You can consider that x is a constant. Thus its second derivate will be null and you'll have an equation with xp
I hope for you it's true that , because this is something i can't do ^^
I really don't know for xc... I don't remember the rules for general solution ! If you're not sure, derivate again to see if you get xc so that xc is solution of d²xc/dt²+5w²xc=0
However, for xp :
xp is a particular solution. Let X be a constant satisfying the equation.
The second derivate of X is 0.
Moo dropped a 5. It was a simple error, easily spotted. As you did. If you understand what Moo was doing to get the particular solution, then all's well. So go back, put the 5 in where it's meant to go. There's the answer.
Life's full of small mistakes, most easily spotted and corrected for. The world keeps turning. Don't have cow, man.