What is Laplace transforms: Definition and 186 Discussions
In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable
t
{\displaystyle t}
(often time) to a function of a complex variable
s
{\displaystyle s}
(complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication.For suitable functions f, the Laplace transform is the integral
Homework Statement
y'' + y = delta(t-2*pi)
The Attempt at a Solution
I have solved this using laplace transforms.
Y = e^(-2*pi*s)/(s^2 + 1) + 10s/(s^2 + 1)
so that
y = sint*u(t - 2*pi) + 10 *cost
what I don't understand is this function is not definied for t<0. According to...
Using Laplace transforms, find the solution of Abel's integral equation:
\int^{x}_{0}\frac{f(u)}{\sqrt{x-u}}du = 1 + x + x^2
I recognized that the integral is a Laplace convolution, leading to:
(f*g)(x) = 1+x+x^2
where g(x)=x^{-1/2}
So:
L(f*g)=L(1)+L(x)+L(x^2)...
Hi all, 2 questions here
1) I've been doing some questions on laplace transforms and have been running into some trouble getting my answers into the same form as the answers given with the questions.
For example:
f(t) = 1 - e^(-t)
Using the linearity property i got 1/s - 1/(s-1) which is...
Hey can anyone tell me the method to solve this, I can do inverse linear eqns and such but i don't know how to solve this type of question:
Use Laplace transforms to solve the following systems of linear equations
(a)dx/dt= 3x(t) − 4y(t),
dy/dt= 2x(t) + 3y(t),
with initial conditions...
Hi hows it going,
Im currently doing a linear maths course, i suppose it'd be introductory. I am using Elementary Linear algebra by Howard anton and find that very good. However the course goes on to deal with laplace transforms and Fourier series, can anyone recommend a good (introductory)...
1. Please see attachment for Question I've put it into microsoft eqn editor.
2. I have not got a clue where to begin i don't want anyone to do it for me but some information on how i can do it for myself would be appreciated.
3. as above
We are supposed to work this using Laplace transforms
U_{tt}=9U_{xx}; -infty<x<infty
U(x,0)=sinx
U_t(x,0)=0
The attempt at a solution
Let L[U]=\hat{U}
L[U_{tt}]=s^2\hat{U}-s(sinx)
L[9U_{xx}]=9\hat{U}_{xx}
s^2\hat{U}-s(sinx)=9\hat{U}_{xx}...
Is it possible to use a laplace transform to solve a problem like
x' + x = T
where x is a function of T and x(0) = 5 and x(4) = 7
Or can you only solve initial value problems?
Use the Laplace transform to solve
\frac{\partial^2 y }{ \partial t^2 } = c^2 \frac{ \partial^2 y }{ \partial x^2 } for x>0, t>0
y(0,t) = t, for t>0
y(x,0) = 0, \frac{\partial y(x,0) }{ \partial t } = A, for x>0So I used the Laplace transform of a derivative, along with the initial conditions...
Homework Statement
Given the equation H'(t) + u H(t - T) = 0 u > 0
Look for solutions of the form e^{rt}
Show that these solutions are exponentially damped if e^{-1} > uT > 0
Find uT for which these solutions for r complex are oscillatory with growing, decaying, or constant amplitude.
The...
Homework Statement
Solve using Laplace Transforms
\frac{dx}{dt}=x-2y
\frac{dy}{dt}=5x-y
Subject to x(0)=0 y(0)=0Okay. So I know that there are more efficient ways to solve this, but I missed the class in systems of Laplace Transforms and would like to get this resolved.
I guess my...
Homework Statement
Inverse laplace transforms
F(s)=\frac{5s-2}{s^{2}(s-1)(s+2)}
Homework Equations
Residue technique
The Attempt at a Solution
F(s)=\frac{5s-2}{s^{2}(s-1)(s+2)} = \frac{k1}{s^{2}} + \frac{k2}{s-1} + \frac{k3}{s+2}
I solved for K1,K2, and K3, which all came to...
Homework Statement
We are given the following LaPlace Transformations and we are to find g(t) and h(t).
L(g)(t) = 1/[(1-e^(-2s))(s-1)]
L(h)(t) = (e^-s)/s(1-e^-s)
Homework Equations
By looking at the Table of LaPlace Transformations, the only one I found that could possibly work...
Homework Statement
Derive the Laplace transform of the following functions, using first principles
3d) u(t - T) \} = 0, \ t<T \ (= 1, t>T)
3e) f(t) = e^{-a(t-T)}u(t-T)
Homework Equations
see above
The Attempt at a Solution
I know I need to derive the transform using by integration...
Find v(t) at t=800ms for the circuit in Figure 1.
Ans: 802mV
Writing a single node equation we have
\frac{v(t)-2tu(t)}{5}+0.1\frac{dv}{dt}=0.
Taking the Laplace transform we have
L\left\{ \frac{v(t)-2tu(t)}{5}+0.1\frac{dv}{dt}=0\right\} .
tu(t)\Rightarrow\frac{1}{s^{2}}...
Homework Statement
(t − 1)^2u(t − 1) Find the Laplace transform.
Homework Equations
L{f(t-a)u(t-a)}(s) = e^-as F(s)
The Attempt at a Solution
The solution manual says take f(t) = t^2, I don't see why? Why is f(t) not (t-1)^2?
Homework Statement
What is the laplace transform of e^(-t) cos 2t u(t-1)
Homework Equations
definition of Laplace transform: LT of f(t) = integral of f(t)e^-st dt, where limits of integration are from 0 to infinity
The Attempt at a Solution
since I have u(t-1) then do I just...
I am new to Derive 6 and Laplace Transforms. Is it possible to solve the closed loop gain A/B of the attached loop using Derive 6. If it is possible, can someone guide me through the steps using Derive, as the problems that lie ahead will be much more complex, and I want to use Derive in the...
Laplace Transforms on Partial Differential Equations - Non-dimensionalization too!
Homework Statement
The experiment described in the previous problem was analyzed from the point of view of long time \left(\frac{D_{AB}\,t}{L^2}\;>>\;1\right). We wish to reconsider this analysis in order to...
I'm posting these as images because it would be difficult to type. If this isn't allowed, please let me know. I really need help with this before tomorrow, when I have a test. Any help appreciated.
Homework Statement
problem here
The Attempt at a Solution
I completed part 1 (solution here)...
Homework Statement
Using improper integrals, find the Laplace transform of f(t)=t, determining the values of s for which the transform is valid.Homework Equations
The Laplace transform F(s) of a function f is defined as
∞
/
F(s)= | f(t)e^(-st) dt...
Homework Statement
Use a table to fing L{f(t)} for the given function f(t) defined on the interval t>=0
Homework Equations
f(t)=sin(3t)cos(3t)
The Attempt at a Solution
Well this type of Laplace transform has got me stumped, I'm not to sure how to proceed. It doesn't seem to...
well, I found this place searching for a method to solve a laplace transform as I'm a little stuck on exactly how to do it, So I'd like to say hello, and I hope I can be of some help for someone else in the future.
also, do posts here go into a moderator queue before appearing , some other...
Hello everyone, I am actually a mechanical engineering major at the University of Texas, but I thought I'd ask you kind folks for help with this problem. I apologize in advance for this being difficult to read because I don't know how to type many of these symbols.
Homework Statement...
Homework Statement
just a small part of a large problem ><a I am not even sure if this is in the right place. the first two questions i have to ask are really minor so i didnt want to make separate threads for each one. the third one is an actual problem itself tho.
(1) i need the laplace...
Can someone show me how to do these laplace transforms of these differentials?
1) y""-4y"'+6y" -4y'+y=0
y(0)=0, y'(0)=1, y"(0)=0, y"'(0)=1
2) y"-2y'+4y=0
y(0)=2, y'(0)=0
3) y"'+2y'+y=4e^-t
y(0)=2, y'(0)=-1
4) y"-2y'+2y=cos(t)
y(0)=1, y'(0)=0
the Laplace transfrom that i got for...
I began studying with a friend of mine for the FE (fundamentals of engineering) test and we began with the math section. We came across a LaPlace transform problem and we had never learned it before. Is this something I can learn relatively quickly (within the week)? I have taken calculus 1 2...
Hey guys, I am supposed to find the Laplace transform of a set of ODEs.
Ive broken it down a bit and I am left with finding the Laplace transform of:
(-2e^-s)/(s(s+4)(s+1))
Is this something I have to use partial fractions for? Or is there another way? I am a bit confused.
Could you please help me to do Laplace transformation ? That is an autonomous and nonlinear function on the right-hand side
L{dx/dt}=L{1/1+x^a}
where a is an integer
other kind is in the form of delay
L{dx/dt}=L{sin(x(t-\tau))}
where \tau is delay, and real number
Hi, I am taking a circuits analysis course currently. We are studying Laplace Transforms.
I was wondering if anybody out there knows a way of bypassing all of the cumbersome complex algebra to find the partial fraction decomposition of a given function with respect to s.
If not maybe...
Ok, this is the question:
Assume that the Laplace transform of x(t) is given as X(s)=s / (2s^(2) + 1).
Determine the Laplace transform of the following function.
g(t)=x(2t-5)u(2t-5)
How do I use the transform they have given me to solve this...I guess my major problem lies using time...
Hi there,
I'm trying to derive an expression for the transient response (for a step input of magnitude V), for a non-ideal inductor modeled in the schematic I have drawn. This non-ideal inductor includes its inductance ( :smile: ), a parasitic parallel resistance and a parasitic capacitance...
I need a large table of Laplace transforms, do you know any good ones? Format HTML, Pdf, Ps, Latex or Word (?)
Extra question: what is the Laplace transform of a Gaussian (normal distribution), with mean m and standard deviation s (assume equal to zero for t<0)?