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

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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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14 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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41 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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1answer
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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24 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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3answers
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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61 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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16 views

A name of abook with exercises of laplace methode [closed]

I really nead a name of abook which contains exercises concern Laplace methods
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24 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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7 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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1answer
43 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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17 views

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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2answers
76 views

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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1answer
36 views

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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24 views

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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41 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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31 views

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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1answer
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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2answers
26 views

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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2answers
166 views

Book on applied mathematics

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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31 views

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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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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34 views

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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46 views

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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74 views

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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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 ...
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28 views

The inverse Laplace transform of $\Gamma\left(\zeta\right) \, W_{\zeta,\mu}(z) $

Someone has a reference that addresses an integral of the followns type $$I = \frac{1}{2i\pi} \int_{\sigma-i\infty}^{\sigma+i\infty} e^{t\zeta} \, \Gamma\left(\zeta\right) \, W_{\zeta,\mu}(z) \, ...
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Confusion in the usage/property of Laplace Transform.

While proving that $$\int^{\infty}_0 \frac{\sin x}xdx$$ I saw the Laplace Transform proof. It used that $$\cal L\left\{\frac{\sin t}{t}\right\}=\int^\infty_0 \cal L\left\{\sin(t)\right\}d\sigma$$ So ...
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52 views

What is the difference between an impulse response and a transferfunction?

An imupulse response, is the output you get when you apply an impulse, like a delta dirac function, to your system (only for LTI?). By knowing the impulse response you know the system. The ...
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zeros/poles of Laplace transforms of Dirac combs (Riemann zeta function)

let's define $p_\alpha(n) = \displaystyle\int_1^n x^\alpha dx$ so that $\left\{\begin{array}{lll} p_0(n) &=& n-1 \\ p_{-1}(n) &=& \ln n \\ p_\alpha(n) &=& \frac{\textstyle ...
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Inverse Laplace transform of $\tan^{−1}\left(\frac{1}{s}\right)$

I'm studying Laplace transformations, but I don't understand where $-\frac{1}{t}$ comes from. And what is the relationship between the corollary and the example?
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61 views

What is the Laplace transform of $\cos(4t+8)$?

Could someone please explain how to transform this to the Laplace domain? I've tried to use the definition of Laplace (not sure this is the easiest way). $$\int_{0}^{t}e^{-st}f(t)\,dt$$ ...
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34 views

Residue theorem with pole on integration path

I have to calculate the inverse Laplace transform of $\dfrac{1}{s^2+1}$ (which I know is sin(x)) by residue theorem: $\int^{i \infty}_{-i \infty}exp(t\cdot s)\cdot \dfrac{1}{s^2+1}\mathrm{d}s$. ...
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29 views

A Partial fraction expansion questions about Laplace transform

I am learning signals and systems. Our teacher give us the following answer, it's about Laplace transform . But I can't figure out the second line, the calculation of k1,k2,k3,k4. why they can be ...
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51 views

complex integration, residues, inverse Laplace transform, calculus

Dear Mathematicians, I kindly ask your expertise on complex integration. The problem is the last step in the solution to a differential equation, using an inverse Laplace transform. I know that the ...
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Application of Initial Value Theorem

Let $$F(s):=\frac{s}{2s-i}$$ be the Laplace transform of some $f(t)$. I have been asked to compute $f(0^+)$ assuming that this quantity, intended as limit, exists. I thought I could apply the IVT ...