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

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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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574 views

inverse Laplace transform of $e^\sqrt{as}$

I am trying to find the inverse Laplace transform of $e^\sqrt{as}$ for $a>0$. So we need to solve $\oint_B dz \: e^\sqrt{az} e^{z t}$ (Bromwich contour), but not sure how to start. How do we even ...
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16 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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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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18 views

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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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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18 views

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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1answer
41 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 in real variable can be extend into complex variable?? [closed]

Let $g(x)$ be a real variable function. if Laplace transform of nonegative random variable $X$ exists on nonnegative real line and the laplace transform $E(exp(-xX))=g(x)$ on $x$ nonnegative real line ...
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17 views

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
33 views

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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6 views

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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45 views

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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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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Inverse Laplace Transform as Bromwich Integral

I am seeking a references that provide a rigorous treatment of the inverse Laplace transform (Bromwich integrals), and how to compute them (beyond using tabled solutions - they don't cover my needs, ...
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How to determine the bounds of integration for an inverse Laplace transform?

I don't completely understand how to find the original time signal when I'm given a Laplace transform and its region of convergence. For instance, if I'm given the Laplace transform: $$ X(s) = ...
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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 ...
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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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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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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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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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19 views

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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2answers
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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

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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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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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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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 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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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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25 views

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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1answer
66 views

Residue Theorem for Laplace Transform

I need to know what's the Residue Theorem for a Laplace Transform. Does anyone know the name or something, so I can search it? I couldn't find anything. For example, if I have this two equations: ...
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Inverse Laplace Transform involving sqrt(s)

Please does anyone know the inverse Laplace transform of the function $$ H(s)=se^{-\sqrt{s}} $$
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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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Compare Fourier and Laplace transform

I would like to clarify main difference between Fourier and Laplace transforms and also understand if exponential factor is main difference between this two method. So Fourier transform is ...
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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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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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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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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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1answer
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Discrete Laplace transform. Analogy to change of basis

Assume $$f=\sum_{k=0}^{N-1} c_k\cdot E^{k}$$ where the vector $E^k$ is $$E^k = (e^{2 \pi i k\cdot M}(0),e^{2 \pi i k\cdot M}(1),\cdots,e^{2 \pi i k\cdot M}(N-1))$$ (M is a constant and e represents ...
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104 views

Calculating Inverse Laplace Transform of stretched exponential

I am trying to solve a Laplace transform problem that has gotten way over my head in terms of complex analysis knowledge. I would like to solve the Inverse Laplace Transform $(s\rightarrow t)$ of ...
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How to find Bilateral Laplace Transform of $e^{at}$ Using Changing of the Time Horizon

Ok, this has me a bit stumped. In my class the teacher "showed" us how to find the bilateral Laplace transform of x(t)=$e^{at}$ where $-\infty<t<\infty$. Breaking them into the two parts ...
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Laplace transform of $1/t$

Does the laplace transform of $1/t$ exist? If yes, how do we calculate it? Putting it in $$\int_0^\infty (e^{-st}/t) dt$$ won't solve. Is there any other way? If not, why? Thanks!
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Wave Equation with outgoing wave boundary conditions

I need some help with this problem: I have a to solve the wave equation with two initial conditions and with outgoing wave boundary conditions; i.e., $$\begin{cases} u_{tt}-u_{xx} & =0\\ u(x,0) ...
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1answer
43 views

Solve 2nd order ordinary differential equation by Laplace transforms and convolution of their inverse functions. (5.6-40)

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 ...
2
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2answers
7k views

Inverse Laplace Transform of s/(s+1)

What is the inverse laplace transform of $\frac{s}{s+1}$? My work was: $$ X(s)=\frac{s}{s+1}\\ X(s)=s\frac{1}{s+1}\\ x(t)=\frac{d}{dt}e^{-t}=-e^{-t} $$ My only issue is that when I check my answer ...