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

learn more… | top users | synonyms

2
votes
2answers
48 views

Solving a differential equation using Laplace transform?

$$y''+2y'+ 10 = b\,δ(t-T),\,\begin{cases}y(0)=3\\ y'(0) = 0\end{cases}$$ I managed to solve this equation. My answer is $$y(t) = 3e^{-t} \cos(3t) - ...
1
vote
0answers
37 views

Find what values of 'b' have bounded solution(differential equation)?

$y′′ + b^2{y} = f(t)$ $ f(t) = t$ for $0 < t < 2\pi$ ($2\pi$ periodic sawtooth wave) This is my solution to the differential equation. $y(t) = C_1\cos(bt) + C_2\sin(bt) + b^{-3}\left(bt - ...
0
votes
2answers
46 views

Use Laplace transformations to solve $y''+4y'+4y=e^{-x}$, so that $y(0)=0$ and $y'(0)=1$

Use Laplace transformations to solve $y''+4y'+4y=e^{-x}$, so that $y(0)=0$ and $y'(0)=1$. I applied the transformation but I don't understand the rest of the process. Can anyone explain me based on ...
0
votes
3answers
162 views

inverse laplace transformation of $\arctan(\frac{4}{s})$

inverse laplace transformation of $\arctan(\frac{4}{s})$ using I was trying use 12 but i couldn't arrive to a solution
0
votes
0answers
132 views

laplace transformation $\cos^2(3t)$ and $\sin(5t)cos(2t)$

it is asked to transform $\cos^2(3t)$ and $\sin(5t)cos(2t)$ using the results from i think the process might be similar for both of them but i don't know wich result to use. can you help me? ...
3
votes
2answers
81 views

Laplace tranform of $t^{5/2}$

It is asked to transform $t^{5/2}$. I did $t^{5/2}=t^3\cdot t^{-1/2}$. Then followed the table result $$L\{{t^nf(t)}\}=(-1)^n\cdot\frac{d^n}{ds^n}F(s)$$ However i got $\frac{1}{2} \cdot\sqrt\pi ...
3
votes
1answer
76 views

Equality of laplace transform

Assuming that Laplace Transforms of two functions $f$ and $g$ are equal, is it true that $f=g$? There is one-to-one correspondence between functions and their Laplace Transforms, so it seems to me ...
1
vote
1answer
41 views

Laplace transform: Convolution

Let $H(s) = \frac{1}{(s^2 + w^2)^2}$ Then $\displaystyle h(t) = \frac{\sin(wt)}w * \frac{\sin(wt)}w = \frac 1{w^2} \int_0^t \sin(w \tau) \sin(w(t-\tau)) \,d\tau$ $\displaystyle = \frac 1{2w^2} ...
0
votes
1answer
53 views

Abscissa of Convergence for the Laplace Transform of $f(t)=e^t \sin(e^t)$

I am trying to solve the following question: Show that the abscissa of convergence for the function $f(t)=e^t \sin(e^t)$ is zero, i.e the unique number $\sigma$ such that the integral $\int_0^\infty ...
6
votes
2answers
156 views

How to find inverse laplace transform of $\frac{2\sqrt s}{2\sqrt s+1}$

How to find inverse laplace transform of $$\dfrac{2\sqrt s}{2\sqrt s+1}$$ I tried to solve it, but I couldn't.
2
votes
2answers
139 views

How to solve IVP using Laplace transform(of matrix)?

$$x' = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & -2 \\ 3&2 & 1\end{bmatrix} x, ~~ x{(0)} = \begin{bmatrix} 2 \\ -1 \\ 1\end{bmatrix}$$ I am having very hard time solving this ...
1
vote
1answer
102 views

finding the inverse Laplace transform of $\frac{1}{z\sqrt{z+1}}$

i know that the inverse Laplace transform is given by $$2\pi i \left\{\sum\space\text{ of the residues at the poles of}\space e^{zt}f(z)\right\}- \frac{1}{2 \pi i}\int \text{ along the branch cut}$$ ...
3
votes
2answers
167 views

Laplace transform of $f(t)=te^{-t}\sin(2t)$

I was asked to find the Laplace transform of the function $\displaystyle f(t)=te^{-t}\sin(2t)$ using only the properties of Laplace transform, meaning, use clever tricks and the table shown at ...
1
vote
0answers
101 views

Pollaczek-Khinchin formula for ruin probability - proof

I got stuck in a specific part of proof of the Pollaczek-Khinchin formula (in book "Stochastic Processes for Insurance and Finance", T. Rolski et al., section 5.3.3, theorem 5.3.4). Namely, why the ...
1
vote
1answer
78 views

Check my answer - simple laplace transform of piecewise continuous function.

I'd just like to check that I got the idea right, first exercise im doing in laplace transforms and am a bit clueless. We are given $f(t)=0$ if $0<t<2$ and $f(t)=t$ if $t>2$. We are asked to ...
1
vote
1answer
79 views

Laplace Transform of the square of first derivative

I want to compute the laplace transformation of this equation: $$4 -0.1{\operatorname{d}\!x(t)\over\operatorname{d}\!t} - 0.01\left({\operatorname{d}\!x(t)\over\operatorname{d}\!t}\right)^2 = ...
2
votes
1answer
58 views

Solution to truncated renewal function

Let's begin with some theory on the renewal process. In a renewal process $N(t)$, let $t$ denote the interarrival time, and $f(t)$ and $F(t)$ denote the PDF and CDF respectively. Let $M(t)=E[N(t)]$, ...
2
votes
3answers
85 views

Inverse Laplace of $\frac 1 {(s^2+a^2)^n}$

How to compute the Inverse Laplace of $\frac 1 {(s^2+a^2)^n}$? I know that to compute Inverse Laplace $\frac 1 {(s^2+a^2)^2}$, the convolution Theorem is useful. but is there an interesting idea for ...
0
votes
1answer
23 views

Inverse Laplace Transformation

I was solving a problem but I am stuck at it. Here is the question : $\frac{7s^2+9s+3}{(s^2-12s+40)(s^2+9)}$ Find inverse Laplace transform. I performed these operation : ...
2
votes
1answer
40 views

Inverse Laplace Transformation

I have a question about laplace transformation. $\frac{8s+4}{s^2+23}$ I tried to split them. $\frac{8s}{s^2+23}$ is the image of a cosine and $\frac{4}{s^2+23}$ is the image of a sine. Here is ...
2
votes
0answers
47 views

Transforming Exponential to Ordinary Generating Functions

I am looking for a particular ordinary generating function, if it exists for the Associated Stirling Numbers of the second kind $$b(1;n,j)=b(n,j)=\sum_{k=0}^j(-1)^k\binom{n}{k}S(n-k,j-k)$$ Where ...
0
votes
0answers
26 views

Laplace transform of $\sin \sqrt x$ [duplicate]

I want to find the Laplace transform of the $\sin \sqrt x$. The first thing it came to my mind is to definition, but I think there are others solutions. Please show me this solution thanks.
1
vote
2answers
852 views

How to solve a linear system in matrix form using Laplace transform?

How to solve this linear system using Laplace transform? $$\mathbf X'(t)=\left[\begin{array}{r,r,r}-3&0&2\\1&-1&0\\-2&-1&0\end{array}\right]\mathbf X(t); ~~~~~~~~\mathbf ...
0
votes
2answers
27 views

Laplace transform of the following: $f(t)=\int\limits_{0}^{t} \frac{\cosh (\tau) - 1}{\tau} d\tau$

I thought it would be a simple one, however, that integral of $f(t)$ cannot be expressed in terms of standard functions...and I'm pretty much confused. What should I do?
4
votes
2answers
49 views

How to find the inverse Laplace transform?

I'm trying to calculate $$\mathcal{L}^{-1}\left(\frac{3s^3-3s^2+3s-5}{s^2(s^2+2s+5)}\right)$$ But I am not sure how to go from here. I would be really grateful for any help. Thanks.
1
vote
2answers
85 views

Tough inverse Laplace transform

I know what the solution is to this inverse Laplace transform, I just have NO idea how to get there. $$\mathcal{L}^{-1}\left(\frac{16s}{\left(s^2+4\right)^2}\right)$$ Basically, my question is what ...
1
vote
1answer
42 views

Differential equation, for which values of 'a' does this have a bounded solution?

Let $f(t) = f(t$) be the 2pi periodic("sawtooth wave"), f(t) = t for $0 \leq t \leq 2\pi$ and consider the equation $$y^{\prime \prime} + a^2y = f$$ For which values of $a$ (here $a$ >0) does this ...
2
votes
1answer
43 views

How to find solution of the integral equation?

$$y(t) + t \int_0^t y(v)dv = 1 + \int_0^t vy(v)dv$$ I found the answer to be $y(t) = \cos{t}$. I have no idea how they go this answer. I would appreciate any suggestions how to solve this.
2
votes
2answers
35 views

Inverse Laplace transform shifting error

I am doing the inverse Laplace transform of the function: $\frac{e^{-s}}{s-1}$. I am solving and receiving the answer: $e^t\mathcal{U}(t-1)$, however the correct answer is $e^{t-1}\mathcal{U}(t-1).$ ...
0
votes
1answer
66 views

Solve laplace equation inside a rectangular

My answer is $U = Acos(nπx/L)e^-nπy/L$ I really have no idea how to solve the particular solution. Please advise me.
1
vote
0answers
24 views

Singularities of complex exponential and asymptotic expansion

Consider the equation $$L[u(x,t)] = \tilde u(s,t) = \frac{e^{-t\sqrt{s^2-1}}}{s-2}$$ I want to find $u(x,t)$ in the form of an integral. I first need to find the poles and singularities of the ...
0
votes
0answers
59 views

Renewal Process with Pareto distributed inter-arrival time

I am analyzing a renewal process $N(t)$ whose inter arrival time $t$ conforms to the Pareto distribution, the PDF $f(t)$ of the Pareto distribution is as follows: $$f(t) = \left\{ \begin{array}{lr} ...
1
vote
2answers
55 views

What's the laplace inverse of this function?

I'm completely stuck on how to do this one. Any help is appreciated. What is the inverse Laplace transform of: $$\mathcal{L} ^ {-1} \left\{ \frac{e^{-2s}}{s-2} \right\} = f(t)$$
0
votes
1answer
25 views

How to find the Laplace transform of $\sin^2 (2t)$?

How to find the Laplace transform of $\sin^2 (2t)$? So far I have: $\sin^2(x) = \frac12 - \frac12 \cos(2x)$.
1
vote
0answers
29 views

Inverse Laplace transformation correct?

I'm actually on the way to solve a little bit complicated differential-equation. Therefore I used the Laplace transformation. I've already solved it but I am actually not sure, whether my solution ...
0
votes
0answers
37 views

Laplace transform of piecewise function

I have a piecewise function f(t), and I'm trying to get it's laplace transform. When I do it manually, i'm getting a different result than with Maple. $$ f(t)=\!\cases{{t}^{2}&$0<t$ and ...
0
votes
0answers
30 views

Approximate the inverse Laplace transform

I am struggling with an inverse Laplace transform for a long time! Assume we have a function $m(t)$ and its Laplace transform is denoted by $M(s)$. I have derived the expression of $M(s)$ by some ...
0
votes
1answer
49 views

Laplace Transformation using Heaviside functions

I'm not very familiar with Heaviside functions so I am struggling with this: I'm supposed to compute $Lu$ where $u''+4u=H(x-0)+H(x-\pi)$ and $H$ is a Heaviside function. Any suggestions are greatly ...
2
votes
1answer
25 views

Laplace transform and differential equations

Given $\frac {d^2y(t)}{dt^2} + a\frac {dy(t)}{dt} = x(t) + by(t)$ Find: a) $ H(s) = \frac{Y(s)}{X(s)}$ b) ROC of the stable function and the correspond h(t) and determine if the stable system is ...
0
votes
0answers
23 views

Approximate ways to compute Laplace transform of $t^{-a}$

I am encountered with the Laplace transform of $\frac{1}{t^{a}}(a>0)$ in a practical experiment. Obviously, the exact Laplace transform of $\frac{1}{t^{a}}(a>0)$ does not exist, but I am ...
3
votes
2answers
41 views

Use Laplace Transform to solve the following IVP:

I know that this is a somewhat simple problem but I have been having trouble coming up with the little "tricks" that help with Laplace. The problem is: $y''+2y' +5y = e^{-t}\sin(2t)$ where $y(0) = ...
0
votes
1answer
190 views

Use the chain rule to convert the Laplace equation in (x,y) coordinates into an equivilent differental equation in (r,theta) coordinates. [duplicate]

use the equations $r=\sqrt{x^2 +y^2}$ and $\theta=\arctan(\frac{y}{x})$. I was able to get the partial derivative of of $r$ with respect to $x$ and $y$ and the partial derivative of $\theta$ with ...
2
votes
2answers
31 views

Inverse laplace transform excercise

I want to find the inverse transform of $$\frac{1}{(2s-1)^3}$$ I first applied a shifting theorem to get $$(e^t)\mathcal{L}^{-1}\left( \frac{1}{(2s)^3} \right)$$ I am just wondering is it possible ...
1
vote
1answer
171 views

Frequency integration theorem (Laplace transform)

In my textbook I have the following theorem about the integration of the frequency (F(s)): Let the Laplace transform of a function $f(t)$ be $\mathscr{L}\{f(t)\}=F(s)$. If $\dfrac{f(t)}{t}$ is the ...
0
votes
1answer
39 views

Laplace Transform of $e^t\cos(3t)\operatorname{heaviside}(t)$

Find the Laplace Transform of $e^t \cos(3t) \operatorname{heaviside}(t)$ Since $\operatorname{heaviside}(t)g(t) = \mathcal{L}(g(t)) $ and $\mathcal{L}(e^t\cos(3t)) = \frac{(s-1)}{(s-1)^2+9} $ ...
0
votes
0answers
32 views

Laplace transform involving two functions of t

I need to solve the following $$ \int_0^{\infty} f(t)g(t)e^{rt} dt$$ Where $$g(t)=t^n$$ Letting r=-s we have the definition of $$ \mathcal{L} [ f(t)g(t) ]$$ and am unsure how to continue.
2
votes
0answers
25 views

How to find Laplace transform of a differential equation?

$y′′ + 3y′ + 2y = f$ , $y(0) = 0$ , $y′(0) = 1$ where $f$ is given by $f(t) = \sum_{n=1}^\infty \delta(t−n)$; find a 1-periodic function $y_*$ with $\lim_{t\rightarrow \infty} |y(t)−y_*(t)| = 0$. I ...
0
votes
2answers
43 views

How to re-write one fraction as two others.

I have the two following fractions. $$ \dfrac{A}{Bx^{\alpha+1}}$$ and $$ \dfrac{C}{Dx^{\alpha+\beta}}$$ The form i want $$ \dfrac{E}{Fx^{\alpha+\beta+1}}$$ I was thinking to do partial fractions or ...
0
votes
1answer
59 views

Trying to show $\int_0^1 e^{-xt}sin(t) dt \sim \frac{1}{x^2}$

I am using Laplace's Method and I am trying to show $$I =\int_0^1 e^{-xt}sin(t) dt \sim \frac{1}{x^2}$$ $h(t) = -t$ has a maximum at $0$ and as it is a simple function there is no need to expand it. ...
2
votes
1answer
28 views

Laplace transform of a differential equation?

$y′′ + 2y′ + 2y = δ(t − \pi) + aδ(t − T)$ , $y(0) = y′(0) = 0$ $a$ and $T$ are positive numbers and $T > \pi$. I need to find values for $a$ and $T$ such that $y(t) = 0$ for all $t \ge T$? I just ...