If you understood (i) and not (ii) then you're probably thinking about (ii) too hard. I would say it's simply because both sides of the equation have to be equal. If and then you must have .
It's the way they're wording it that's confusing you, they are saying it "satisfies" the equation, but we often refer to zeros as trivial solutions. If I have a diff eq. like for instance then one obvious "solution" is since you can plug it into the diff eq. and it's true. It's just uninteresting. We're more interested in the solns. obtained by separating variables and integrating.