So on this practice exam paper problem #5 (page 7)

http://www.math.lsa.umich.edu/course...inalW10sol.pdf
I am told to find the Laplace Transform of:

tx''-3x'+x=0 where x(0)=0

Now I've always transformed tx'' in the following way:

Transform of tx'' -> -[(s^2)*X(s)-x'(0)]'

So I would get -[(2*s*X(s))+((s^2)*X'(s)))]

But the solution gives tx'' -> -(s*X(s)-x(0))'

Which gives -[(s*X'(s))+x(s)]

Why?

Even on my textbook: Differential Equations: Computing and Modelling by Edwards & Penney they are using the method that I have always used.