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
Exponential polynomials are linear combinations of terms like ##t^ne^{at}\cos{bt}## and ##t^ne^{at}\sin{bt}##, where ##n## is a nonnegative integer, ##a## is real and ##b>0##. Like the proper rational functions, these are presumably subspaces of ##\mathbb R^{\mathbb R}##.
The proof goes like...
Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function.
How can Laplace transforms be introduced so that students are...
Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function.
How can Laplace transforms be introduced so that students are...
I was wondering how you work out what values of s a Laplace transform exists? And what it actually means? The example given in class is an easy one and asks to calculate the Laplace transform of 3, = 3 * Laplace transform of 1 = 3 * 1/s. Showing this via the definition, where does the range of s...
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I attended Oregon State U. and majored 3 years in Electrical Engineering. Then I switched to a Math major for my final years and graduated with a B.S. in Math (1967). Developed several Apps for Engineers & Scientists.
My first job was with Lockheed Aircraft Co...
Take the Laplace Transform of the equation:
$\displaystyle \begin{align*} s\,Y\left( s \right) - y\left( 0 \right) + 11\,Y\left( s \right) &= \frac{3}{s^2} \\
s\,Y\left( s \right) - 5 + 11\,Y\left( s \right) &= \frac{3}{s^2} \\
\left( s + 11 \right) Y\left( s \right) &= \frac{3}{s^2} + 5 \\...
This requires the convolution theorem:
$\displaystyle \int_0^t{f\left( u \right) \,g\left( t- u \right) \,\mathrm{d}u } = F\left( s \right) \,G\left( s \right) $
In this case, $\displaystyle g\left( t - u \right) = \mathrm{e}^{-3\,\left( t - u \right) } \implies g\left( t \right) =...
Start by taking the Laplace Transform of both equations, which gives
$\displaystyle \begin{cases} s\,X\left( s \right) - s\,x\left( 0 \right) + X\left( s \right) + 6\,Y\left( s \right) = \frac{6}{s} \\ s\,Y\left( s \right) - s\,y\left( 0 \right) + 9\,X\left( s \right) + Y\left( s \right) = 0...
The Heaviside function suggests a second shift, but to do that, the entire function needs to be a function of $\displaystyle t - 4$.
Let $\displaystyle u = t - 4 \implies t = u + 4$, then
$\displaystyle \begin{align*} \mathrm{e}^{5\,t} &= \mathrm{e}^{5\left( u + 4 \right) } \\ &=...
hello if someone could please tell me if i am incorrect and where , and how to type it into a math program so it can understand it many thanks stephan2124
L -3e^{9t}+9 sin(9t)
L-3e^{9t}+L 9 sin (9t)
-3 Le^{9t}+9 L sin(9t)
-3 (1/s-9) +9 (9/(s^2+9^2))
-3 (1/s-9) +9 (9/(s^2+81))
into a math...
[Solved] Solving PDE using laplace transforms
Hey, I'm stuck on this problem and I don't seem to be making any headway.
I took the Laplace transform with respect to t, and ended up with the following ODE:
$\frac{\partial^2 W}{\partial x^2}-W(s^2+2s+1)=0$
and the boundry conditions for $x$...
Hi, i need some help here. Can you help me?:sorry:
Here is the problem.
Exercise statement: The switch have been closed for a long time y is opened at t=0. Using Laplace's transtormation calculate V0(t) for t ≥ 0
This is what i made to solve it:
1) I know while the switch is closed, the...
Homework Statement
I am given this equation:
and asked to solve using Laplace transforms
The Attempt at a Solution
This is what I did:
This seemed logical to me, I used partial fractions and it stayed pretty simple.
This is what the solutions my prof posted do:
Is my answer equivalent...
Homework Statement
A step voltage of 120v is applied to a series CR circuit. R = 20KΩ, C = 4µF
1. Deduce, using Kirchoff's voltage law and Laplace Transforms, an expression for the transient circuit current.
2. Using the equation obtained in 1. deduce the equations for the transient voltages...
Homework Statement
The Attempt at a Solution
At this point, usually I would replace the values and sometimes separate into partial fractions, and then take the inverse Laplace transformation. So I know that the inverse Laplace needs to give me 6+12e^-2t.
I am given the answers in my...
Homework Statement
I want to invert a function from Laplace transform space to normal space.
Homework Equations
In Laplace transform space, the function takes the form $$ \bar f (s) = \frac{\exp\left[ x (-a +\sqrt{a^2+ b +c s} )\right]}{-a +\sqrt{a^2+ b +c s}}.
$$
Here, ##s## is the Laplace...
Homework Statement
Homework Equations
V=IR
All of them actually
The Attempt at a Solution
So I Started off by transforming the voltage source into the 's' domain
vs(s) = (4/s) -(4/s)*e-.5t
I know the initial conditions are zero, in other words at t=0, the voltage and currents at the...
As the denominator is a function of s + 3, it suggests a shift had to have been utilised. As such, we also need the numerator to be a function of s + 3...
Let $\displaystyle \begin{align*} u = s + 3 \end{align*}$, then $\displaystyle \begin{align*} s = u-3 \end{align*}$ and thus...
Homework Statement
Given the Laplace transform
$$F_L(s) = \frac{1}{(s+2)(s^2+4)},$$
by using the complex inversion formula compute the inverse Laplace transform, ##f(t),## for the following regions of convergence:
(i) ##Re(s)<-2;##
(ii) ##-2<Re(s)<0;##
(iii) ##Re(s)>0.##
Homework Equations...
Let's say you have a function y(t). You know how derivatives of y have their own Laplace transforms? Well I was wondering if powers of y such as y^2 or y^3 have their own unique Laplace transforms as well. If so , how do you calculate them (because plugging them into the usual integral doesn't...
Homework Statement
Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teacher is against using it..)
y'' + 2y' + 2y = 2 ; y(0)= y'(0) = 0
Homework Equations
Lf'' = ((s^2)*F) - s*f(0) - f'(0)
Lf' = sF - f(0)
Lf = F(s)
The...
Homework Statement
Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teach is against using it..)
y'' - 4y' + 3y = 0 ; y(0)=2 y'(0) = 8
Homework Equations
Lf'' = ((s^2)*F) - s*f(0) - f'(0)
Lf' = sF -...
Homework Statement
I uploaded the question as a picture and attached it.
Homework Equations
Unit step function -
u_c (t) =
\begin{cases}
1 & \text{if } t \geq c \\
0 & \text{if } t < c
\end{cases}
Impulse function -
δ(t) = \displaystyle\lim_{Δ\rightarrow 0} δ_Δ (t)
Multiplication Property...
Homework Statement
I uploaded the problem statements as a picture as well. I have completed these and was wondering if someone could check my work, and let me know if it is correct.
Problem 1.3:
Find the expression for the transfer function of this linear time-invariant causal system with...
Homework Statement
Here is an imgur link to my assignment: http://imgur.com/N0l2Buk
I also uploaded it as a picture and attached it to this post.
Homework Equations
u_c (t) =
\begin{cases}
1 & \text{if } t \geq c \\
0 & \text{if } t < c
\end{cases}
The Attempt at a Solution
Question 1.1 -...
i have read many of the answers and explanations about the similarities and differences between laplace and Fourier transform.
Laplace can be used to analyze unstable systems.
Fourier is a subset of laplace.
Some signals have Fourier but laplace is not defined , for instance cosine or sine...
Homework Statement
Homework Equations
Laplace Transforms
The Attempt at a Solution
Using basic physics knowledge I got
m1a1=-k1x1+k2(x2-x1)
and
m2a2=-k3x2-k2(x2-x1)
Sub in values and use laplace transforms and rearrange partial fraction and I found that
By doing this I am assuming...
I've been given this:
x''+ x = 4δ(t-2π)
The question asks:
With initial conditions of x(0) = 1 and x'(0) = 0, find x(t) using Laplace transforms.
I can easily get it to this:
4(sin(t-2π)u(t-2π))
But the question says "express your final solution without use of the unit step function". This is...
I should state, from the outset, that this tutorial is NOT going to go into any great detail about the theory and applications of Laplace transforms. Some of the aforementioned will be discussed in a cursory way, but the aim here is merely to provide a selection of proofs for common transforms...
Homework Statement
f(s) = 6/s^2-9
Homework Equations
I think
f(t) = (1/b-a)(e^-at-e^-bt)
The Attempt at a Solution
Replace 6/s^2-9 with 6/(s-3)(s+3)
a=-3
b=3
Plug in
(1(6)/3-(-3))(e^-(-3)t-e^-3t)
Final Result
e^3t-e^-3t
Homework Statement
f(s) = -5s/S^2+9
Homework Equations
I think
f(t) cosωt = f(s) s/s^2+ω^2
The Attempt at a Solution
ω=3
Answer
-5cos(3t)
Can anyone tell me if I did this correctly? I think I did but just want to make sure, if not can you tell me what I did wrong?
Thanks
Homework Statement
The coordinates ##(x,y)## of a particle moving along a plane curve at any time t, are given by
\frac{dy}{dt} + 2x=\sin 2t,
\frac{dx}{dt} - 2y=\cos 2t.
If at ##t=0##, ##x=1## and ##y=0##, using Lapace transform show that the particle moves along the curve
4x^2+4xy+5y^2=4...
Ok, so I start out with the basics and find k.
F=ma=kx
(1.5 kg)(9.8 m/s^2) = k (4.9 m)
k = 3 N/m
I also know from the problem that c=0 (no damping) and x(0) = 2m and x'(0) = 0.
ƩF = ma
ma = -cx' - kx, a = x''
mx'' + cx' + kx = 0 and since c=0
mx'' + kx = 0
Nothing out of the...
Hi. We are learning about Laplace transforms at uni and I must say that this is a real pain. I have one questions concerning the concept of Laplace transforms, and also a question concerning a specific transform. The task is to make a Laplace transform of: t*sin(2t). I could do an integration by...
Homework Statement
Use Laplace transforms to derive an expression for the current flowing in the circuit shown in the figure, given that i = 0 when t=0
Homework Equations
Expression for the current in an LR series circuit
I(t) = V / R (1-e-Rt/L)
V - Volts (6V)
R - Ohms (10Ω)
L...
Homework Statement
A capacitor of 0.1 F and a resistor of 5 Ω are connected in series; the combination is applied to a step voltage of 20V. Determine the expression for the:
(a) current that flows in the circuit and
(b) the voltage across the capacitor in time domain.
Homework Equations...
Homework Statement
$${ { L } }^{ -1 }\{ \frac { s }{ { ({ s }^{ 2 }+1) }^{ 2 } } \} +{ { L } }^{ -1 }\{ \frac { 1 }{ { ({ s }^{ 2 }+1) }^{ 2 } } \}$$
Homework Equations
The Attempt at a Solution
I used ##{ { L } }\{ { t }^{ n }f(t)\} ={ (-1) }^{ n }\frac { { d }^{ n } }{ d{ s...
Hey guys, i have read many posts on physics forums but this would be my first post. I am quite stuck so any help would be much appreciated.
Homework Statement
Use Laplace transforms to solve the initial value problem:
f''(y) + 4f'(y) +8y = u(t-1) where y(0) = 1 and y'(0) = -1
Solve...
Homework Statement
##\int_0^\infty \frac{a}{a^2+x^2} dx##
Homework Equations
All the basic integration techniques.
The Attempt at a Solution
So, I saw this problem and wanted to try it using a different method then substitution, which can obviously solve it pretty easy. Since it is a very...
Homework Statement
Solve the DE for y(t) with the IC's
y(0)=20.8m/s and y'(0)=0
if the input is a step function scaled by the desired velocity Vo.
vd(t)=Vou(t).
Assume the desired velocity Vo=27.8m/s
Homework Equations
y''(t) + (D/M)y'(t) + (K/M)y(t) = (K/M)vd(t)
M = 1,000kg
D = 100kg/s
K...
Here is the question.
Here is a link to the question:
Can you show the steps of this differential equations problem? - Yahoo! Answers
I have posted a link there to this topic so the OP can find my response.