Laplace transform Definition and 752 Threads

Given $\begin{aligned} & {{u}_{t}}={{u}_{xx}},\text{ }x>0,\text{ }t>0 \\ & u(x,0)={{u}_{0}}, \\ & {{u}_{x}}(0,t)=u(0,t). \end{aligned} $ I need to apply the Laplace transform to solve it. I'll denote $u(x,s)=\mathcal L(u(x,\cdot))(s),$ so for the first line I have $s\cdot...

Homework Statement Homework Equations The Attempt at a Solution We went over the Laplace transformation today in my DE course. We only covered essentially, "how to do it", so: L(f(t)) = \int_{0}^{\infty}f(t)e^{-st}dt (also, how do you make the fancy f and curly brackets in Latex?)...

Hi everyone, I have a problem with finding the transfer function of a linear system. It happens that in some terms the input and the output are multiplying and I have no idea how to do the Laplace transform of this. The system is a car and I am only studying its speed. The input is the...

Homework Statement Find the inverse laplace transform of \frac{3s + 7}{s^{2} - 2s + 10}Homework Equations completing the square. e^{at}sin(bt) = \frac{b}{(s-a)^{2} + b^{2}} e^{at}cos(bt) = \frac{s-a}{(s-a)^{2} + b^{2}} The Attempt at a Solution F(s)= \frac{3s + 7}{s^{2} - 2s + 10} F(s) =...

Homework Statement I'm trying to find the Laplace transform of tJ''0(t), it's from bessels equation, but that doesn't matter too much at the moment, I just need to integrate (e^-st)*t*J''0(t) but am unsure how to go about this with the J''0(t) in there.

Sorry for thumb terminology, I just would like to grasp the main idea, as a physicist, without unnecessary complications, associated with system of axioms and definitions. Fourier transform can be seen as rotation of basis in space of all complex-valued functions from basis of delta-functions...

Using the laplace transform, find the solution to the differential equation: y'' + y' + y = 0 , y(0)=0, y'(0)=1 Using the laplace transform and its properties I end up with: f(s) = 1/(s2+s+1) How can I find the inverse of this/ does anyone know the inverse of it? Setting y=eax I got a...

How do I find the Laplace transform of 1 using MATLAB? >> laplace(1) gives the error ? Undefined function or method 'laplace' for input arguments of type 'double'.

Find the inverse Laplace transform of $\displaystyle \frac{2s+7-e^{-2s}}{(s+1)^2}$.

Find the Laplace transform of $\displaystyle f(t)=1$ if $\displaystyle 1\le t\le 4$; $\displaystyle f(t)=0$ if $\displaystyle t<1$ or if $\displaystyle t>4$.

Homework Statement I'd like to solve a DE using Laplace transform. \ddot y (t)+\omega ^2 y(t)=f(t) for all t>0. Initial conditions: y(0)=\dot y (0)=0. The dot denotes the derivative with respect to t. Homework Equations \mathbb{L}( \dot y )=s\mathbb{L}( y )-y(0). Convolution: if h=f*g...

Hi all I have been working on some unique solutions to advection-diffusion type problems. One inverse Laplace transform that I seem to continue to encounter is the following: Inverse Laplace[F(s)] where F(s)=[(1/(((s-α)^2)+β)*exp(-x*sqrt(s/D))] In their classic 1959 text, Carslaw...

Homework Statement y''+6'+10y=0 y(0)=2 y'(0)=1 Homework Equations The Attempt at a Solution Laplace everything and I get s^2*Y(s)-2s-1+6s*Y(s)-12+10Y(s)=0 isolate Y(s) Y(s)=(2s+13)/(s^2+6s+10) split into 2 terms, bottom can be rearranged by completing the square...

Homework Statement Given transfer function H(s)=s^2+4 and input x(t)=sin(2t), find the ouput y(t) in time domain, and show whether bounded or unbounded. Okay, so I know L^{-1}[sin(2t)]=2/(s^2+4)=2/[(s+j2)(s-j2)] and that Y(s)=H(s)X(s)=2 Therefore, y(t)=2\delta{(t)} However, I am a...

Homework Statement f(t)= 1 if 0≤t≤1 ; 0 is t>1 find the laplace transform Homework Equations The Attempt at a Solution I know u(t)= 0 for t<0 and 1 for t≥0 I know I have to shift it and get u_a(t)=u(t-a)= 1 if 0≤t≤a, 0 if a>1 am I even going the right way? then I think...

Homework Statement f(t) = (1/(a^2))(1-e^-at - ate^-at) Homework Equations f(t) = (1/a)(1-e^(-at)) F(s) = 1/s(s+a) f(t) = t e^(-at) F(s) = 1/(s+a)^2 The Attempt at a Solution F(s) = (1/s(s+a^2)) - (1/(s+a)^2)Totally lost now, but I think this is wrong anyway, can anyone help me please. I...

Homework Statement f(t)=te^t, find laplace Homework Equations The Attempt at a Solution I started doing integration by parts and after doing it three times I wasn't sure if I was going in the right direction/making any progress. I'm not supposed to use a table to solve this (I...

Homework Statement Use convolution theorem to solve: \mathfrak{L} \left \{t\int_{0}^{t} \sin \tau d\tau \right \} Do not solve the integral. Homework Equations "Convolution Theorem" in textbook is stated as: \mathfrak{L}\left \{ f*g \right \}=F(s)G(s)f*g=\int_{0}^{t} \ f(\tau )g(t-\tau...

Good day to you people, I have just started learning control theory at Uni, as part of my course, and I have to admit it is quite difficult to grasp. I am starting from the basics, and I am having difficulty understanding what 's' is supposed to represent as regards the Laplace transform...

Solve $\begin{eqnarray*} {{u}_{t}}&=&{{u}_{xx}},\text{ }0<x<\infty ,\text{ }0<t<\infty \\ u(0,t)&=&\sin t,\text{ }0<t<\infty \\ u(x,0)&=&0,\text{ }0\le x<\infty . \end{eqnarray*}$ I need to apply the Laplace transform to solve this, so by applying it I get...

Hi guys, Would you help me in finding how Laplace transform is done for equation (1) in the attached image? Equation (2) is the correct solution but I don't know how we get it! I know that we can split the integration in (1) into two parts; one is from t0 to 0 and the other part is from 0...

IVP Laplace Transform Problem -- Tricky Inverse Laplace Transform Homework Statement Solve x"+x'+x=1, given x(0)=x'(0)=0 Homework Equations The Attempt at a Solution Plugged in transforms: s2*Y(s)-s*y(0)-y'(0)+s*Y(s)-y(0)+Y(s)=1/s Plugged in initial value points, simplified...

Homework Statement Hello all, Having difficulty with this one question that involves complex roots. Here it is: F(s)=\frac{s+3}{s^3+3s^2+6s+4} I tried two different ways to tackle it. First method I divided it right away: F(s)=\frac{s+3}{s^3+3s^2+6s+4}\rightarrow{s^2+6-\frac{14}{s+3}} Is there...

Hi guys, would you please help me in my question which is in the attached image? Thanks

Hello all, Say one wants to find the inverse Laplace Transform of a function, and the method for attaining the solution is executed via partial fractions. Do the real numbers go with the complex numbers when determining the constants of partials? Perhaps this is wordy. I'll provide a theoretical...

Homework Statement I understand how to do initial value problems but I'm slightly stuck when the initial values are y(0) = y'(0)=0 The question is Solve: y''+3y''+2y=f(t), y(0)=y'(0)=0 where f(t) is a square wave. Homework Equations \Im{y'} =s\Im{y}-y(0)...

I tried doing this problem soooo many times for several days and keep failing and then I'm so worn out I can't keep thinking straight and I can't move on because of this! I'm attaching my work and it would be great if I can get back and forth feedback if necessary to find my mistake(s). The...

Homework Statement show that the Laplace transform of e^(At) = (sI - A)^(-1) \mathcal{L}\left\{ e^{At} \right\}(s) = \left(sI - A \right)^{-1} The Attempt at a Solution I find \left( e^{At} \right)_{ij} = \sum_{k=0}^{\infty} \frac{(A^k)_{ij}t^k}{k!} and since \mathcal{L}\left\{...

Homework Statement L^{-1}\{\frac{1}{(s^2+4)^2}\} Homework Equations The Attempt at a Solution I have no idea how to solve this. Any idea to being solving the problem would be appreciated.

If yes, how can we find it?

Homework Statement Homework Equations The Attempt at a Solution I don't know if it is possible to use Derivative of a transform while it is on unit step function procedure. How can these questions be solved? Thank you.

The link below shows a circuit and the s-domain equation for I1(s) that I have derived; My question is on the portion of the equation containing the "(s+a)" terms. I am trying to arrive at the time-domain equation for I1(s) but instead of arriving at the Laplace transform pair shown in the...

Homework Statement Find the laplace transform of f(t)=(t-5)e^{-17(t-5)}u(t-5) The Attempt at a Solution The answer I got was \frac{e^{-5(s+17)}}{(s+17)^{2}}. I thought I finally understood the process, but when I plugged it into wolfram alpha, the answer I got was different...

Homework Statement Compute the inverse Laplace transform of Homework Equations http://img267.imageshack.us/img267/696/66451772.png The Attempt at a Solution I tried using partial fraction but no luck.

So the question is: A curve rise from the origin of the xy plane into the 1st quadrant. The area under the curve from (0,0) to (x,y) is 1/5 the area of the rectangle with these points as opposite vertices. So I'm solving for f(x): So far what i have is: Area(D)=1/5 xy=integral 0 to x...

Homework Statement Say you have: EQ1: y1''*t+y1'*t+y2=0 and EQ2: y2''*t+y2'*t+y1=0 y1(0)=0,y1'(0)=0,y2(0)=0,y2'(0)=0 Homework Equations The Attempt at a Solution I can get it so far, but having both y1 and y2 really gives me fits: Eq1: Y1(-2s-1)+dY1/ds(-s2-s)=-Y2...

Hello all, Next semester I will be taking a Network Analysis course in my EE degree. Moreover, we will be utilizing numerous mathematical concepts I have not yet seen. If anyone has (preferably free) access to any of the concepts to follow that they would be willing to share, I would be...

f(t)=\int^{c+i\infty}_{c-i\infty}F(s)e^{st}ds Why we suppose that all singularities are left from line Re(s)=c?

Homework Statement Laplace Transform of u(t-∏/2)et (u is unit step function) Homework Equations Laplace Transform Table (any) The Attempt at a Solution I tried using the Laplace transform for the unit step function and the exponential function. L{u(t-∏/2)} = e-(∏s)/2...

Homework Statement Let f(t) = t if 0<t<3 et if t>3 a. Is f(t) piece-wise continuous? b. Is f(t) of exponential order α? Either prove it by producing an M, T and α that satisfies the definition, or show that no such constants exist. c. Does the Laplace transform of f(t) exist...

Homework Statement I'm attempting to find the inverse laplace transform of (5/((2x+3)(4+x^2))) The Attempt at a Solution Here's the solution: http://www.wolframalpha.com/input/?i=inverse+laplace+transform+%285%2F%28%282x%2B3%29%284%2Bx%5E2%29%29%29 There's 3 terms in the solution and 2 are...

Homework Statement f(t)=e^(t+7)Homework Equations £{f(t)}=∫e^(-st)f(t)dtThe Attempt at a Solution so i insert my f(t) into the formula, came up with ∫e^(-st+t+7)dt using u substitution, u=t(-s+1)+7, du=(-s+1)dt so it follows that 1/(-s+1)∫e^(u)du=e^(u)/(-s+1) so I plug u back in, and should be...

Hello, What does laplace transformation exactly 'do'? If I have PDE of second order and use LT on it, what do i get to solve? ODE? or if I have ODE of second order, what do I need to solve afet transformation? How does this work? is there any rule?!

I'm working on a LaPlace transform problem. Part of it was this: -ty' I elected to do this first: (-1)\frac{d}{ds}L(y') Which I then expanded to: -\frac{d}{ds}(-y(0)+sL(y)) By the given initial conditions y(0)=0 -\frac{d}{ds}(sL(y)) So next I need to expand this out: -(L(y)+sL(y)') Now I'm...

Hello I'm struggling to understand some basics here with the laplace transform.. I'm given the laplace transform of 2/(s + 4)^4 and I need to take the inverse of this to get back to y(t) Looking at my tables the only transform similar to this is 1/(s + a)^2 I understand I can pull...

Hello everyone, I'm currently enrolled in Control Theory at my University, and part of the coursework requires differential equations; which wouldn't be a problem, if not for the fact it's been 2 years since I've taken D.E. Anyway, over the course of the problem I ran into this little function...

Hello everyone, I'm currently enrolled in Control Theory at my University, and part of the coursework requires differential equations; which wouldn't be a problem, if not for the fact it's been 2 years since I've taken D.E. Anyway, over the course of the problem I ran into this little function...

Homework Statement F(s) = 3/(s(s^2 +2s + 5)) Homework Equations The Attempt at a Solution I have used partial fraction using coefficients. F(s) = (3/5)s - (3/5) ((s+2)/(s^2 +2s + 5)) and reduce s^2 +2s +5 by completing the square F(s) = (3/5)s - (3/5) ((s+2)/((s+1)^2 + 4))...

Homework Statement the questions asks to determine inverse laplace transform of s/(s+4)^4 Homework Equations The Attempt at a Solution this can supposedly be done just using laplace transform tables so I am guessing i need to simplify that to something that's workable but i...

Can someone explain the steps of this solution? In my linear systems class, we are doing Laplace transforms using transform tables and the properties. I can usually do the problems that closely resemble the table, but when they involve heavy algebraic manipulations, I don't know what to do...