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

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Find the original function by using convolution theorem

Seems like I don't know how to apply convolution theorem on this problem properly, I would appreciate some help and a brief explanation how did you solve it if you do it. ...
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29 views

Find the inverse Laplace transform of $L(s)= \frac{s}{s^2 + 25} e^{-\pi s}$

$$L(s)= \frac{s}{s^2 + 25} e^{-\pi s}$$ I never seen such function. Can exponential function appear in Laplace transform? Help required
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20 views

Laplace Transform Injectivity

Intuitively how can the Laplace transform be injective? You are taking an integral with limits $0$ and $\infty$. So you don't care about the function before $0$. Define $g(x)=x^2$ for $x>0$ and ...
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How and why an integral Transform is created?

I don't know if what I'm going to ask will make any sense, but I was just wondering about integral transforms. I am talking about, for example, Mellin Transform, or Laplace Transform or Hilbert ...
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23 views

Problem with convolution, insecure

$$f(t)= t^2\cdot u(t),\quad g(t)=t^4\cdot u(t)$$ I know that I need to use convolution theorem to solve this problem, but I really don't know what to do with step functions. Do I need to include ...
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How to calculate inverse laplace of $e^{a\sqrt s}$?

I was using Laplace to find solutions for $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions $$u(0,t)=1 \\ u(1,t)=1 \\ u(x,0)=1+ \sin \pi x$$ I used ...
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18 views

Laplace transform second shifting rule

I have found seccond shifting rules with and without the Heaviside function. Even my professor taught us the one without Heaviside. For example if $f(t)=2e^-e(t-3)$ 1)Rewrite this expression as ...
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Polar System with Short Answers, How $U(0, \theta)=\pi$ will be calculated?

I read some notes on Laplace. I ran into a short answer question as follows. We have a Laplace equation in Polar Systems: $\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial u}{\partial ...
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31 views

Laplace Transform of $e^{t^3}$

I have to find the Laplace transform of $$e^{t^3} u(t)$$ and I know that $u(t)$ will just change the integral from negative infinity to positive infinity to $0$ to positive infinity, but I'm stuck ...
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Laplace transform for $-t\cos(2t)$

This Laplace transform exercise is giving me a headache. I was trying to use the definition of the Laplace transform but when I make the $u$ and $dv$ substitutions for the integration by parts I never ...
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50 views

Region of Convergence

I know to find the Region of Convergence you find the poles of the denominator, but I'm unsure what to do for the case where there is no denominator (for a and d) and what to do if the poles are ...
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Different answer when using the 'method of undetermined coefficients' compared to Laplace transform

I have an ordinary differential equation: $$ \frac{\mathrm{d}^2u}{\mathrm{d}t^2} + u = \mathrm{e}^{-t}\cos(t)$$ with $u(0) = u_0$ and $\dot{u}(0) = v_0$, when using the method of undetermined ...
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15 views

Initial value Laplace Transform exercise

I'm having trouble with the following exercise $$ y'' +4y - (4/e^x) = 0 $$ with the initial values: $$ y(0) = 1 y'(0)=5$$ I used the formula $$ y'' = s^2Y(s) − s*f(0) − f'(0)$$ and got to: $$ ...
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46 views

Why is Laplace Transform used for ODEs

This part is taken from differential equations with applications and historical George simmons. According to the given information , there are another integral transformation.I wonder why is the ...
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20 views

Moment generating function and convergent random variables

denote by $X$ and $X_n$, $n\in \mathbb{N}$, random variables and $r\in\mathbb{R}_+$ with $E=\mathbb{E}\left[ e^{rX} \right] < \infty$ and $E_n=\mathbb{E}\left[ e^{rX_n} \right] < \infty$ for all ...
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17 views

Laplace transform and value in x(0)

Somebody told me that if i have something like this: $x''(t) + x'(t) = -2x(t) + u$ $x(0) = 7$ and use laplace transform on it i will get $s^2X(s) + sX(s) = -2X(s) + U(s)$ next i'm getting ...
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26 views

Laplace transform of a definite integral

I'm having some troubles with what follows. I am interested in finding the Laplace transform w.r.t. $x$ of some real-valued, positive, continuous (in general well-behaved) function $f(x,t),x,t>0$. ...
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61 views

Evaluation of $\int_{0}^{\infty}t^3e^{-3t}dt$

I have to evaluate the integral $\int_{0}^{\infty}t^3e^{-3t}dt$ using complex analysis techniques (the laplace transform). Can you check my steps, please? $$\int_{0}^{\infty}t^3e^{-3t}dt =\Rightarrow ...
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64 views

Show that $\int\limits_0^\infty\frac{1}{t}(\cos(at)-\cos(bt))dt=\ln(b/a),\,a,b>0$.

Show that $$\int\limits_0^\infty\frac{1}{t}(\cos(at)-\cos(bt))dt=\ln(b/a),\,a,b>0.$$ Thanks to wikipedia I know that $$\int\limits_0^\infty\frac{1}{t}(\cos(at)-\cos(bt))\,dt ...
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On Laplace transforms - Applications in Probability Theory

I'm trying to find good bibliography on Laplace transforms for Applications in Probability Theory. I can't understand deeply the importance of this tool; nor I was taught very much on the subject. ...
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28 views

Existence of solutions in time and Laplace domains

I have not made use of Laplace transforms for many years since my education and I am a bit rusty on applying the various theorems associated with the transform. I have an equation $f(t)=0$ and I am ...
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9 views

Interpretation of diagonal detail in Haar Wavelet Transforms

I am a statistics grad student, and I have just begun exploring the topic of wavelet regression (specifically, Haar wavelets for discrete functions). I understand the generalization from a one ...
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45 views

fourier transform of $f(x) = x^2+\frac{1}{1+2x^4}$

I really have no thought on this. I can't seem to use residue thm., nor could I find a inverse transform for it. by some Fourier Calculator I know it's solvable but how?
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Differential Equation by Laplace Transform [closed]

I was solving normal IVP problmes but I have no idea as how to solve this problem with $u(t)$ present in the question. Please help with this one.
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Can Laplace solve every lineair differential equation?

I'm learning about laplace tranform method to solve lineair differential equations but i'm wondering if laplace transformations can be used to solve every linear differential equations there is. Or ...
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express as contour integral $ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $

Let $0 < x < 1$, I have to compute this Laplace transform: $$ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $$ I am not 100% this interal is defined. If $t > ...
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28 views

Why is causality important for laplace transformations? [closed]

Could someone please explain why causality is important for laplace transformations?
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Verify second order Cauchy Riemann equations

How do I differentiate the equations in 12? I understand the hint, but I'm not sure how to act on it.
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19 views

Where are the particular and homogeneous solution of the ODE when using Laplace?

When solving an ODE with Laplace, it seems as if there is no distinction between the homogeneous and particular solution. As if you calculated both at once. Is this correct? How does it come? Where ...
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48 views

What are disadvantages/limitations of Laplace?

I was curious about what limitations the famous Laplace theorem for solving ODE had and what drawbacks it may have. PS: I am NOT familiar with Fourier
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Comparison between Laplace, operator calculus and system of first order ODE

I am trying to understand those three methods to solve differential equations. I would like to know what actually are the differences between the three: Laplace calculus operator conversion to a ...
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33 views

Given integral equation, find $y(1)$

Let $y(t)$ be a continuous function on $[0,\infty)$ whose Laplace transforms exists. If $y(t)$ satisfies $$\int\limits_0^t(1-\cos(t-\tau))y(\tau)d\tau=t^4\to(1)$$ then $y(1)=$ I was able to find ...
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To solve given differential equation using laplace transform

I am solving following diff eqn using laplace transform: \begin{eqnarray} y''+y= \begin{cases} 0, & \text{if 0<t<2 $\pi$}\\ \sin t, & \text{t>$2\pi$} ...
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Book on applied mathematics/analysis

My Applied Mathematics course covers these subjects: -Calculus of Variations -Laplace Transform -Fourier Analysis -Special Functions -Integral Equations And as an introduction to the subject it has ...
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To find the value of a constant when the Laplace transform of a function is given

This question is regarding my previous post Find the value of a constant when the Laplace transform of a function is given where the hint was given by Moo to find the laplace transform of ...
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Laplace transform of $g(x)=\begin{cases}0.5\sin{4t}&\text{if }t<4\\0&\text{if }t\geq 4\end{cases}$ using second shift theorem

Using the Second Shifting Theorem to compute the Laplace transform of $$g(t)= \begin{cases} 0.5\sin{4t}&\text{if }t<4\\ 0&\text{if }t\geq 4 \end{cases}$$. I try to write ...
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32 views

Find the value of a constant when the Laplace transform of a function is given

I am given that $F(s) = \tan^{-1}{s} + k$ is the laplace transform of some function $f(t)$ $t\geq 0$ . I have to find the value of $k$. What I get is: $F(s) = L(f(t))$ $\Rightarrow L(f(t)) = ...
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25 views

Laplace Transform of Shifted Function

Why do we need to multiply the shifted function $f(t - a)$ by the shifted step function $u(t - a)$ to obtain the Laplace transform? $$ \mathcal{L\{f(t - a)\}} = \int_0^\infty u(t - a)f(t - ...
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27 views

Convolution of complex functions (Laplace Domain)

Convolution of functions in the time domain is equivalent to multiplication in the frequency domain. However, I am interested in multiplication of functions in the time domain, which is convolution in ...
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1answer
23 views

Using Laplace Transform to solve this ODE

How to solve this ODE, with Laplace Transform: $$ \begin{cases} 20y'(x)+y(x)+4y''(x)=20\\ y(0)=10\\ 4y'(0)=-2 \end{cases} $$ Thanks in advance. My work: ...
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62 views

Solve the IVP $xy'' + y' + 4xy = 0, y(0) = 3, y'(0) = 0$

It has to be solved with Laplace transform and then converted to Bessel equation. $L(xy'') = -\frac{dL(y'')}{ds}$ $L(4xy) = -\frac{4dL(y)}{ds}$ $L(y'') = s²L(y) - sy(0) - y'(0) = s²L(y) -3s$ ...
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12 views

The need for two laplace transforms

So I have recently come across Laplace transforms, but I have seen one sided and two sided laplace transforms, my question is why do we need two kinds of transforms, when do we use which transform?
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2-sided Laplace transform of $\exp(-(t + e^{-t}))$

I'm having trouble finding an analytic solution to the 2-sided Laplace transform of; $$f(t) = \exp(-(t + e^{-t}))$$ Integration by parts doesn't seem to help. Any pointers appreciated. It seems like ...
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Laplace trasform

i am trying to do this exercise but i do not get it. The laplace trasform is: \begin{equation} T(f)(s)= \int_{0}^{\infty} f(t)e^{-st} dt \end{equation} The exercise is: a) If $f$ is the ...
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Solve $y''-xy'+y = 1 , y(0)=1, y'(0) = 2 $ with Laplace transform

What's making me get stuck is the Laplace transform of $xy'$. I'm aware of different methods of solving this, but it's asking specifically for Laplace transform.
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Inverse Laplace transform of $1/\sqrt{s^2-a^2}$ using complex integration

I want to find the inverse Laplace transform of $$F(s) = \frac{1}{\sqrt{s^2-a^2}}$$ preferably using the Bromwich integral: $$f(t) = \frac{1}{2\pi i}\int_{\beta -I \infty}^{\beta +i ...
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24 views

Calculating Laplace inverse

I'm having difficulties calculating a simple Laplace inverse : $$ \frac{S-4}{S^2-2S-11} $$ I'm new at this and couldn't find good examples for this case. could you please guide me ?
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1answer
19 views

Closed-loop transfer function in the time domain

In a simple linear system with feedback (figure 1), the closed-loop transfer function $H(s)$ can be written as $$ H(s)=\frac{X_o(s)}{X_i(s)} = \frac{G(s)}{1+G(s)F(s)} $$ by solving the equations $$ ...
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How to get the Laplace transform of $t \cdot f(t) \cdot e^t$

Is there a formula to get the Laplace transform of $t \cdot f(t) \cdot e^t$ ? I tried integration, but that got me nowhere, because I'm probably missing something. Any ideas?
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51 views

How can I find the Fourier transform of constant value like $1$.

The textbook told me that $\mathbb F[1] = \delta(f)$ and $\mathbb F[\delta(t)]=1$. It is easy to prove that $\mathbb F[\delta(t)] = 1$. $$ \mathbb F[\delta(t)] = \int_{-\infty}^\infty ...