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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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Solve initial value problem $y'' + 9y = f(t)$ using Laplace transform [on hold]

$y'' + 9y = f(t),$ $y(0) = 0$ and $ y'(0) = 1$ $f(t) = 1$ if $0 \leq t < 3\pi $ $f(t) = 0$ if $3\pi \leq t < \infty $
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Initial value problem , system of equations involved , solve using Laplace transform

Can anyone help me solve this problem, I could not find a similar solution on the web. Solve the following problem : Initial value problem, System of equations involved , Using Laplace transform $...
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0answers
17 views

PDF for the ratio of two random variable

Let $X_1$ and $X_2$ two random variable with pdfs $$ f_{X_i}(t)=\oint_{c}^{}g_i(t)(a_it)^{-s}ds $$ for $a_i>0$. I would like to dirive the pdf of $$ Z=\frac{X_1}{X_2} $$ I found the following ...
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0answers
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Calculate Laplace transform of $I_0(\sqrt{x}) \ln(I_0(\sqrt{x}))$

Calculate the Laplace transform of the function $$I_0(\sqrt{x}) \ln(I_0(\sqrt{x})) $$ where $I_0(u)$ is the modified Bessel functions of zero-th order. I have been trying to solve this but It ...
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1answer
19 views

Possible ways of solving second order ODE IVP.

The problem is the following: y''(t) + 2y'(t) + 5y(t) = H(t-2) - H(t-1) (Where H(t) is the Heaviside function) y(0)=0, y'(0)=0 (Initial values) I started working with the Laplace transform and ...
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16 views

Finding the Laplace transform of the Laguerre polynomial

I have tried to find the Laplace transform of $L_n(t)$, which is the $n$th order Laguerre polynomial defined by $$L_n(t) :=e^t \frac{\mathrm{d}^n}{\mathrm{d}t^n}(t^n e^{-t}).$$ I have managed to get ...
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38 views
+50

Clarification of Solution of PDE Laplace Transform Problem

I have the following problem: The transverse displacement $u(x, t)$ of a semi-infinite elastic string satisfies $$\dfrac{\partial^2{u}}{\partial{t}^2} = c^2 \dfrac{\partial^2{u}}{\partial{x}^...
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0answers
16 views

I want MATLAB codes that can plot the graph of θ =2.6575*exp(-0.8875*y)-3*erf(0.942*y)-4.3332*erf(0.942*y) with boundary conditions. [on hold]

I obtained this MATLAB codes [θ=2.6575*exp(-0.8875*y)-3*erf(0.942*y)-4.3332*erf(0.942*y)] from a temperature equation ∂θ/∂t=1/Pr((∂^2 θ)/(∂y^2 ))+MEc u^2 which I solved in fluid mechanics using ...
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2answers
38 views

Solving integral equation $y(t) = e^t ( 1 + \int_0^t e^{-\tau} y(\tau)d \tau )$ with Laplace transform

Solve the integral equation $$y(t) = e^t \bigg(1+ \int_0^t e^{-\tau}\ y(\tau) d \tau \bigg)$$ The apporach that stands out to me is using the convolution theorem and laplace transforms. Im able to ...
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24 views

Inverse Laplace Transform Mixing Sine and Exponential

I have a doubt about Laplace transforms. Most concretely, about the existence of an inverse. I know already these facts $$\mathcal{L}^{-1}(\exp(-\sqrt{s}))=\frac{\exp({-\frac{1}{4 t}})}{2 \sqrt{\pi ...
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0answers
21 views

Inverse Laplace transform of $\frac{1-s^2e^{2s}}{(s-2)^2(s^2+is+2)}$ [on hold]

Inverse Laplace transform of $\frac{1-s^2e^{2s}}{(s-2)^2(s^2+is+2)}$ can't confirm my result with Wolfram
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1answer
35 views

Using Convolution Theorem to Find the Inverse Laplace Transform of $\frac{s}{(s - 2)^{3/2}(s^2 + 1)}$

I am trying to use the convolution theorem to find the inverse Laplace transform of $$\dfrac{s}{(s - 2)^{3/2}(s^2 + 1)}$$ The convolution theorem states that $$f * g = \int_0^t f(\tau)g(t - \tau) ...
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1answer
50 views

Inverse Laplace transform of $\left(\frac{1+\alpha s}{1+\alpha(s-s_0)}\right)^p$

I am hoping to identify the function $f(t)$ that has the following Laplace transform, $$ \tilde f(s)=\int_0^\infty f(t)e^{-st}dt=\left(\frac{1+\alpha s}{1+\alpha(s-s_0)}\right)^p $$ where $\alpha,...
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Differential Equations: Switching from a sum of unit step functions into a piecewise function

I've completed a question involving unit-step functions and would like verification on my working out and answers, particularly for the last part, as we are asked to express the solution explicitly ...
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0answers
20 views

Please can someone help me find the inverse Laplace transformof u(y,s)=e^(-y√(Ps+F )) [on hold]

I solved the partial differential equation ∂u/∂t=1/P ((∂^2 u)/(∂y^2 )-Fu) using Laplace transform and obtained the general solution u(y,s)=e^(-y√(Ps+F)) . I now want the inverse Laplace transform of ...
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1answer
40 views

The Laplace transform of $t^{-3/2}e^{-1/t}$ is denoted $F(s)$. Show that $\frac{dF}{ds} = -\frac{F}{s^{1/2}}$.

The Laplace transform of $t^{-3/2}e^{-1/t}$ is denoted $F(s)$. Show that $$\frac{dF}{ds} = -\frac{F}{s^{1/2}}$$ not sure how to get this. I tried differentiating $F(s) = \int_{0}^{\infty}t^{-3/2}...
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1answer
43 views

Laplace Transform of $t^{t^2}$ [closed]

I have been trying to model a dynamic system and I came up with a differential equation which involved me finding the laplace transform of $t^{t^2}$. I have tried all the theorems I am familiar with, ...
1
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1answer
34 views

Finding Inverse Laplace Transform of a product series

I need to compute the inverse Laplace transformation of the following equation. \begin{align*} f(s)&=\frac{A}{\prod_{i=1}^{L}(s+a_i)^m} \\ &=\frac{A}{(s+a_1)^m\,(s+a_2)^m\cdots (s+a_L)^m}. \...
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0answers
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Compute the inverse Laplace transformation [closed]

Need help computing the inverse Laplace transformation of the following equation. \begin{align*} f(s)&=\frac{A}{\prod_{i=1}^{L}(s+a_i)^m} \\ &=\frac{A}{(s+a_1)^m\,(s+a_2)^m\cdots (s+a_L)^m}. ...
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47 views

Improper Integral using Laplace

So I want to calculate the following integral using the laplace transform: $$\int_0^\infty \frac{dt}{(t^2+1)^2} $$ I used the theorem: $$\int_0^\infty f(t)G(t) dt=\int_0^\infty \mathcal{L}[f(t)](x)...
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0answers
17 views

How to understand the Laplace transform of function?

Assume, we have the function as below $$I_{m} = \sum_{j=1}^{3} \sum_{d_{i}\in \Phi}P_{D}h_{i}G{j}P(||d_{I}||),$$ where $d$ is the Euclidean distance. Using the equation above, the Laplace transform ...
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0answers
18 views

Inverse Laplace Transform in Thermal Problem with Newton Cooling Condition

I need to compute the inverse Laplace transform of the following function: $$\hat u(x,s)=\frac{\exp(-\sqrt{s}x)}{s(1+\Lambda^{-1}\sqrt{s})},$$ where $x\geq0$ and $\Lambda>0$. This function comes ...
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2answers
40 views

Inverse Laplace transform of $\frac{40.5}{s(s^2-9)}$ using convolution theorem

Find the inverse Laplace transform of $$\frac{40.5}{s(s^2-9)}$$ using the convolution theorem. I see how you can solve this using partial fractions, but apparently it's supposed to be easier if you ...
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1answer
68 views

Using Laplace for PDE

The transverse displacement $u(x, t)$ of a semi-infinite elastic string satisfies: $$\frac{\partial^2u}{\partial t^2} = c^2 \frac{\partial^2u}{\partial x^2}$$ $ x > 0, t > 0$ with initial ...
0
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0answers
50 views

Laplace transform and Fourier transform

I used the Fourier and Laplace transforms to solve a series of equations. Now I have to use the inverse of these conversions to get the wave function. My question is $$L^{-1}(\frac{d}{dk}\psi(k,s))=?\...
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0answers
39 views

How to evaluate the inverse Laplace transform of $F(p) = \lambda\tanh (\lambda)/p$ where $\lambda=\left(\frac{p(1+p)}{2(1+2p)}\right)^{1/2}$

I am trying to solve a system of partial differential equations arising from a mathematical physical problem using the Laplace transform approach. In order to retrieve the solution in the time domain, ...
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On the solution of a linear system of differential equations for the unknown series coefficients

In a mathematical physical problem, I came across the following linear system of differential equations (obtained upon using Fourier series expansion): \begin{align} \frac{\mathrm{d} \rho_n}{\mathrm{...
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1answer
22 views

What is the Laplace transform of this convolution-like integral?

We have the following integral in the time domain : $$\int_0^\infty e^u\cos{(2(u-t))}y(u)du$$ This is very similar to the convolution integral. I'm not sure if the $2$ scaling in the cosine function ...
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1answer
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Laplace transform of zero …how it will be solved [closed]

What is Laplace transform of zero? How it will solved? Explain with simple example
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14 views

Prove that a function could be a Laplace transform of another function

I'm preparing for some exams and this is a question that is usually asked. Given a function $F(s)$ prove that it can be a Laplace transform of a function $f(t)$ While we've been given the conditions ...
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0answers
21 views

Laplace transform signal

Is this correct ? $$x(t)= \begin{cases} e^{-5t}, & 0≤t<5 \\ 0, & \text{otherwise} \end{cases}$$ \begin{align} L[x(t)] &=\int_{-\infty}^∞x(t)e^{-st}dt \\ &=\int_{0}^5e^{-5t}e^{-st} ...
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Laplace transfom of $\int_t^{\infty} \frac{f(u)}{u}du$

I want to know the Laplace transform of the function: $\mathcal{L} \int_t^{\infty} \frac{f(u)}{u}du$. I tried the following method $\int_0^{\infty} e^{-ts}\int_t^{\infty} \frac{f(u)}{u}du dt=\int_0^{...
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2answers
38 views

Can the Inverse Finding the Laplace transform of $\frac{2s + 1}{s(s + 1)(s + 2)}$ without using partial fractions?

I'm wondering if we can perhaps using the convolution theorem to find the inverse Laplace transform of $\dfrac{2s + 1}{s(s + 1)(s + 2)}$? I can find it using partial fraction decomposition, but it is ...
2
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2answers
23 views

Inverse Laplace Transform of $s^{-2}(s^2 + 1)^{-1}$ Using Convolution Theorem?

I am trying to find the inverse Laplace transform of $s^{-2}(s^2 + 1)^{-1}$. I could multiply these together and use partial fraction decomposition, but, unless I am mistaken, I think there is another ...
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1answer
29 views

Proof of Laplace Transform of *Real Number* Powers

As I understand it, the theorem and proof for the Laplace transform of positive integer powers is as follows: Theorem Let $t^n : \mathbb{R} \to \mathbb{R}$ be $t$ to the $n$th power for some $...
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2answers
22 views

Inverse laplace transform of $\frac{1}{(s+a)(s+b)}$ [closed]

Been trying to find the inverse laplace transform of $$\frac{1}{(s+a)(s+b)}$$ where $a$ and $b$ are constants. Most obvious thing to do is try a partial fraction decomposition, but it just becomes a ...
3
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1answer
59 views

“Damped” wave equation with Fourier method

The problem I was trying to solve is the following PDE problem $$\begin{cases} \partial_{tt}^2 u = \partial_{xx}^2 u -\gamma\partial_t u \\[5 pt] u(0,t)= u(\pi, t) = 0 \\[5 pt] u(x,0) = (\sin2x)^4 -...
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1answer
44 views

Generalized Fresnel Integral using Laplace

So I wanted to solve the following integral: $$\int_0^\infty \sin{(x^2) dx}$$ I did it by using the Laplace transform of the function: $$I(t) = \int_0^\infty \sin{(tx^2) dx}$$ $$\mathcal{L} [I(t)]...
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Inverse Laplace Transform of irrational function $\frac{1}{{a{s^\alpha } + bs + c}}$

I'm looking for a way to inverse this transformant $$F(s) = \frac{1}{{a{s^\alpha } + bs + c}}$$ where $a,b,c,\alpha $ are real numbers and $\alpha > 0$. Is there any technique to do it? ...
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1answer
21 views

Confused About Change of Integration Limits in Convolution Proof

My textbook has proved the convolution theorem as follows: Theorem If $f(t)$ and $g(t)$ are two functions of exponential order (so that their Laplace transforms exist), and writing $\mathcal{...
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2answers
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Questions About Textbook Proof of Convolution Theorem

I am reading a textbook on Laplace transforms. In the proof of the convolution theorem, the author starts by writing the following: $$\mathcal\{ f(t) * g(t) \} = \int_0^\infty e^{-st}\int_0^t f(\...
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2answers
66 views

Using contour integration for the inverse Laplace transform to find the inverse transform of $\dfrac{s}{s^2 + a^2}$

I am trying to use the contour integration formula for the inverse Laplace transform, find the inverse transform of $\dfrac{s}{s^2 + a^2}$. My textbook says that the solution is $\cos(at)$, but it ...
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2answers
31 views

Inverse Laplace transform of $\dfrac{s}{(s + 1)^2 - 4}$

I am trying to find the inverse Laplace transform $\mathcal{L}^{-1} \left\{ \dfrac{s}{(s + 1)^2 - 4} \right\}$. My textbook says that the solution is $e^{-t} \cosh(2t) - \dfrac{1}{2}e^{-t}\sinh(t)$. ...
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1answer
17 views

Laplace transform and delay

Hi guys if i have to transform $e^{t-1}u(t-1)$ , why the result is $\frac{e^{-s}}{(s-1)}$ and not $\frac{e^{-(s-1)}}{(s-1)}$ if the $e^{t-1}$ multiply all the "function" ?
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1answer
17 views

$\mathcal{L} ( (t+5)U(t-1) ) $

$\mathcal{L} ( (t+5)U(t-1) ) = U(t-a)f(t-a) = e^{-as} F(s) $ a= 1 I am having trouble dealing with $f(t-a)$ and finding $F(s)$ $f(t-1) = e^{t-3} = e^{(t-1)-3+1} = e^{(t-1)-2} $ $f(t) = e^{t-2} $ ...
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0answers
28 views

Help Solving Textbook Heat Conduction Laplace Transforms PDE Problem 2

This problem is related to this question. If you can answer this, then you might be able to also answer the other question, so please have a look at it. I am trying to solve the following problem: $$...
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2answers
67 views

Solve DE by Laplace transformation: $y'' + y = 8e^{-2t} \sin t$?

The differential equation is as follows $$\begin{cases} y'' + y = 8e^{-2t} \sin t \\ y(0)=0 \\y'(0)=0 \end{cases}$$ How do I solve it by Laplace transformation? In my solution I've taken the ...
1
vote
1answer
58 views

What are practical applications of Laplace Transforms in the real world [closed]

In lay man's terms what really in the real world are practical examples of how Laplace transforms are used to solve basic mathematical problems in mechanical engineering I would really appreciate a ...
6
votes
2answers
96 views

Help Solving Textbook Heat Conduction Laplace Transforms PDE Problem

I am trying to solve the following problem: $$\dfrac{\partial{\phi}}{\partial{t}} = \dfrac{\partial^2{\phi}}{\partial{x}^2} - \cos(x), \ x > 0, t > 0$$ $$\phi(x, 0) = 0, \ x > 0$$ $$\phi(0,...
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vote
1answer
35 views

What Is the Laplace Transform of $\frac{\partial{\phi}}{\partial{t}} = \frac{\partial^2{\phi}}{\partial{x}^2} - \cos(x)$, where $\phi = \phi(x, t)$?

I was wondering what the Laplace transform of $\dfrac{\partial{\phi}}{\partial{t}} = \dfrac{\partial^2{\phi}}{\partial{x}^2} - \cos(x)$, where $\phi = \phi(x, t)$, is? I know the Laplace transform of ...