Finding Inverse Laplace Transform with Fractional Powers and Convolution

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Homework Help Overview

The discussion revolves around finding the inverse Laplace transform of a function involving fractional powers and convolution. The subject area includes Laplace transforms and potentially convolution operations in the context of differential equations.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to compute the inverse Laplace transform for a given expression but expresses uncertainty about the application of convolution. Some participants suggest rewriting terms to facilitate the transformation process, while others question the accuracy of the initial algebraic manipulations.

Discussion Status

The discussion is active, with participants exploring different approaches to the problem. Some guidance has been offered regarding the manipulation of terms, and there is acknowledgment of differing results from computational tools, indicating a productive exchange of ideas.

Contextual Notes

Participants are navigating the complexities of Laplace transforms and convolution, with some uncertainty about the algebra involved and the implications of their manipulations. There is a mention of potential discrepancies between manual calculations and results obtained from software.

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Homework Statement

Find the inverse laplace transform of:

1/(s-2)3 + 25/(s+1)(s-2)2 + s/(s-2)2


The attempt at a solution

I get (1/2)e2tt2 + (25/7)e-t-(25/7)e2t+(75/7)te2t for the first two, but I'm not even sure where to start for s/(s-2)2. I was thinking it might use convolution, but I'm not sure that I understand how convolution works... I'd really just like a hint to get me started on it...
 
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1/(s-2)2 looks like a shifted form of 1/s2, so you want to write numerator in terms of (s-2), i.e. s = (s-2)+2.
 
Okay, so then I would have (s-2)/(s-2)2 + 2/(s-2)2

which is 1/(s-2) + 2/(s-2)2

so then I'd get e2t + 2e2tt
 
That's right. You may want to check your algebra on your earlier work. Mathematica gets a different answer than you do.
 

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