How to determine the order and degree of this differential equation, (y''')^2-2(y')^6=x-2

1st step: go to your DE book and look it up, or go online. Google is your friend.

order: highest derivative
degree: highest power of highest derivative

I can't believe you could not look that up for yourself.

www.ul.ie/cemtl/pdf%20files/bm2/DegreeOrder.pdf