivalent to this: $\int_s^\infty F(\sigma) \,d\sigma$ . What I've done so far is ...
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1answer
41 views

How does one find the Laplace transform for the product of the Dirac delta function and a continuous function?

As an example, what is the Laplace transform for the following: $$g(t)=\delta(t-2\pi) cos t$$ I've worked through a few examples that required finding $\mathcal{L}\{\delta(t-t_0)\}=e^{-st_0}$, but ...
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Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} ...
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1answer
45 views

The Laplace Transform of Piecewise Function

Write the following as an unit step function and find the Laplace transform. $f(t)=\begin{cases}{t}&0 \leq t < 3\\ 3&3 \leq t < 4\\ 11-2t& 4 \leq t < 5.5 \\ 0&t \geq 5.5 ...
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1answer
67 views

Use Laplace transform to solve initial value prob.

The problem is: $y" + 9y = e^t$, with the initial conditions $y(0) = 0, y'(0) = 0$. I'm stuck at the inverse Laplace transform part. Do I have to use partial fraction expansion or can I just split ...
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1answer
50 views

Solving Laplace $\nabla^2 \phi=0$ in $x,y \geqslant 0$

I'm trying to solve $\nabla^2 \phi=0$ in $x,y \geqslant 0$ $\phi(x,y)=0 $ as $x^2 +y^2 \rightarrow \infty$ $\phi_x(0,y)=0$ and $\phi(x,0)= \frac{1}{1+x^2}$ I know the solution is ...
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Why is this laplace identity true $\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$?

I was wondering why this laplace identity is true? Does it follow from definition? $$\int_{\Bbb{R}^+}\frac{f(t)}{t}\,dt = \int_{\Bbb{R}^+}\mathcal{L}\{f\}$$ I'm trying to understand the first ...
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Laplace Transform of an Piecewise Function

Write $f(t) = \begin{cases} 5,& \mbox{if} \quad 0 \leq t \lt 3 \\ -4,& \mbox{if} \quad 3 \leq t \lt 7 \\ 0,& \mbox{if} \quad t \geq 7 \end{cases}$ as a unit step function and find ...
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inverse laplace tranform

I have a simple question, There are some functions f(t), g(t) and lets say F(s) and G(s) for the form of Laplace transform of f(t) and g(t), respectively. While I am solving differential equation ...
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Describe the diffrence between the following two problems and give an example of a physical situation which may be modeled by each equation

$y'' + y =\mu_\pi \big(t\big)$ $y''+y= \delta (x- \pi )$ wih initial conditions: $y \big(0\big) =0$ $y' \big(0\big) =0$ It is obvious to me that the first equation is a Heaviside distribution ...
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Show that s * exp (- s * inf) = 0 ? (s complex)

Reading on control theory and the Laplace transform of the unit step function, I came upon the following in my textbook. The Laplace transform defined as: $$Y(s)=\int_{0}^{\infty}y(t)e^{-st}dt$$ s ...
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1answer
92 views

Solve the Integral Equation Involving Laplace Transforms

I want to solve $\int^\infty_0x'(T)x(t-T)dT=6t^3$ where $x(0)=0$ I did the Laplace transform to both sides, and the left side is a convolution, so I then have $X(s)x(s)=\frac{36}{s^4}$, but here I'm ...
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Laplace transform of a linear input vs output line?

I think my intuition of the Laplace transform and transfer functions is broken. Suppose I have a linear function which relates two quantities r to x as such: $$ r(x) = -100x + 25 $$ i.e. a ...
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1answer
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Is multiplication commutative in the laplace domain?

I'm studying control theory and saw this picture explaining some of the basic rules. My question is if we could also say that Y(s) = (G2(s) * G1(s)) * U(s) Or Y(s) = U(s) * G2(s) * G1(s) I'm ...
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Laplace Transform to evaluate an integral

Compute $\displaystyle\int_{0}^{\infty} \frac{\cos(x)}{x^2 + a^2} \mathrm{dx}$, for $a\in \mathbb{R}$ using the Laplace Transform. I'm not sure on how to start with this problem. I tried to first ...
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Find the Inverse Laplace Transforms

Find the inverse Laplace transform of: $$\frac{3s+5}{s(s^2+9)}$$ Workings: $\frac{3s+5}{s(s^2+9)}$ $= \frac{3s}{s(s^2+9} + \frac{5}{s(s^2+9)}$ $ = \frac{3}{s^2+9} + \frac{5}{s}\frac{1}{s^2+9}$ $ ...
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Solve integral (convolution) equation

Given a function: $u(t) = \exp\left( -\frac{At^2}{1+t}\right),$ $A>0, t>0,$ and an equation: $\frac{d u(t)}{dt} = \int^{t}_0 \phi(t-\tau) u(\tau) d \tau .$ How to find a closed expression for ...
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59 views

Transfer function of differential eqaution

I'm trying to find out the transfer function of simple differential equation: $$a_0\dot y + a_1y=b_0x+b_1$$ The problem is i have no idea what to do with $b_1$. If we apply the Laplace transform ...
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Maximising a net present value function

I am looking at an equation for profit derived from fishing operations. This is defined in terms of a bounded integral (with an upper bound of $+ \infty$), so it's a Laplace transform really. It gives ...
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Using the Laplace Transform solve $y''+6y'+5y=e^t$

The initial conditions are $y(0)=0$ and $y'(0)=1$. I began the process and ended up with $Y=1/(s-1)(s^2+6s+4)$. Since the second factor in the denominator does not factor so I have a feeling I messed ...
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1answer
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Laplace, Correct Use of the Second Shift Theorem

I have invested some time now trying to understand how to use the Second Shift Theorem, mostly by doing the full integration first. What threw me off at first, I discovered, is that almost all books ...
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1answer
41 views

Inverse Laplace tranform via the table formulas

In my inverse Laplace table there is this inversion "formula": $(1) \frac{1}{s-a} \rightarrow e^{at}$ I understand that $\mathcal{L}^{-1}[\frac{1}{s+4}] = \frac{1}{2}\sin(2t)$ But why can I not do ...
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1answer
28 views

Laplace Transform

Let us assume that complex-valued differential equations as follows $\dot{z}(t)=-Az(t)+Bz(t-\tau)$, $z\in \mathbb{C}$ How to find the solution of the above equation by using Laplace transform.
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Discrepancy with the book's solution and mine of Laplace transform of a piecewise defined function

Determine the Laplace transform of $f(t)$ below: $$ f(t)= \begin{cases} 0, & \text{if } t < 2 \\ (t-2)^2, & \text{if } t \geqslant 2 \end{cases} $$ So my answer is $$ 2e^{-2s}/s^3 $$ ...
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1answer
94 views

Transfer function for double cart system

System: Define X2 = Y2; I've described the system with the following diff equation: $$f_{tot} = m_1\ddot{x_1} + k(x_2-x_1)+m_2\ddot{x_2}+B(\dot{x_2}-\dot{x_1})$$ where m1, m2, k and B are Cart ...
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Exponential Order: $\forall t>M$ or $\forall t>0$?

The following comes from the discussion of Laplace transformation in ODE. Let $f(t)$ be piecewise continuous on $[0, \infty)$ and of exponential order. Prove that there exist constants $K$ and ...
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Laplace Transform for a difficult function

The Laplace Transform I'm having trouble with is: $$f(t) = 6te^{-9t}\sin(6t)$$ I'm not sure what the protocol is for multiplying t into it. The Laplace Transform for $f(t) = 6e^{-9t}\sin(6t)$ is ...
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Partial fraction expansion for non-rational functions

With regard to this answer to an inverse Laplace transform question, I derived the following result: $$\frac1{i 2 \pi} \int_{c-i \infty}^{c+i \infty} ds \, e^{s t} \Gamma(s)^2 = 2 K_0 \left ( 2 ...
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Solve for inverse Laplace transform using non-repeating complex partial fractions. (5.7-4)

Synopsis: Please check my work. I do not have a text "answers to odd problems" for reference as this is an "even" numbered problem. The following documents in good detail the steps taken to solve for ...
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Probability proof of inversion formula for Lapalce transform

Let $f:[0, \infty[\longrightarrow \mathbb{R}$ be bounded and continuous and define $L(\lambda)=\int_0^\infty e^{-\lambda x}f(x)dx$. Let $X_n$ be a sequence of independent random variables with ...
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Finding the Laplace transform of the solution of the given IVP

Find the the Laplace transform $Y(s)$ of the solution of the given initial value problem $$y''+y=\begin{cases}t & 0 < t < 1 \\ 0 & 1 < t < \infty \end{cases}$$ $$y(0)=0$$ ...
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Evaluate Integral with $e^{ut}\ \Gamma (u)^{2}$

I am trying to integrate this integral: $$f(x)=\frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty}x^{-s}\sigma ^{ms-m}\left [ \frac{\Gamma \left ( \frac{s}{\beta} \right )}{\Gamma \left ( \frac{1}{\beta} ...
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laplace transformation of a function using definition

I want to find the laplace transformation of $x^ne^{ax}$ using the definition. I'm stuck with the integral. How shall I proceed the integral and find the final answer with $n$?
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What is the laplace transform and how is it performed? (detailed explanation)

I am a high school student and I became interested after someone mentioned it. Although I am not quite at the level where I am taught this it just captured my attention. Could someone give me an ...
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Different proofs of uniqueness of the Laplace transform

How many different types of proof do you know for the so-called Lerch's theorem, i.e., uniqueness of the Laplace transform? I have found the following references for proofs. New books, in general, do ...
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102 views

Solve 2nd order ordinary differential equation with unit-step driving function by Laplace transforms and convolution theorem. (5.6-42)

Synopsis: Please check my work. I do not have a text "answers to odd problems" for reference as this is an "even" numbered problem. The following documents in good detail the steps taken to solve for ...
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0answers
26 views

Inverse Laplace transform of $ \frac{7s-6}{s^2-s-6}$

IT is asked to find the inverse Laplace transformation of $$\frac{7s-6}{s^2-s-6}$$ Writing it with partial fractions $$\frac{7s-6}{s^2-s-6} =\frac{4}{s+2}+\frac{3}{s-3}$$ Ive found that the ...
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Partial Fraction Decomposition for Laplace Transform

As part of trying to solve a differential equation using Laplace transforms, I have the fraction $\frac{-10s}{(s^2+2)(s^2+1)}$ which I am trying to perform partial fraction decomposition on so that I ...