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

learn more… | top users | synonyms

3
votes
1answer
26 views

After Laplace Transformations, how do initial conditions come into play?

The Laplace transform is relatively simple to derive but my solution was wrong. I am positive it is because of the initial conditions. Find the laplace transform f(t)={0 for t < 3, (t-3)^2 for t ...
0
votes
3answers
61 views

Determine the Laplace Transform $(t-1)^4$

How do I find the Laplace transform of $ (t-1)^{4} $? by using properties and a table of common Laplace transforms? I know I could just turn the equation into a polynomial and use $ t^{n} \to ...
1
vote
1answer
42 views

Take Laplace Transform of the integral J_0

I was just wondering how to use tables from Spiegal to solve $\int_0^\infty J_0(2\sqrt{ut}) J_0(u) du$ At the moment, I see similar transforms on page 244, but I don't actually know how to combine the ...
0
votes
1answer
232 views

inverse Laplace transform of $e^\sqrt{as}$

I am trying to find the inverse Laplace transform of $e^\sqrt{as}$ for $a>0$. So we need to solve $\oint_B dz \: e^\sqrt{az} e^{z t}$ (Bromwich contour), but not sure how to start. How do we even ...
1
vote
3answers
69 views

How do i find the inverse laplace?

$$ F(s) = \frac{2s-1}{s^2(s+1)^3} $$ If I try to use partial fractions, I end up with 8 constants to solve for! Is there some shortcut I'm not seeing? Am I supposed to simplify it first? Am I even ...
2
votes
1answer
102 views

Integral Equation without solution?

working on a physical problem I arrived at the following equation $$ y(x) + A \int_{0}^{x} e^{\lambda (t-x)} y(t) \mathrm{d}t = 0$$ and after some struggling (not that easy to apply the basic Laplace ...
0
votes
1answer
44 views

Finding the Inverse Laplace transform using the Step and Shift theorems

I want to find the Inverse Laplace Transformation of the function given above. I used the step and shift theorems to come up with an answer. Can someone simply verify the answer. This is my first ...
4
votes
1answer
70 views

Laplace transformations in differential equations

Using Laplace transformations solve the differential equation $$y''+3y'-17y=e^{-3x}$$ $$y(0)=4 , y'(0)=7$$ I'm having some trouble understanding the basic concept and seeing an example done I'm ...
1
vote
1answer
80 views

Laplace Transform Property for faster solution.

$$ f(t):= \begin{cases} 3\cos(2t) & t<5 \\ 4\sin(3t) & 5 \leq t \leq 9 \\ 0 & t>9. \end{cases} $$ The above function I know how to solve using straight Laplace Transformation by ...
4
votes
1answer
53 views

Laplace Transform Csch(x) (1/Sinh(x))

I need to find the Laplace Transform of $Csch(x)=\frac{1}{\sinh(x)}$. Wolfram Alpha and Mathematica say $-H(\frac{s-1}{2})$, where H(n) is the $n$-th Harmonic Number. I hope someone have a nice hint ...
0
votes
2answers
109 views

Find the inverse Laplace transformation of $\frac{e^{-s}}{s+2}$

My question is: Find the function $f(t)$ that has the following Laplace transform $$F(s)=\dfrac{e^{-s}}{s+2}$$ Thanks . my try:I have find this Find the inverse Laplace transformation of ...
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
1answer
56 views

Laplace Transformation spring question

Here is the question: http://i.imgur.com/XAH2UnX.jpg I can't seem to get the answer. Are those values in the writing like 1N/m even relevant? Can someone give me some direction? Thanks!
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 ...
1
vote
0answers
55 views

Transformed Laplace “solution space”

From my own knowledge I can tell that when we take the Laplace transformation of a function we are in essence transforming our f(t) into a F(s). I've looked at several Q/A here asking for the ...
0
votes
1answer
846 views

How to prove that inverse Fourier transform of “1” is delta funstion?

$\mathscr{F}\{\delta(t)\}=1$, so this means inverse fourier transform of 1 is dirac delta function so I tried to prove it by solving the integral but I got something which doesn't converge.
1
vote
0answers
24 views

Mellin transform and equality of two functions

If say, $\mathcal{M}f(s) = \mathcal{M}g(s)$ in a strip say $0 < a < \Re(s) < b$, then can we say that $f$ and $g$ are identical? Does the Mellin inversion theorem imply this?
1
vote
0answers
20 views

What polynomial transformation is this and how is it related to the original polynomial?

Let $f(X) = a_n X^n + \dots+a_0$. Then the Laplace transform of $f$ is $g(s) = \mathscr{L}\{f\}(s) = \frac{n! a_n}{s^{n+1}} + \dots + \frac{a_0}{s}$. If you now define the polynomial transform of ...
0
votes
1answer
43 views

How do you generalize the Laplace transform to more variables?

Just what the title says. How can I take the Laplace transform of $f(x,y,z)$ ?
1
vote
1answer
258 views

Laplace transformation of a polynomial function involving square root (or approximation of the integral)

Could somebody suggest how to solve this Laplace transform: $$ \int_0^\infty{e^{-at}\over\sqrt{A+Bt+Ct^2}}{\rm d\,}t $$ ? The real coefficients $A,B,C$ are chosen such that the roots of $A+Bt+Ct^2$ ...
-3
votes
1answer
507 views

How to find laplace transform of (t.sint.e^t) [closed]

How to find laplace transform of $\;f(t)=t e^{2t} \sin3t.$
2
votes
1answer
70 views

How to approach/solve this integral?

Could somebody suggest how to approach or solve this integral: $$ \int_{0}^\infty e^{-a t}{2+t-2\sqrt{1+t}\over t^2}{\rm d\,}t, $$ where $a>0$ ? It is not a homework. I tried to use residuum ...
0
votes
1answer
118 views

Laplace transform of floor$(x)$

One way to compute the Laplace transform of floor$(x) = \left \lfloor{x}\right \rfloor$ (defined as the greatest integer $\leq x$) for $x$ positive is to use the definition of the transform. Write ...
1
vote
2answers
138 views

Laplace transform of trig + Heaviside

So I am trying to take the laplace transform of $\cos(t)u(t-\pi)$. Is it valid for me to treat it as $((\cos(t)+\pi)-\pi)u(t-\pi)$ and treat $\cos(t)-\pi$ as $f(t)$ and use the 2nd shifting property, ...
0
votes
1answer
373 views

Laplace Transform Involving Heaviside Step Function

I'm trying to find the Laplace transform of $7 e^{-3t} u(t-3)$, where $u$ is the heaviside step function. However, we've never really gone through what the Laplace transform of the heaviside step ...
0
votes
1answer
25 views

On Laplace transforming an integrand

Suppose I have the following function: $$ M(s) = \int_{0}^{\infty}\frac{f(E)}{s+2\pi i /T(E)}\text{d}E \tag{1} $$ Where $f(E)$ is some potentially complex function, $T(E)$ is some real function and ...
0
votes
1answer
48 views

Laplace Transform using t-shift

$$f(t)=\begin{cases}cos(πt), & 1\leq t < 4 \\ 0, &elsewhere \end{cases}$$ Okay, I attempted to write it in terms of step functions and I got $$ f(t) = cos(πt)u(t-1)-cos(πt)u(t-4)$$ But ...
4
votes
1answer
153 views

Inversion of Laplace transform $F(s)=\log(\frac{s+1}{s})$ (Bromwich integral)

I am looking for the inversion of Laplace transform $F(s)=\log(\frac{s+1}{s})$. I started by using the general formula of the Bromwich integral: $\displaystyle \lim_{R\to\infty} \int_{a-iR}^{a+iR} ...
0
votes
1answer
61 views

Laplace Transform using t-shift (second shift)

$$f(t) = tu(t-π)$$ I know I have to get t in terms of $$(t-π)$$ and to do that I have done $$ t = a(t-π) + b$$ $$ t = at-aπ + b$$ $$ t = (a-π)t + b$$ $$ (a-π) = 1$$ and $$b = 0$$ Then I think I ...
2
votes
0answers
71 views

Existence of the Laplace Transform

I'm seeing Laplace transforms for the first time, and I'm having trouble understanding the criteria for deciding when they exist. I've read a few websites and books that seem to say that we only ...
0
votes
1answer
1k views

Laplace Transform, Transfer Functions, Calculate the new output when input changed

Consider the initial value problem for $0<t<∞$: $ay′′+by′+cy=f(t)$,$y(0)=0$,$y′(0)=0$, where $a$,$b$,$c$ are constants and $f(t)$ is a known function. We can view this problem as defining a ...
2
votes
0answers
218 views

How Heaviside step function changes limits of integration

This question involves the Laplace transform of the convolution of two functions. The derivation in my textbook has a step that really confuses me. First I'll lay out their argument. $$ f(t) = f_1(t) ...
-1
votes
1answer
161 views

Solve IVP using Laplace transform?

Solve the IVP using Laplace transform: $$y'' + 4y = g(t); \hspace{5 pt}y(0) = 1, \hspace{5 pt} y'(0) = 3$$ and $$g(t) = 3 sin (t), 0 \leq t < 2\pi; \hspace{10 pt} 0, 2\pi \leq t$$ Take step ...
1
vote
2answers
32 views

Find inverse laplace of the following?

Find the inverse Laplace of $$\frac{e^{-2s} - 3e^{-4s}}{s+2}.$$ I tried splitting up the function into two, so I get $$\frac{e^{-2s}}{s+2} - \frac{3e^{-4s}}{s+2}.$$ Then we have ...
0
votes
1answer
43 views

Find the Laplace transform?

Find the Laplace transform of $f(t) = 1 + (1 - t)u_1(t) + (t-2)u_3(t)$. Obviously each term of the function must be of $f(t - c)u_c(t)$ or be clearly transformable. Thus we have for our first term ...
-1
votes
1answer
233 views

How to use Complex Inversion Theorem to find the Inverse Laplace Transform?

How to use Complex Inversion Theorem to find the Inverse Laplace Transform for the given $F(t)=L^{-1} \{s^{-1/2} e^{-1/s}\}$ ? Hint: make the radius $\epsilon$ of the inner circle $t-1/2$ rather than ...
0
votes
1answer
54 views

Showing an inequality

I wish to show $$|{(Re^{i \theta})^{-\frac{1}{2}}}\exp(\frac{-1}{Re^{i \theta}})| < \frac{M}{R^k}$$ for some M, k > 0 I've managed to reduce it to $$|R^{-\frac{1}{2}}| |\exp(\frac{-1}{Re^{i ...
0
votes
0answers
30 views

How to find the Laplace-Stieltjes transform of a joint distribution?

Consider two r.v.s (not necessarily independent) $X$ and $Y$ distributed exponentially with rate $\lambda$ and $\mu$ and having LSTs $E(e^{-sX})=\frac{\lambda}{\lambda+s}$ and $E(e^{-\theta ...
0
votes
2answers
79 views

Why is there an exponential in Fourier's defining integral?

I am having a hard time relating integration with Fourier series. Basically, I just get lost where there is an exponential in the integration to convert into the frequency domain. If someone can ...
1
vote
1answer
75 views

Find the inverse Laplace transform $f(t)=L^{-1}\left\{F(s)\right\}$ of the function $F(s)=\dfrac{7s−22}{s^2−6s+13}. $

Find the inverse Laplace transform $f(t)=L^{-1}\left\{F(s)\right\}$ of the function $F(s)=\dfrac{7s−22}{s^2−6s+13}. $ $f(t)=L^{-1}\left\{\frac{7s-22}{s^2-6s+13}\right\}$. I was trying to break ...
2
votes
2answers
352 views

Finding Inverse Laplace Transform using Taylor Series

Find the inverse Laplace transform $F(t)=\mathcal{L}^{-1}(s^{-\frac{1}{2}}e^{-\frac{1}{s}})$ using each of the following techniques: Expand the exponential in a Taylor series about s=∞, and take ...
1
vote
1answer
136 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
32 views

Use the relation of Laplace Transform and its derivative to figure out $L\left\{t\right\}$,$L\left\{t^2\right\}$,$L\left\{t^n\right\}$

If $F(s) = L\left\{f(t)\right\}$, then $F'(s) = -L\left\{tf(t)\right\}$ Use this relation to determine $(a)$ $L\left\{t\right\}$ $(b)$ $L\left\{t^2\right\}$ $(c)$ $L\left\{t^n\right\}$ for any ...
1
vote
2answers
46 views

What is the name of this function similar to convolution?

The functions seems to be very near convolution function, but the only difference is that you integrate by $du$ in convolution, in contrast to $ds$ in this example: $g(t,u) ...
1
vote
2answers
36 views

Verify that the transform of $y(t) = t^2e^{at}$ is $Y(s) = \frac{2}{(s-a)^3}$

I made the distinction to amplify "=" 3 times for easier readability. I tried: $$F(s) === \int_0^\infty t^2e^{(a-s)t}dt === \frac{1}{a-s}e^{(a-s)t}t^2\Big|_0^\infty \ - \frac{2}{a-s}\int_0^\infty ...
0
votes
1answer
39 views

inverse transform of $Z(\omega) =\frac{a}{\alpha-i\omega}$

I am stuck at calculating the inverse transorm of $Z(\omega) =\frac{a}{\alpha-i\omega}$. Can someone help me please? thanks
1
vote
1answer
114 views

Laplace transform (differential equation containing several functions)

I have a differential equation which looks like: $$ \dfrac{dT}{dt} = \dfrac{P}{\rho Ac_ph} + \dfrac{q (T_{in} - T)}{Ah} - \dfrac{U\pi D(T - T_{a})}{\rho Ac_p} $$ where $P$, $h$, $q$, $ T_{in}$ are ...
0
votes
0answers
45 views

Laplace's Initial value theorem: discontinuity in $0$

The Laplace's Initial value theorem: $$\lim_{t\to 0}f(t)=\lim_{s\to \infty}sF(s)$$ This is a demonstration: $$ \lim_{s\to \infty} sF(s)=\lim_{s\to \infty}[\int_{0^-}^{\infty} \frac{d}{dt}f(t) ...
0
votes
1answer
94 views

How to solve this Reaction-Diffusion problem by FEM?

I want to solve this by Finite Element Method numerically, since the exact solution is too hard. Separation of variables does not help me here. Epsilon is positive so cannot be Helmholtz equation. ...
2
votes
1answer
80 views

Inverse Laplace Transform Problem

I have this problem $\frac{1}{(s^2+1)^3}$. I have to find its Inverse Laplace Tranformation. I already try using partial fraction but it didn't work because I found it will back to the problem form. ...