ODE please help rush!!?

Question

Find the Laplace transform of f(t)=[e^(-2t)]cosh(3t)

Find the Laplace inverse transform of F(s)=(s+2)^2/(s+3)^5

And last one is Use the Laplace transform to solve the equation
y^(2)-2y^(1)+y=f(t), y(0)=y^(1)=0 where f(t) ={t^2, 0<(or equal)t<1 and 2t-1, t>(or equal) 1


I think I am just missing a step in each of these that is causing me to get a very strange answer for them. Help on any of these would be appreciated.

Answer 1

The Laplace transform of cosh(3t) is s/(s² - 9). Then, by the shifting theorem, the Laplace transform of e^(-2t) cosh(3t) is (s+2)/((s+2)² - 9)

For (s+2)²/(s+3)^5, write this as ((s+3)-1)²/(s+3)^5

This is the Laplace transform of f(t)e^(-3t) where the Laplace transform of f(t) is

(s-1)²/s^5

Expand the above:

(s-1)²/s^5 = (s² -2s + 1)/s^5 = 1/s^3 - 2/s^4 + 1/s^5

So f(t) = t²/2! - 2t³/3! + (t^4)/4! = t²/2 - t³/3 + (t^4)/24

so the desired inverse Laplace transform is

e^(-3t)[t²/2 - t³/3 + (t^4)/24]

I'm still working on the last problem...