Questions tagged [laplace-transform]

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

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$y''+y=x^2+1, y(\pi)=\pi^2, y'(\pi)=2\pi$ - By Laplace Transform

I am solving the following IVP by Laplace Transform: $$y''+y=x^2+1,\qquad y(\pi)=\pi^2, \qquad y'(\pi)=2\pi$$ Let $f(x)=u_{\pi}(x)y(x-\pi).$ Then, $$f''(x)+f'(x)=u_{\pi}(x)(x-\pi)^2+u_{\pi}(x), \qquad ...
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A Laplace transform of a certain family of rational functions.

I was always interested in computing Laplace transforms. It was already during the course of my studies in the subject of electric circuits that I encountered this technique. Again, by using Laplace ...
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What's the formal (matrix) expression for Laplace transform?

This answer gives the formal expression for Fourier transform: ${\mathcal {F}}=e^{{\frac {\pi i}{4}}(D^{2}-x^{2}+1)}$. What's a similar expression for Laplace transform?
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Generalizing $\int_{0}^{\infty }\frac{f(t)}{t}dt=\int_{0}^{\infty }\mathcal{L}\left \{ f(t) \right \}ds$.

Here, I saw the following formula: $$\int_{0}^{\infty }\frac{f(t)}{t}dt=\int_{0}^{\infty }\mathcal{L}\left \{ f(t) \right \}ds$$ Say we have the integrand only $f(t)$, not $\frac{f(t)}{t}$, then I ...
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Heat equation using Laplace transform with IC equal to zero

I was trying to solve the following heat equation using the Fourier transform $\frac{du}{dt}=\frac{d^2u}{dx^2}$ With initial boundary condition $\hspace{1cm}u(x,0)=0\hspace{1cm}$ for $\hspace{1cm}0<...
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Laplace transform of natural logarithm of a function

Let's assume that we have a function $i(t)$ and it has a Laplace transform $I(s)$. Can we calculate the Laplace transform of $\ln(i(t))$ in terms of $I(s)$ ?
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Time Laplace transform for Lévy process?

For which Lévy processes do I know the time Laplace transform? So if $p(t,x)$ is the density function of the process, when do I know \begin{align*} \int_0^\infty e^{-\lambda t} p(t,x) dt, \quad \...
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What's the distribution of the integrated OU process?

Suppose that we have an OU process $U(t)$ satisfying the following SDE $$ {\rm d}U(t)=-aU(t){\rm d}t+b{\rm d}B_t, $$ where $B_t$ is Brownian motion. Given $t$, what is the distribution of $\int_0^tU(...
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Solving integral equation with Laplace transform [closed]

Using Laplace transformation, solve : y(t)=sin2t+int_0^t y(u)sin(t-u)du Answer..?
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Zero crossings of the function $f_x(t) := \sum_{i=1}^nx_i e^{-\lambda_i t}$

Let $\lambda_1,\lambda_2,\ldots,\lambda_n$ be positive real numbers. For any $t \ge 0$ and $x=(x_1,\ldots,x_n) \in \mathbb R^n$, define $f_x(t) := \sum_i x_ie^{-\lambda_i t}$. Question. Given $s_1,\...
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Extend an operator from exponential functions to all functions?

I have an operator on the half line $\mathbb{R}^+$, I know its behavior on exponential functions. In particular, given a function $g$ \begin{align*} \mathcal{A}_g e^{-tk}=g(k) e^{-tk}, \quad t>0. \...
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Laplace problem with t-axis translation [closed]

"Be $G(s)$ the Laplace transform of the function: $$g(t) = u(t-3,5) + t u(t-2,4)$$ What is the value of $G(1)$ ?" Pretty sure it involves some Translation on the $t-$axis, but couldn't go ...
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Laplace transform of homogenous differential equation with no initial values

I have the following differential equation: $$ \frac{d^2y}{dt^2}+5\frac{dy}{dt}+6y=0 \\ y(0)=0\\ \frac{dy}{dt}(0) = 0\\ $$ To solve this differential equation I want to use the Laplace ...
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Inverse Laplace transform via inverse Fourier transform?

Laplace transform can be represented via Fourier transform the following way: $$\mathcal{L}_t[f(t)](x)=\frac{i}{2} \mathcal{F}_t[i t f(i t) \text{sgn}(t)](x)$$ But how to represent inverse Laplace ...
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Inverse Laplace without Partial Fractions [closed]

How do I find Inverse Laplace of $s^3/(s^4+4a^4)$ without using Partial fractions. I solved it using Partial Fractions but I wonder if there is some way solving it using properties of Laplace ...
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Critically damped system with zeros

I have asked these 2 questions in some other forums before, but it always have been buried deep with my other questions, resulting to it being overlooked. However, I think that these 2 questions ...
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A question on laplace transform

I try to solve the following question on Laplace transform $$L(\{ \int_0^{t}e^{-x^2}\})$$ I solved as following: $$L(\{ \int_0^{t}e^{-x^2}\})=\frac{1}{s}L(\{e^{-x^2}\})=\frac{1}{s}L(\{\sum_0^{\infty}\...
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Complex Fraction magnitude

I'm having a problem with finding the magnitude of the transfer function. $$G(s) = \frac{s+20}{(s+1)(s+100)}$$ I know that if we have a simple complex in the rectangular form $$Z = a + jb$$ The ...
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simple pendulum equation, why it cannot be solved with laplace transform (the general solution)

Usually to solve the simple pendulum equation: $\qquad \ell {\ddot \theta }+g\sin \theta =0\,$ Using the first term of Taylor series is used as approximation, but although $\sin \theta$ can be ...
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Working out Laplace transform of $t^2g'(t)$

I am stuck on the LT of $t^2g'(t)$. I have uploaded the lecturers working, but I don't know how they get from $$\frac{d^2}{ds^2}(sL[g(t)](s) - g(0))$$ to the final answer of $$s\frac{d^2}{ds^2}(L[g](s)...
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Partial fractions with products in numerator and denominator

Given Laplace Transform $$Q(s)= \frac{a_{1}\left(s-b_{1}\right)\left(s-b_{2}\right)}{\left(s+a_{1}\right)\left(s+a_{1}+a_{2}\right)} $$ I need to find the partial fractions decomposition of the sort $ ...
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band-pass filter poles and zeroes of even function

let $ \omega_H>\omega_L>0$ be frequencies. I need to create an LTI non ideal band-pass filter so that$$ Y\left(s\right)=\begin{cases} 1 & \omega_{L}<\left|\omega\right|<\omega_{H}\\ 0 &...
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Find a sequence $(b_n)_n$ such that $\sum_n b_n = 0$ and $g(t) \equiv \sum_n e^{-2\lambda_n t}a_n^2 + \sum_n e^{-\lambda_n t} a_n b_n + \sum_n b_n^2$

Fix a sequence of real numbers $(a_n)_n \in \ell^2$ and a sequence of positive numbers $(\lambda_n)_n \in \ell^2$. Question. What is a crisp characterization of functions $g:[0,\infty) \to [0,\infty)$,...
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Simplification Error Inverse Laplace Transform

I am trying to build a model of a electronic circuit and to solve the differential equation using the laplace transformation. With the help of numeric simulation I know what one factor has to be and ...
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Laplace transform of a random variable with a constant

Let $W_0=p\delta_0 + (1-p)\text{exponential}(1-p)$, where $0<p<1$, $\delta_0$ is the point mass at zero and exponential($\cdot$) is the exponential distribution. How do I calculate the Laplace ...
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Convolution Equation Solving [closed]

Solving the equation $$y'(t)=\sin(5t)-25 \int\limits_{0}^{t} y(u)\,\mathrm du$$ with $y(0)=0$, we obtain the convolution $$y(t)=\sin(5t)\cdot g(t)$$ For some function $g(t)$. What is the value of $g\...
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Use Residues to find the inverse Laplace transform $F(s)=\frac{2s^3}{(s^2-4)}$

Use Residues to find the inverse Laplace transform $F(s)=\frac{2s^3}{(s^2-4)}$. The answer from the text book is $f(t)=\cosh^2(t)+\cos^2(t)$. But my result is $2\cos^2(t)\cdot \cosh^2(t)$. Which is ...
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Inverse Laplace transform of $\dfrac{1}{s(e^s+1)}$

The original problem is to solve $$\mathcal{L}^{-1}\left\lbrace\frac{e^s}{s(e^s+1)}\right\rbrace.$$ Doing partial fractions $$\frac{e^s}{s(e^s+1)}=\frac{1}{s}-\frac{1}{s(e^s+1)}$$ the problem reduces ...
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With what classes of functions the equality $\int_0^\infty f(x)\,dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx$ leads to paradoxes?

The following operators keep the area under the convergent integrals unchanged: $$\int_0^\infty f(x)\,dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx=\int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)](x)\,...
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get first ODE general solution using integrating factor and laplace transform

$y' + ay = h(t) , y(0)= b $ is the question. I get $e^{at}$ as integrating factor, but I don't know where it uses while doing laplace transform.
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Visual interpretation of the Laplace-transform

I was wondering if there is a visual interpretation for the Laplace-transform. For example, you can visualize integrals by sketching the area under the graph. That way it has a visual meaning. I’m ...
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In what cases the transform $\mathcal{L}_t[t f(t)](x)$ preserves the ordering?

Let's consider the set of integrable functions $f:[0,\infty)\to(-\infty,\infty)$ with countable number of singularities. Let's define order in such a way that $f>g$ if and only if $\int_0^\infty (f(...
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Laplace transform of $te^{-2t}\sin(2t)u(t-3)$

Laplace transform of $te^{-2t}\sin(2t)u(t-3)$ I do know the following properties of Laplace Transform: A) $t f(t) = \frac {dF(S)}{ds}$ B) $e^{at} f(t) = F(S+a)$ But from what I see a part of the ...
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What is inverse Laplace transform of Dirac delta?

I know that the Bromwich integral to get inverse Laplace transform, which is given by $$ \mathcal{L}^{-1}(F(s))=\frac1{2\pi i}\lim_{M\to+\infty}\int_{\sigma-iM}^{\sigma+iM}F(s)e^{st}ds $$ , $\sigma$ ...
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Solving $\int_0^x\int_0^t f(u)\,du\,dt$ using Laplace transform

So I've been trying to solve this problem with Laplace transform. But the problem is I don't know the function and therefore couldn't even get close to the answer! $$\int_0^x\int_0^t f(u)\,du\,dt=\...
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Trouble with ODE by Laplace transform with boundaries

From section 5.2 of Zill's book Differential equation with boundaries problems, I have to resolve the next equation with Laplace transform \begin{equation} \frac{d^{2}}{dx^{2}}\left(EI\frac{d^{2}y}{...
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why is the Laplace transform of local integrable function with support on $[0,\infty)$ analytic?

There is a proposition about Laplace transform, but I don't know how to prove it. Let $f \in L^1_{loc}(\mathbb{R})$, $\operatorname{supp}(f) \subset[0, \infty)$, such that $a$ is the abscissa of ...
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What's wrong with this Laplace transform?

The following operators keep the area under the convergent integrals unchanged: $$\int_0^\infty f(x)\,dx=\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx=\int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)](x)\,...
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absolut convergence of the Laplace transform of a intensity measure

Let $\xi$ be a point process and $\mu$ its intensity measure, i.e. $\mu(\cdot)=\mathbb{E}[\xi(\cdot)]$. The Laplace transform of $\mu$: $\mathcal{L}\mu(z)=\int_{0}^{\infty}e^{-zx}\mu(dx)$ converges ...
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Modified Bessel function and laplace transform?

WILLING TO PAY Hi, I'm doing revision for Laplace transforms and I have absolutely no idea how to complete this question properly. THOUGHTS So I'm just following what they have told me to do and ...
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Find a function $g$ such that $(1*g)(t)=t^{-1/2}$.

Is there some function $g: (0,+\infty) \to \mathbb{R}$ satisfying $$(1*g)(t)=t^{-1/2}$$ for all $t>0$? Here the sign $*$ represents the convolution. I tried apply the Laplace transform both site of ...
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Final value theorem for non-rational transfer functions

One version of the Final Value Theorem often seen in controls textbooks: Suppose $f(t)$ has (one-sided) Laplace Transform $F(s)$ and further suppose that every pole of $s F(s)$ is in the open left-...
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Laplace transform of measure on complex plane

I construct Laplace transform of measure on complex plane,i.e. $F(t)=\int_\mathbb{C}e^{-\lambda t}d\mu(\lambda), t \in \mathbb{R},\mu(\mathbb{C})=1$ and have bounded support. Suppose that $F_n(t)=\...
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laplace transform of two different functions

Is it possible for two different functions to have the same laplace transform?.In this sense how do we know that the inverse laplace transform gives exclusively one function?.if it is not possible ...
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A strange trigonometric inequality

I am trying to check that: $$f(t) = t^2(t-3) + 2 e^{-t/2} + e^{t/2}\left(\sqrt{3} \sin \frac{\sqrt{3}t}{2} + \cos \frac{\sqrt{3} t}{2} \right) - e^{-t/2} \left(\sqrt{3} \sin \frac{\sqrt{3}t}{2} + 3\...
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Good source to study the Laplace transform.

I am studying the theory of semigroups and its links with the spectral theory and the Laplace transform turns out to be the intermediary between the two. Any suggestions for good sources?
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Inverse Laplace: apply successively shift and scaling

I am looking for a function $f(t)$ whose Laplace transform is given by $F(s)$: \begin{equation} F(s)=\frac{(1+\rho s)^\alpha}{(1+\rho s)^\alpha-1} \end{equation} Now, I know from this table the ...
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Bound of growth of function from its Laplace transform

This question is a follow up question on Decay of a function from its Laplace transform. Let $f:[0, \infty) \to \mathbb{R}$ be a continuous function whose Laplace transform $$ F(s) = \int_0^\infty f(t)...
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Finding Inverse Laplace Transform of exponential and quadratic term together?

I'm having trouble finding the inverse Laplace transform of: $$G(s) = \frac{1}{2s^2}e^{-Cs}$$ where C is a constant. I know that the inverse Laplace transform of $s^{-2}$ on its own is $t$, and that ...
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Expressing the convolution integral of scaled and translated arguments

I was reviewing the Fourier transform where I came across the convolution integral. If the convolution $x(t)*y(t) = \int_{-\infty}^{\infty}x(\tau)y(t-\tau)d\tau$ be defined, how the convolution of $x(...
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