Laplace transform Definition and 779 Threads

Homework Statement Derive he Laplace Transform of the third derivative of f(t). Homework Equations The Attempt at a Solution So, I'm not at all sure how to do this. I think I can start with: L{f'''(t)} = But I'm honestly not sure how this works. Any guidance would be...

I am reviewing some material on Laplace Transforms, specifically in the context of solving PDEs, and have a question. Suppose I have an Inverse Laplace Transform of the form u(s,t)=e^((as^2+bs)t) where a,b<0. How can I invert this with respect to s, giving a function u(x,t)? Would the inverse...

Homework Statement The Attempt at a Solution I know that u(t) is a unit step function and holds a value of either 0 or 1. In laplace transform, when we integrate f(t) from 0 to infinity, we take u(t) to be 1. In this case, since u(t) is u(-t), does this mean it holds a value of 0? Does not...

Homework Statement Find the inverse Laplace transform of e^(-3pi*s)/(s^2+2s+3). Homework Equations I know that you're supposed to factor out the e^(-3pi*s) and the other part becomes 1/(s+1)^2+2 but how do you get the answer? I'm confused. The Attempt at a Solution The answer is...

Homework Statement Find the Inverse Laplace Transform of \frac{1}{(s^{2} + 1)^{2}} Homework Equations The Attempt at a Solution I tried using partial fractions but it didn't work. It looks like a cosine transform, but I don't know what else to do. Help please :(

Hi All, in a previous post on the physical meaning of Laplace's Transform I found the following statement " The fundamental Laplace transform pair is H(t), the Heaviside step function, and 1/s, its spectrum of damped sinusoids. Note that the spectrum is weighted towards low frequencies...

Homework Statement Find the inverse Laplace transform of F(s)=(8s^2-4s+12)/(s(s^2+4)). Homework Equations A/s+(Bs+C)/(s^2+4) 8s^2-4s+12=A(s^2+4)+(Bs+C)(s)=As^2+4A+Bs^2+Cs=s^2(A+B)+Cs+4A 8=A+B C=-4 A=3 B=5 L^-1 (3/s)+L^-1 ((5s-4)/(s^2+4)) =3+ (Now I'm stucked.) The Attempt...

Homework Statement Find the inverse Laplace transform of F(s)=(2s-3)/(s^2-4). Homework Equations I don't want to find the answer by looking at the Table. F(s)=2s/(s^2-4)-3/(s^2-4) The Attempt at a Solution The answer is f(t)=2 cosh 2t - (3/2) sinh 2t.

I attached the problem as a word document. I'm stuck trying to determine the laplace transform for t-tU(t-1). I know I'm supposed to work with 1/s^2(s+2) and solve for A, B,C. I got B=1/2, A=-1/4, and C=1/4 when 1=(As+B)(s+2)+Cs^2. The answer to the problem is y= 1/4 + 1/2t +1/4 e^-2t -[1/4...

Having difficulty with L-1 {1/(s^2+s-20)}: Should I make it L-1 {1/(s+5)(s-4)}? I'm stuck.

what is the inverse laplace transform of (2s)(1/(s-2))? could i use the identity ∫f(T)g(t-T)dT=F(s)G(s)? i was hesitant so i figured i'd just ask before i continue..

In the attachment that I added I highlighted the portion I am questioning. I will define L[f(t)](s) to be the laplace transform of the function f(t). f(t) = e^t L[f(t)](s) = 1/(s-1). The laplace transform is defined for all values s≠1. L[f(t)](2) = 1. Question: "What do they mean by...

Homework Statement y(t) solves the following IVP y''(t) + 2y'(t) + 10y(t) = r(t) y(0) = 2 y'(0) = 3 r(t) = 0 if t < 0 t if 0 ≤ t ≤ 1 0 if t > 1 Demonstrate that the laplace transform of y(t) is Y(s) = \frac{2s+7}{s^{2}+2s+7} + \frac{e^{-s}}{s(s^{2}+2s+7)} +...

Homework Statement (6-t)heaviside(t-2) This is just one term of the real problem I'm working, but it will serve to help me figure this out. Homework Equations The Attempt at a Solution http://www.wolframalpha.com/input/?i=laplace+transform+%7B%286-t%29heaviside%28t-2%29%7D...

Homework Statement \dfrac{\partial^2 \varphi }{ \partial x^2} - \dfrac{\partial ^2 \varphi }{\partial t^2} = 1 Initial Conditions: \varphi (x, 0) = 1; \varphi_t (x, 0) = 1 Boundary Condition: \varphi (0, t) = 1 On 0 \leq x < \infty, 0 \leq t < \infty...

Homework Statement Use the Laplace transform approach to find the renewal function for a renewal process with interrenewal p.d.f. as follows: g(x) = (c^2)xe^(-cx) , x > 0 The Attempt at a Solution M*(s) = G*(s)/(1-G*(s)) where M*(s) and G*(s) denote laplace transforms I have that G*(s) =...

Google seems to provide not much information on this. In essence, I am asking about the eigenfunctions of the Laplace transform when λ=1? Anyone have any insights on this rather unusual problem? BiP

Homework Statement Evaluate the laplace transform of {t2e7tsinh(3t)} Homework Equations Laplace transform of {tnf(t)}=(-1)ndn/ds2 * F(s) The Attempt at a Solution I've replaced it with (-1)2d2L{e7tsinh(3t)} I'm not sure how to proceed, though, as I don't really see how to take...

Hey all, Learning the Laplace transform and I get the point that it is a transformation but I would like to know what are some of the merits of the Laplace transform or more general why perform transformations in the first place. Any examples would be helpful.

I've been wondering whether the Laplace transform is injective. Suppose I have that \int^{∞}_{0}e^{-st}f(t)dt = \int^{∞}_{0}e^{-st}g(t)dt for all s for which both integrals converge. Then is it true that f(t) = g(t) ? If so, any hints on how I might prove it? Thanks! BiP

I'm not sure about the answers to parts of the limit step of the work. Can you explain what the answer to that step should be?

1. The limit as b approaches infinity always shows up as undefined on my calc so I don't know what to put for that section of the work. 2. What pat of the work is supposed to need L'hopital's rule? The integration?

Homework Statement Find the inverse transform of the function F(s) = log\frac{s-2}{s+2} Homework Equations L(\frac{f(t)}{t}) = \int^{∞}_{s}F(x)dx f(t) = tL^{-1}(\int^{∞}_{s}F(x)dx) The Attempt at a Solution I missed the lecture on this and while I was able to figure out...

Homework Statement L-1{\frac{s}{s^2+4s+5}} Homework Equations \frac{s-a}{(s-a)^2+k^2} \frac{k}{(s-a)^2+k^2} The Attempt at a Solution I completed the square for the denominator and got: L-1{\frac{s}{(s+2)^2+1}} (a= -2, k=1) But how do I get rid of the s in the numerator? Or do I have...

Hey guys! I'm stuck on a Laplace transform. Following is the problematic function: [cos(t)]^3 Seems simple, but I'm having issues doing the Laplace transform on odd trigonometric functions. When I use the half-angle formula, I get this, which I can't seem to solve: 1/2cos(t) +...

Here is the question: Here is a link to the question: Consider the initial value problem for y; Laplace? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement ODE: y'' + 4y' + 3y = f(t) f(t) = (?? HELP - What's the mathematical term to describe these? I can't seem t o find it in my notes :cry: ) 1, 0 ≤ t < 2 t², 2 ≤ t < 3 0, t ≥ 3 Write a brief description on how you would solve this ODE using Laplace transforms. Also use the...

Homework Statement Differential equation: \frac{d^2x}{dt^2}+4\frac{dx}{dt}+4x=6e^{-2t} with initial conditions x(0)=-2 and \frac{dx}{dt}(0)=8 Use the laplace transform to solve for x(t) Homework Equations http://www.atp.ruhr-uni-bochum.de/rt1/syscontrol/node9.html...

Homework Statement A system is characterized by the equation y' + 3y = r' . When the input is r(t) = u(t) - u(t-1), find y(t) by taking the inverse Laplace transform of Y(s).Homework Equations The Laplace transform integral The Laplace transform of a derivative sF(s) - f(0) The transfer...

Hi I was wondering what the Laplace transform of a squared differential is. With that I mean the Laplace of (y' )^2 (this being y'*y' and not the second order derivative). So for example velocity squared.

Homework Statement Homework Equations The Attempt at a Solution The answer is y(t) = t^{4}+\frac{t^{6}}{30} Don't know what to do next any advices please

Homework Statement Find the Laplace transform of the following function t[SUP]2[SUP] - 2t The Attempt at a Solution \[\begin{gathered} f\left( t \right) = {t^2} - 2t \hfill \\ l\left( {f\left( t \right)} \right) = F\left( s \right) = \int_0^\infty {{e^{ - st}} \cdot \left( {{t^2} - 2t}...

Homework Statement Find the laplace transform of: v(t) = sin(7t)u(t-3) Homework Equations u(t) is step responseThe Attempt at a Solution I know that taken alone the laplace of: sin(7t) is 7/(s^2+49) u(t-3) is (1/s)*(e^-3s)I don't understand how I would figure out the answer to this...

Homework Statement f(t) = cos (pi*t) if 1\leq t <4 and 0 elsewhere using unit step functions to find Laplace TransformHomework Equations The Attempt at a Solution I came up with the unit step function f(t) = cos(pi*t) u(t-1) - cos(pi*t) u(t-4) in order to use the second shifting theorem f(t)...

Homework Statement find the inverse laplace transformation of \frac{5s+4}{s^2} e^{-2s} Homework EquationsThe Attempt at a Solution I have tried to partial fractions \frac{5s+4}{s^2} and I got \frac{5}{s}+\frac{4}{s^2} and I know that the answer must have u(t-2) because of second shifting (...

I am doing a laplace transform as part of a coursework assignment. I have some example transpositions that are relevant to the question I am answering but I can't see how the author has got from one arrangement to the next. [b]2. Homework Equations He has given 1/(s^2(τs+1)) =...

I have a question asking for the inverse laplace transform of (e^(-s))/(s^2+pi^2). I split it up to (e^(-s))/s x s/(s^2+pi^2) and got u(t-1)cos(pi(t-1)),but the correct answer is (sin(pi(t-1)/pi)u(t-1). So here it was split up to (e^(-s))/pi x pi/(s^2+pi^2) and I don't understand where the...

Homework Statement What is the inverse laplace transform of (1/(s+s^3))? Homework Equations The Attempt at a Solution I looked it up on wolframalpha and got 1-cos(t), but I don't understand how they got that answer. I looked up a basic laplace transform chart and didn't see anything...

Homework Statement Find inverse Laplace transform \mathcal {L}^{-1}[\frac{1}{(s^2+a^2)^2}]Homework Equations The Attempt at a Solution I try with theorem \mathcal{L}[f(t)*g(t)]=F(s)G(s) So this is some multiple of \mathcal{L}[\sin at*\sin at] So \mathcal {L}^{-1}[\frac{1}{(s^2+a^2)^2}]=\propto...

Hello, I am trying to figure out in my notes how my professor did L[(e^-3t)(sin2t)] = 2/(s+3)^2 +4 I think she just did it in her head and wrote it, so I don't actually know how to solve it. I am looking at my table of laplace transforms and there is none for a product of an exponential and...

I'm having trouble with this question. Can anyone please guide me. My Attempt : = 1/(s-1) * L{t^2*e^-t} = 1/(s-1) * (2/(s+1)^3) = 2/((s-1)(s+1)^3)) but that's not the answer , its 2/((s-1) s^3) somehow. The question is attached below.

Homework Statement Find the inverse laplace transform of (2/(s+2)^4) using the given table of identities: Homework Equations Here are the given identities: The Attempt at a Solution Alright, I realize that there is a simple identity that I can use with a factorial symbol, but this...

I thought it would be obvious, but I can't find a series representation of the Laplace transform. I'm looking for something analogous to the Fourier series and how it can be used to derive the Fourier transform. I though it would simply be f(x) = \sum_{s=-\infty}^{\infty}{C_{s} e^{sx}} , but...

This is a conceptual question on the region of convergence (ROC) and the inverse Laplace transform (ILT). Here the bilateral laplace transform (LT) and the ILT are given by F(s)=L\{f(t)\}=\int_{-\infty}^{+\infty} f(t) e^{-st} dt and f(t)=L^{-1}\{F(s)\}=\frac{1}{i...

why was laplace transform developed i have googled it and found that it is something about shaping a family of exponential and vector projections etc i couldn't get it. some simply said that it was used to make a linear differential equation to algebraic equation but i couldn't understand how...

Homework Statement L[t^{2} - t^{2}δ(t-1)] Homework Equations L[ t^{n}f(t)] = (-1^{n}) \frac{d^{n}}{ds^{n}} L[f(t)] L[δ-t] = e^-ts The Attempt at a Solution My teacher wrote \frac{2}{s^{3}} -e^{s} as the answer. I got \frac{2}{s^{3}} + \frac{e^-s}{s} + 2 \frac{e^-s}{s^2} + \frac{2e^-s}{s^3}

Homework Statement Use Laplace transform to the system: \frac{dy}{dt} + 6y = \frac{dx}{dt}3x - \frac{dx}{dt} = 2\frac{dy}{dt} x(0) = 2 ; y(0) = 3 The Attempt at a Solution I've tried everything on this one. I first solved \frac{dy}{dt} + 6y = 2\frac{dy}{dt} and I got y = 3e^{6t} ...

Homework Statement Problem 8-19 in Matthews and Walker's book on mathematical physics. A straight wire of radius a is immersed in an infinite volume of liquid. Initially the wire and the liquid have temperature T=0. At time t=0, the wire is suddenly raised to temperature ##T_0## and...

Homework Statement I must solve the following diff. eq. ##tx''-(4t+1)x'+(4t+2)x=0## with the initial condition ##x(0)=0## and the relations ##\mathcal {L }[tx]=-\frac{d \mathcal{L}[x]}{ds}##, ##\mathcal {L} [tx']=-\frac{d [s \mathcal {L}[x]]}{ds}## and ##\mathcal{L}[x']=s \mathcal...

Homework Statement this one stumped me.. d^2y/dt^2 +ωy=ksin((√ω)t) y(∏/4)=0, y'(∏/4)=0 The Attempt at a Solution → (s^2 + ω)U(s)= LT {ksin((√ω)(T+∏/4)} is as far as i can get (i know what to do with the left hand side once i get the LT of the right hand side but i don't know what to do with...