Finding Inverse Laplace Transform with Fractional Powers and Convolution

[Sol-chan]

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)e^2tt² + (25/7)e^-t-(25/7)e^2t+(75/7)te^2t 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...

 

[vela]

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.

 

[Sol-chan]

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 e^2t + 2e^2tt

 

[vela]

That's right. You may want to check your algebra on your earlier work. Mathematica gets a different answer than you do.