The Laplace transform is a widely used integral transform, similar to the Fourier transform.

learn more… | top users | synonyms

2
votes
3answers
857 views

Inverse Laplace transform of fraction $F(s) = \large\frac{2s+1}{s^2+9}$

Is there a general method used to find the inverse Laplace transform. Are there any computational engines that will calculate the inverse directly? For example, can a procedure be followed to find ...
13
votes
1answer
703 views

Compute the inverse Laplace transform of $e^{-\sqrt{z}}$

I want to compute the inverse Laplace transform of a function $$ F(z) = e^{-\sqrt{z}}. $$ This problem seems very nontrivial to me. Here one can find the answer: the inverse Laplace transform of ...
5
votes
1answer
283 views

$\mathcal{B}^{-1}_{s\to x}\{e^{as^2+bs}\}$ and $\mathcal{L}^{-1}_{s\to x}\{e^{as^2+bs}\}$ , where $a\neq0$

http://en.wikipedia.org/wiki/Integral_transform#Table_of_transforms claims than the integral form of inverse bilateral Laplace transform and inverse Laplace transform are both the same. But are they ...
1
vote
1answer
178 views

Convolution Laplace transform

Find the inverse Laplace transform of the giveb function by using the convolution theorem. $$F(x) = \frac{s}{(s+1)(s^2+4)}$$ If I use partial fractions I get: $$\frac{s+4}{5(s^2+4)} - ...
4
votes
2answers
377 views

Laplace transform:$\int_0^\infty \frac{\sin^4 x}{x^3} \, dx $

I have a trouble with a integral: Using this Laplace trasform equation: $$\begin{align} \int_0^\infty F(u)g(u) \, du & = \int_0^\infty f(u)G(u) \, du \\[6pt] L[f(t)] & = F(s) \\[6pt] ...
1
vote
1answer
274 views

find the inverse Laplace transform of complex function

It would be appreciate if someone help me to obtain the inverse Laplace transformation of the complex function $F(s)=\frac{e^{-\frac lc\sqrt {s(s+r_0)}}}{\frac lc\sqrt {s(s+r_0)}}$ where $r_0,l,c$ are ...
0
votes
0answers
91 views

About evaluating $\mathcal{L}^{-1}_{s\to x}\left\{\dfrac{F(s)}{s}\right\}$ by considering contour integration with different entire functions $F(s)$

Detailedly compare the difficulties of different entire functions $F(s)$ where $F(0)\neq0$ when evaluating $\mathcal{L}^{-1}_{s\to x}\left\{\dfrac{F(s)}{s}\right\}$ by considering contour integration, ...
3
votes
2answers
4k views

Laplace transform of $ t^{1/2}$ and $ t^{-1/2}$

Prove the following Laplace transforms: (a) $ \displaystyle{\mathcal{L} \{ t^{-1/2} \} = \sqrt{\frac{ \pi}{s}}} ,s>0 $ (b) $ \displaystyle{\mathcal{L} \{ t^{1/2} \} =\frac{1}{2s} \sqrt{\frac{ ...
0
votes
0answers
172 views

About the inverse laplace transform of sinc function

How to calculate $\mathcal{L}^{-1}_{s\to x}\{\text{sinc}(s)\}$ ? Note: $\text{sinc}(s)=\dfrac{\sin s}{s}$ when $s\neq0$ . Also note that $\lim\limits_{s\to\pm\infty}\dfrac{\sin s}{s}=0$ .
6
votes
1answer
308 views

Inverse Laplace Transform of $\bar p_D = \frac{K_0(\sqrt[]s r_D)}{sK_0(\sqrt[]s)}$

I solved the following partial differential equation using Laplace Transform: $\LARGE \frac{1}{r_D}\frac{\partial}{\partial r_D}(r_D\frac{\partial p_D}{\partial r_D})=\frac{\partial p_D}{\partial ...
5
votes
1answer
797 views

How to figure of the Laplace transform for $\log x$?

I was looking at a table of common Laplace transforms of functions when I came across the transform for $\log x$. Apparently, the transform is as follows: $$\mathcal{L} \left\{ \log ...
5
votes
5answers
741 views

How can I evaluate $\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx$?

How can I solve this integral: $$\int_{-\infty}^{\infty}\frac{e^{-x^2}(2x^2-1)}{1+x^2}dx.$$ Can I solve this problem using the Laplace transform? How can I do this?
1
vote
2answers
64 views

Laplace question - help needed

I am currently studying the Laplace transformation and came across this question: I have no idea of how to start this and am completely lost. If anyone could help I would be really grateful. ...
11
votes
3answers
4k views

Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
4
votes
0answers
5k views

Relationship Between The Z-Transform And The Laplace Transform

Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an ...
5
votes
1answer
922 views

inverse Laplace Transform: $ L^{-1} \{\log \frac{s^2 - a^2}{s^2} \}$.

I am styding Laplace transforms and for some reason I have stuck in the followning exercise. Find the inverse Laplace Transform $ L^{-1} \{\log \frac{s^2 - a^2}{s^2} \}$. Any help? Thank's in ...
2
votes
1answer
297 views

Special Laplace Inversion

Use complex analysis to show $$\frac1{2\pi i}\int_{a- i\infty}^{a+i\infty} e^{st}/s^{1/2} ds = \frac1{\sqrt{\pi t}}\ ,\quad a >0, t> 0 .$$ This is a special case of Bromwich's integral for the ...
0
votes
2answers
83 views

Laplace question continued (partial fractions)

Last night I attempted and successfully finished (with the help of stackexchange) the first part to this question on laplace transformations: Laplace question - help needed The second part to this ...
0
votes
1answer
210 views

Using Convolution Theorem to find the Laplace transform

In previous questions I have used Laplace transform to find the inverse Laplace transform. I have worked through this work booklet ...
5
votes
1answer
120 views

Show that $\int_{0}^{\infty}\frac{\cos(at)-\cos(bt)}{t} =\ln\frac{b}{a}$ [closed]

It should be using Laplace transform. I found similar problems already solved but I need this to be shown using Laplace transforms: $$\int_{0}^{\infty}\frac{\cos(at)-\cos(bt)}{t} = \ln\frac{b}{a}$$
3
votes
1answer
108 views

$\int_0^\infty x e^{-\mathrm i x\cos(\varphi)}\mathrm dx=-\frac{1}{\cos (\varphi )^2}$ is that correct?

Good day. This integral looks very simple, yet I don't know how to start. $$\int_0^\infty x e^{-\mathrm i x\cos(\varphi)}\mathrm dx$$ I know that if the lower integration limit was $-\infty$ it would ...
2
votes
1answer
91 views

Functional form of a series of a product of Bessels

This question arises from my answer to an inverse Laplace transform question. The result I got took the form $$ f(t)= e^{-r_0 t/2} H(t-a) \left [ J_0\left(\frac{1}{2} a r_0\right) ...
2
votes
2answers
1k views

Solving partial differential equation using laplace transform with time and space variation

I have a equation like this: $\dfrac{\partial y}{\partial t} = -A\dfrac{\partial y}{\partial x}+ B \dfrac{\partial^2y}{\partial x^2}$ with the following I.C $y(x,0)=0$ and boundary conditions ...
2
votes
1answer
195 views

$\frac{dx}{dt}=-\lambda x +\epsilon x(t-a)$ series solution via Laplace method

Consider the following equation $$\frac{dx}{dt}=-\lambda x +\epsilon x(t-a), \quad x(0)=1,\quad |\epsilon|\ll1$$ where $a$ and $\lambda$ are positive constants and $x(t-a)$ means the function $x(t)$ ...
2
votes
2answers
566 views

Calculate the Laplace transform

Help me calculate the Laplace transform of a geometric series. $$ f(t) = \sum_{n=0}^\infty(-1)^nu(t-n) $$ show that $$ \mathcal{L} \{f(t)\} = \frac{1}{s(1+\mathcal{e}^{-s})} $$ Edit: so far I ...
1
vote
1answer
26 views

Laplace transform of integral equation

Use Laplace transforms to solve the integral equation $$y(t)-\frac{1}{2}\int_0^ty(t-v)~dv=1$$ First find the Laplace transform $Y(s)$ of $y(t)$
1
vote
1answer
137 views

Inverse Laplace Transform for $F(s) = (9s-24)/(s^2-6s+13)$

Find the inverse Laplace transform of $\displaystyle F(s) = \frac{9s-24}{s^2-6s+13}$. I have tried factoring out a $3$ from the top and putting it into the form of $\displaystyle\frac{b}{(s-a)^2+b^2}$ ...
1
vote
1answer
71 views

Laplace transformation problem

There is a timely unchanged continuous function : $$H(s)=\frac{s-1}{s+1}$$ At the entry of the system exists a $x(t)$ which Laplace's transformation is: $$X(s)=\frac{(5s^2 - 15s + ...
1
vote
2answers
64 views

Laplace transform of $f(t)=10te ^{-5t}$

Find the Laplace transform of $$f(t)=10te ^{-5t}$$
1
vote
2answers
652 views

Finding the inverse laplace transform of $s$ [closed]

How do I find the inverse laplace transform of $s$, i.e. $$L^{-1}\{s\}=\ ?$$
1
vote
1answer
132 views

Solve linear system of ODEs using Laplace transform

I need to solve the following initial value problem via Laplace transform \begin{align*} \dot{\mathbf{x}} = \begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix} \mathbf{x} + \begin{pmatrix} \sin t ...
1
vote
3answers
163 views

Laplace Transformations of a piecewise function

This is a piece wise function. I'm not sure how to do piece wise functions in latex. $$ f(t) =\begin{cases}\sin t &\text{if } 0 \le t < \pi,\\ 0&\text{if } t \ge \pi.\end{cases} $$ So ...
1
vote
0answers
64 views

Inverse Laplace transform is required

I shall be very very thankful if some one can find the inverse Laplace of the function given below. I really need it as early as possible. $$ \frac{1}{s-a}\exp ...
1
vote
1answer
147 views

Lower bounds of laplace transform of characteristic functions

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
1
vote
1answer
1k views

Finding the inverse Laplace of $e^{-3s}\frac{1}{(s-1)^2}$

I know I can use the following: $$\mathcal{L}^{-1}\{e^{-as}F(s)\} = u(t-a)f(t-a)$$ $$\mathcal{L}^{-1}\{\frac{n!}{s^{n+1}}\} = t^n$$ $$\mathcal{L}^{-1}\{F(s-a)\} = e^{at}f(t)$$ but I'm confused as how ...
0
votes
1answer
38 views

Laplace Transformation Impulse Question

An object with mass $m$ receives 11 impulses of strength $p$ at 1 second intervals at $t=0,1,2,\ldots,10$. The differential equation describing the motion of this object is $$m\frac{dv}{dt} = ...
0
votes
5answers
71 views

How to find the Laplace transform of $t\cos{t}$?

I need to find the Laplace transform of $f(t) = t \cos{t}$. I tried using the Taylor series expansion for $\cos{t}$ but I got stuck since the resulting expression is again a series which I could not ...
0
votes
1answer
167 views

Laplace transform, proof that $L \{ \frac{1}{k}f(\frac{t}{k}) \}= F(ks)$

Let $L \{ f(t)\}=F(s)$, show that for all $k \in \mathbb{R}$, $k \neq 0$ $$L \{ \frac{1}{k}f(\frac{t}{k}) \}= F(ks)$$ if, $u=\frac{t}{k}$ $L \{ \frac{1}{k}f(\frac{t}{k}) \}= \int_0^{\infty} ...
0
votes
2answers
80 views

Laplace of $x^2\frac{d^2y}{dx^2}$

How does one evaluate the Laplace of functions like $t^2\frac{d^2y}{dt^2}$ ? I wanted to solve a differential equation using Laplace Transform resembling: $$x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} + y ...
0
votes
1answer
69 views

Inverse Laplace transform and Lagrange basis polynomials

While reading a textbook on Laplace transform, I found this sentence. The inverse Laplace transform of ...
-1
votes
1answer
46 views

Did I do this Laplace transform correctly?

1) $w'' + w = t^2 + 2$; $w(0) = 1$, $w'(0) = -1$ 2) $s^2W(s) - sw(0) - w'(0) = \frac{2 + 2s^2}{s^3}$ 3) $s^5W(s) - s^4w(0) - s^3w'(0) = 2 + 2s^2$ 4) $ W(s) = \frac{2 + 2s^2 - s^3 + ...