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

.

Therefore

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.