Question

use the laplace transform to solve y''-y=-20(delta)(t-3)

Answer 1

What are the initial conditions?

Answer 2

L(y") = s²Y(s) - sY(0) - Y'(0) and L(δ(t-3))=exp(-3s)
So your equations becomes : (s²-1)Y(s) - sY(0) - Y'(0) = -20exp(-3s)
And @SithsAndGiggles is right, you need the initial conditions (Y(0) and Y'(0)) to solve this.