How Do You Solve a Laplace Transform with Partial Fractions?

[ihatelaplace]

Homework Statement

Got stuck doing the transform on a partial fraction that's giving me trouble

Homework Equations

y" + 2y' + 2y = 0 IC: F(0) = 1 F'(0) = -3

The Attempt at a Solution

Im getting stuck with a partial fraction of (s-1) / (s^2 + 2s + 2)
the only thing i can think of is As+b = 1s-1 A = 1 B = -1 and i know if i change D(s) to (s+1)^2 +1 i can get e^(-t).
(s - a)^2 + b^2 means a = -1 and b = +/- 1 then As+b / s^2 + 2s + 2 = [b - (s + a)] / [(s-a)^2 +b^2] and i get e^-t * [sin(-t) - cos(-t)] I don't think it is right tho

 

[tiny-tim]

Welcome to PF!

Hi ihatelaplace! Welcome to PF!

(try using the X² tag just above the Reply box )

Your solution is nearly right, but I don't follow how you got there.

Why not just use the standard -b ± √(b² - ac) solution to the characteristic equation …

that'll give you roots of the form p ± iq, for which the solutions are … ?