Laplace transforms to solve initial value DE / partial fractions

[CrazyCamo]

Hey guys, i have read many posts on physics forums but this would be my first post. I am quite stuck so any help would be much appreciated.

Homework Statement

Use Laplace transforms to solve the initial value problem:

f''(y) + 4f'(y) +8y = u(t-1) where y(0) = 1 and y'(0) = -1

Solve this problem using laplace transforms, showing all steps in your reasoning. State the solution y(t) for each of 0<t<1 and t>1, then sketch it over the range 0<= t <= 10, noting its main features.

Homework Equations

The Attempt at a Solution

I have gotten up to F(Y) = (e^-s + s^2 + 3s)/(s(s^2 + 4s+8))

However, from here i am not sure what to do. I tried taking the partial fractions of:

1/(s(s^2 + 4s+8))

but am getting very confused. Again any help would be much appreciated. Cheers

 

[SteamKing]

Why don't you break F(Y) down into the sum of its various components?

E.g., D = (s(s^2+4s+8))

F(Y) = (e^-s)/D + s^2/D + 3s/D

You can tackle each term individually.

PS: finding the PFE of 1/D doesn't help.

 

[HallsofIvy]

The denominator is $$s(s^2+4s+ 8)= s(s^2+ 4s+ 4+ 4)= s((s+ 2)^2+ 4)$$ so you can use "partial fractions to write that as $$\frac{A}s+ \frac{Bs+ C}{(s+2)^2+ 4}$$.