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

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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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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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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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40 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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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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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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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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39 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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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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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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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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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 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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66 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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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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23 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 ...
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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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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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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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46 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 ...
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31 views

Existence of Inverse Laplace Transform

Let $F(s)$ the Laplace transform of a function $f(t)$ . Under which conditions on $f(t)$ there exist a unique $g(t)$ such that $g(t) = \mathcal{L}^{-1}\{e^{- F(s)}\}(t) \quad $ ? ...
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134 views

Solve 2nd order ordinary differential equation by Laplace transforms and convolution of their inverse functions. (5.6-39)

Synopsis: I cannot duplicate the answer in my text although it does appear very similar. This tells me that my method is correct but I am making another kind of error -- perhaps in my integration? ...
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How do you solve a homogeneous ODE using Laplace Transform?

Suppose I want to solve $$y'' - 4y = 0$$ all I.C. zero Taking the laplace transform I get $$(s^2 -4)Y(s) = 0$$ So y(t) can be anything. Why am I running into this problem and how can I get around ...
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Uniqueness of Laplace Transform

The theorem for Uniqueness of Laplace transform is as below: Suppose $f$ and $g$ are continuous functions. If $Lf(s)$ = $Lg(s)$ then $f(t)$ = $g(t)$. I am following the proof given in the notes in ...
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Method for taking this inverse laplace transform

i am having trouble taking the answer to this problem that i found a book called "Differential equations with applications and historical notes": $$ e^{-x} = y(x) + 2 \int_0^x\cos(x-t)y(t) dt $$ ...
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28 views

Using a Laplace transform to solve piecewise functions that are also an infinite sums

Consider $y'' + 2y = f(t), y(0)=y(L)$ $$f(t) = \begin{cases}0 \quad 0<t<L/4 \\ 1 \quad L/4<t<3L/4 \\ 0 \quad 3L/4 < t < L \end{cases}$$ Suppose that we can wrtie $f(t)$ and $y(t)$ ...
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43 views

Are all function transforms special cases of Gelfand's transform?

Reading about Gelfand-Naimark theorem I've seen that the Fourier transform is a special case of Gelfand transform for the space $L^1(\mathbb{R})$ with the convolution product. In a related question on ...