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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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Variance of a Laplace Transform

I have a function $F(s)$ which is the Laplace transform of $f(t)$ (which is in itself a normally distributed random process), but I don't know what $f(t)$ is (this comes from solving a differential ...
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35 views

Finding $y(t)$ in a causal system given an input-output relationship

I want to find $y(t)$ in a causal system with input-output relationship $$\frac{dy(t)}{dt} + 3y(t) = x(t)$$ where $$x(t) = e^{2t} \cdot u(-t).$$ Here, $u$ is the Heaviside function. To try and ...
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Taking the Laplace transform of the derivative of some function

I need to find the Laplace transform of $\dfrac{dh(t)}{dt}$ where $h(t) = e^{-100t} \cdot u(t)$ and $u(t)$ is the Heaviside function. I have two ideas for solving this problem but I am unsure which ...
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Solving O.D.E and Initial Values Problem using Laplace Transform

I have this ODE: $$ y'' + y = \begin{cases} \cos t, &\text{ if }0\le t \lt \pi\\ t-\pi,&\text{ if }\pi \le t \lt \infty \end{cases} $$ The initial values are: $$ y(0)=0 \\ y'(0)=0 $$ I ...
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Inv. Laplace $\frac{1}{s}\frac{1}{\frac{\sqrt{sAB}}{A} \sinh \sqrt{sAB} \frac{C}{\sqrt{sCD}} \sinh \sqrt{s CD}+\cosh{\sqrt{sAB}} \cosh {\sqrt{sCD}} }$

What would be the inverse Laplace of the following function? $\frac{1}{s}\frac{1}{\frac{\sqrt{sAB}}{A} \sinh \sqrt{sAB} \frac{C}{\sqrt{sCD}} \sinh \sqrt{s CD}+\cosh{\sqrt{sAB}} \cosh {\sqrt{sCD}} }$ ...
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Inverse Laplace transform of $F(s)=\frac{1}{s\sqrt{s+b\sqrt{as}\tanh{(\sqrt{as})}}}$ using complex integration

I want to find the inverse Laplace transform of $$F(s)=\frac{1}{s\sqrt{s+b\sqrt{as}\tanh{(\sqrt{as})}}}$$ I tried to use the Bromwich integral $$f(t)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\...
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Laplace Transform with Multiplied Terms

I have to take the laplace transform of $te^{3t}\cos(3t)$ So I started out with the property $L\{t^nf(t)\} = (-1)^n\frac{d}{ds} L\{f(t)\}$ Which yielded $-s^2-b^2+2s^2/(s^2+b^2)^2$. If I did it ...
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Solve system differential equation using Laplace transform

Question : Solve system differential equation using Laplace transform : $\cases{x'-3x+2y= \sin t\\4x-y'-y=\cos t\\x(0)=0,y(0)=0}$ My try : Take Laplace we find : $\cases{(s-3)X(s)=\frac{1}{s^...
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Numerical inverse Laplace

I have this function \begin{equation} F(s) = \exp\left(a s + \frac{b}{s+\gamma}\right), \end{equation} for which I need to compute the inverse Laplace transform, at least numerically. I can compute ...
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57 views

Get the equation of the plot of the response of the transfer function

In the case of permanent magnet DC motor position control, I wanted to rotate my motor shaft along a reference trajectory. Then I designed a piecewise function for reference and simulated the response ...
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20 views

Partial integro-differential equation using Laplace transform

Is it possible to solve the linear PDE analytically \begin{equation} \frac{\partial u}{\partial z} + a \frac{\partial u}{\partial t} + \int_{0}^{t} e^{-\beta (t-t')} u(z,t') dt'=f(z,t), \end{equation} ...
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Find inverse Laplace Transform of : $\frac{1}{(s^2+a^2)^2}$

Question : Find inverse Laplace of : $$\dfrac{1}{(s^2+a^2)^2}$$ My try : $$\dfrac{1}{(s^2+a^2)^2}=-\frac{1}{2s}\frac{\mathrm d}{\mathrm ds}\left( \frac{1}{s^2+a^2}\right)$$ I need use this ...
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Find inverse Laplace transform of : $\ln(\frac{s^2+a^2}{s^2+b^2})$

Question : Find inverse Laplace transform of : $$\ln \left(\frac{s^2+a^2}{s^2+b^2}\right)$$ My try : I'm trying use this identity : $f(t)=-\frac{\mathcal{L}^{-1}(\frac{dF(s)}{ds})}{t}$ ...
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Solving integral equation via Laplace transform

I want to solve the following integral equation $$\int_0^\tau \ddot{\Psi}(t-t')g(t')dt'= g(\tau^-)\dot{\Psi}(\tau-t)+g(0^+)\dot{\Psi}(t),$$ where $\Psi$ is a even function, $g(\tau^-)\neq g(0^+)$ ...
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Find Laplace transform of $t-\pi$

I am dealing with an Initial Value Problem of a step function: $$ y'' + y = \begin{cases} \cos t, &\text{ if }0\le t \lt \pi\\ t-\pi,&\text{ if }\pi \le t \lt \infty \end{cases} $$ I am ...
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2answers
41 views

Inverse Laplace transform of $f(s)={\frac{1}{s^{3/2}}}$ using complex integration

I want to find the inverse Laplace transform of $$f(s)={\frac{1}{s^{3/2}}}$$ Refer to the Laplace transform table, and I found that the result is $$F(t)=2\sqrt{\frac{t}{\pi}}$$ But I do not know how ...
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Partial fraction expansion of $\frac{1}{(s+1)^{2}(s-1)(s+5)}$

I'm seeking a partial-fraction expansion of $A=\frac{1}{(s+1)^{2}(s-1)(s+5)}.$ I was solve equation differential using Laplace transform, but I need use partial fraction of $A$.
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Inverting a Laplace Transform with the Residue Theorem

I'm trying to invert the following Laplace Transform using the Residue Theorem: $$ F(s) = \frac{K_1(\sqrt s)}{\sqrt s K_0(\sqrt s)} $$ where $K_0()$ and $K_1()$ are the Modified Bessel Functions of ...
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Clasify the output $y(t)$ knowing that $x(t)=e^{-t}$, $t>0$

Given the transfer function $$G(s)=10\frac{s+1}{s^2-2s+5}$$ clasify the output $y(t)$ given the input $x(t)=e^{-t}$, $t>0$. I know that $$y(t)=\mathscr{L}^{-1}[Y(s)]=\mathscr{L}^{-1}[X(s)G(s)].$$ ...
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2answers
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Inverse Laplace transform of $\frac{π\cosh(\sqrt s)}{2 s^{3/2} \sinh(\sqrt s)}$ using complex integration

I want to find the inverse Laplace transform of $$F(s)=\frac{π\cosh(\sqrt s)}{2 s^{3/2} \sinh(\sqrt s)}$$ using the Bromwich integral $$f(t)= \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{π\...
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1answer
141 views

Asymptotic behaviour of Laplace transform

If the functions $x(t)$ and its derivatives $x'(t), x''(t), \ldots, x^n(t)$ are continuous* and $x(0^+) = x'(0^+) = x''(0^+) \ldots = x^{n-1}(0^+)=0$ ($0^+$ denotes the right side limit when the ...
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Solving the heat equation via Laplace Transform

Question: Let $u=u(y,t)$. Solve the following PDE (heat equation) in the region $y,t>0$: \begin{align} & \frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial y^2} = \cos(t) \\ & u(...
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2answers
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What is the solution to linear ODE $\dot x = Ax + b$?

Suppose I have a system of linear ODE $$\dot x = Ax + b$$ where $A \in \mathbb{R}^{n \times n}, b \in \mathbb{R}^n$ I know that when $b = 0$, using Laplace transform we have $x(s) = (sI-A)^{-1} x(0)$...
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1answer
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A proof for the Final Value theorem using Dominated convergence theorem

I'm going over the proof for the Final Value theorem using the Dominated Convergence theorem on Wikipedia, I don't understand how from the equation $$sF\left(s\right)=\int_{0}^{\infty}f\left(\frac{t}{...
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Mittag-Leffler function and Fox-Wright function

I find the following identity in many special functions books without proof. This identity is called the Laplace transform of the Mittag-Leffler function with three parameters. The result is in the ...
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1answer
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Laplace of the square root of trig functions?

I want to find the laplace of 1/rootcos(t) Laplace calculators don't give an answer with this as an input. I know nothing about laplace, so can someone explain why this happens? Is it impossible to ...
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Alternative version of the Final Value theorem for Laplace Transform

Let $f:[0,\infty) \to \mathbb{C}$ be a continuous and bounded function such that the limit $$\lim_{T\to\infty} \frac{1}{T} \int_{0}^{T}f(t)dt = d \quad\text{exists}.$$ Let $\hat{f}$ be the Laplace ...
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1answer
28 views

differentiation of Laplace transform solution

I am wondering if there is a solution to the differential equation (of sorts): $$\frac{d}{ds}\mathcal{L}\left[y(t)\right]-\mathcal{L}\left[\frac{d}{dt}y(t)\right]=0$$ Using the fact that: $$\frac{d}{...
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Laplace transform of a sum of Heaviside functions

The questions is, find: $$\mathcal{L}\left[\sum_{n=0}^\infty(-1)^n\text{H}(t-n)\right]$$ I started by saying: $$\mathcal{L}\left[\sum_{n=0}^\infty(-1)^n\text{H}(t-n)\right]=\int_0^\infty\sum_{n=0}^\...
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2answers
39 views

Self convolution function?

Given two positive real functions $h$ and $g$ and, $$g(t) \simeq \int\limits_0^t h(\tau) \cdot g(t-\tau) d\tau$$ i.e., $g$ asymptotically approximately equals the convolution of itself and another ...
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35 views

Finding the Inverse Laplace Transform of $-\frac{1}{s(s+1)}+\frac{(s+1)(s+2)}{s\left((s+1)^2+1\right)}$

I am trying to find the inverse Laplace transform of $$F(s)=-\frac{1}{s(s+1)}+\frac{(s+1)(s+2)}{s\left((s+1)^2+1\right)}.$$ I proceeded as follows: \begin{align} F(s)&=-\frac{1}{s(s+1)}+\frac{(s+...
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Inverse Laplace transform help- inverse heat conduction

I've been stuck on an inverse Laplace transform for my research. Would be greatly appreciative of any help solving the inverse Laplace transform of $$ \overline{T}=\frac{2}{k_{1}} \frac{1}{s} \left(\...
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1answer
43 views

Using Laplace transform to find $\int_{0}^{\infty}\frac{\sin^2 t}{t^2}dt$ [closed]

Find $$\int_{0}^{\infty}\frac{\sin^2t}{t^2}dt$$ using Laplace transforms.
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Unilateral Laplace Transform Uniqueness and Dirac Delta “function”

Answering this question, I've tried to alert the OP about the misleading definition of $\delta(t)$ - used in one of the answers - as: $$ \delta(x) = \left\{\begin{array}{cc} \infty & x = 0 \\ 0 &...
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3answers
101 views

What is the correct answer for $\mathcal{L}^{-1}\left(\frac{p^2}{(p-3)^2}\right)?$

What is the correct answer for $$\mathcal{L}^{-1}\left(\frac{p^2}{(p-3)^2}\right)?$$ Using partial fraction technique I got the answer as: $\delta(t)+e^{3t}9t+6e^{3t}$ and uing shifting thorem for ...
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Laplace Transforms with Non-repeated Irreducible quadratic factors

I'm solving the following differential equation with a Laplace Transform: $$x″+ 9x = \cos(t) + \delta(t-\pi)$$ The initial conditions are that x(0) and x'(0) are equal to 0. After after the ...
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what is the Laplace transform of exponential function of dependent variable?

Given the following equation, $\dot x(t) = e^{x(t)}$ The Laplace transform of $\dot x $ is $ sX(s) - x(0) $. a) What is the Laplace transform (natural transform as well) of $e^{x(t)}$ ? b) or ...
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Diffusion equation solution using Laplace transform

Consider the operator $L=k\frac{\partial ^{2}}{\partial x^{2}}-\frac{\partial }{\partial t} $ with domain : $D(L)={{u} \in \Re \times [0,+\infty ):u(x,0)=g(x), \forall x\in \Re , \underset{x\...
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3answers
55 views

Show that the inverse Laplace transform of F(s) = logs is given by f(x) = -1/x

Here's what I did, but I'm not sure if it's right/ valid. F(s) = log(s) L(xf(x)) = -1 So, L(xf(x)) = (-1)(1/s) Therefore, xf(x) = -1 So, f(x) = -1/x
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1answer
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Laplace inverse as a double integral

Show that $$\mathcal{L}^{-1}\left[\frac{f(s)}{s^2}\right]=\int_{0}^{t}\int_{0}^{x}F(x)dxdy.$$ I tried using the formula $$\mathcal{L}^{-1}\left[\frac{f(s)}{s}\right]=\int_{0}^{t}F(x)dx.$$
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How to calculate the Laplace transform of $\sum_{k=0}^{+\infty}(-1)^{k}\delta (t-k)$?

On my midterm, I had the following question: Calculate the Laplace transform of $$\sum_{k=0}^{+\infty}(-1)^{k}\delta (t-k)$$ I was wondering how I should calculate it. I know that the transform ...
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Do Inverse Laplace transforms satisfy the Convolution Theorem too?

The Laplace transform of a function $f$ is defined as $$ F(s) \ = \ \mathcal{L}[f](s) \ = \ \int_0^\infty dt \ f(t) e^{-st} $$ Whereas we can write the inverse Laplace transform in terms of the ...
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1answer
143 views

Breaking up a Convolution Integral

For a diffusion problem in a semi-infinite domain with a transient boundary condition, the temperature profile can be obtained from Duhamel's Principle as $$ \theta(r,t)= \int_0^t \frac{\partial \phi}...
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Does a time function has several Laplace transforms

I was wondering: Does a given time function has several Laplace transforms? For example, take a simple exponential function: $f(t)=e^{-at}$ When the inverse Laplace transform was done with the ...
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Decompose Inverse Cubic into Partial Fractions for Inverse Laplace Transform

I'm trying to perform an inverse Laplace transform on the transfer function: $H(s) = \dfrac{15}{s^3+6s^2+15s+15}$ I know the roots of this equation and for simplicity's sake, I've let the factors ...
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1answer
30 views

Integral of Dirac-delta function from convolution theorem

In a question I have been lead to use the convolution theorem to find the inverse Laplace transform, as shown below: $$\omega(t)=\mathscr{L}^{-1}\left[e^{-bs}\frac{s}{s^2+a^2}\right]$$ From the ...
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1answer
41 views

Output response from closed loop transfer function using MATLAB

This transfer function is to control the position of Permanent Magnet DC motor. I was able to get the transfer function and now I need to analyze the output for the tuned closed loop for a given input ...
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42 views

Inverse Laplace transform of $F(s) = \exp(-a\sqrt{s})/s$, $a > 0$

Show that the inverse Laplace transform of F(s) = $e^{-as^{1/2}}/s$, $a > 0$, is given by $$f(x) = 1 - \frac{1}{\pi}\int^{\infty}_{0} \frac{\sin(a\sqrt{r})}{r}e^{-rx}dr$$ Note that the integral ...
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1answer
30 views

What kind of integral is the one in Laplace transform?

In a book about circuits I found a chapter where it is defined the Laplace transform of a function $f:\mathbb{R}\rightarrow\mathbb{R}$ as $$L[f(t)]=\int_0^{+\infty}f(t)e^{-st}dt$$ where $s\in\mathbb{C}...
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37 views

How to find the laplace transform of $\cos(\sqrt t)$?

I tried solving for the transform using the same method the book uses to find laplace transform for $\sin(\sqrt t)$ which is, by writing the Maclaurin's expansion for $\sin(\sqrt t)$ and then using ...