Warning: idiosyncratic opinion bombastically expressed incoming. Really this is just the vigorously expressed opinion of an amateur, FWIW.
:)
I wouldn't worry about it, Hume's problem of induction is way overblown and really falls out of the hoary old Empiricist stance on "impressions," or the "the Way of Ideas." If you believe that we are in immediate contact only with our "impressions," and only mediately in contact with things (or not), then and only then does Hume's problem (or any modern "sceptical" philosophical problem really) have any bite. Another way of putting the "Way of Ideas" would be that if we know only seemings-to-be, then we are blocked off from knowing actually-ises (or: we have a problem getting to is-es via seemings-to-be). But of course that's nonsense, because things can both seem to be
and actually be as they seem to be; we're not permanently cut off from the way things really are by an opaque inner phenomenological screen of seemings-to-be ("impressions").
Induction works on the basis of things having definable identities, natures or essences and behaving consistently according to their nature. So if you peg a thing's nature - IOW: if either (if it's a new thing) you posit an identity for it that pegs the nature it actually has, or (if it's an already-known thing) you identify it correctly - then you can expect behaviour from it that's consistent with that nature under given circumstances. (In particular, you can expect certain consequences for experience if you poke and prod it, or "interrogate" it, scientifically. This is the same generate-and-test method as science generally, hence science
is induction.)
That's really all induction is. If a thing has identity A, then (under given circumstances) it necessarily behaved in accordance with identity A yesterday, and it necessarily will behave in accordance with identity A today and tomorrow. Percentage certainties arise just because practically speaking there's always room for error in identification, IOW, you can't be fully certain that you've pegged anything's identity properly (this is really what was highlighted by Nelson Goodman's "new problem of induction" - the stuff about "grue" and "bleen" and all that). IOW, all other circumstances being equal, if the thing you're calling "sun" really is the sun (as we understand it now: the star at the center of our solar system, but earlier, more approximate identifications had the same implications for it rising and setting), then it will rise tomorrow with certainty. But it might be something else - an alien construct, a mischievous projection on everyone's retina by a mad scientist, etc., etc., etc. Or all things might not be equal (i.e. circumstances might have changed, unbeknownst to you - e.g. the law of gravity might have changed). But everything being equal, if it's truly the sun,
then it cannot possibly not rise tomorrow.
Another way of saying this would be: logical necessity is a feature of things (or of semantics), not of how we talk about things (or of syntax). That was the classical understanding, which changed with "modern" philosophy, which made logical necessity a feature of talk about things (or of syntax); but that was a mistake.
Aristotle said "induction is easy" - again, all an induction is, is a deduction arising from the
positing of an identity/nature/essence (which is then - the identification - confirmed/disconfirmed by experience), or from the correct
identification of something whose identity has already been settled. The reason why induction has seemed to be problematic is because of a series of confusions arising during the Scholastic period, when some comments about induction in a particular logical context by Aristotle were taken out of context and misinterpreted (it's to do with something called "induction by enumeration", not sure I fully understand the confusion myself, it's quite complicated). The empiricist discussions culminating in Hume's problem only further muddied the waters.
Really, "modern" philosophy was a complete mess that we're only just now (after the later Wittgenstein and a few others) starting to recover from. The only modern philosopher worth a toss, ultimately, was Thomas Reid, and
maybe (on a certain minority interpretation) Kant (but since, if that interpretation is correct, he's been largely misinterpreted, it hardly matters, in that the real Kant wasn't the famous Kant in philosophy books). That's not to say modern philosophy wasn't wonderful in the sense of being an intellectual adventure, or that its thinkers weren't great and profound and sometimes said important, true things; also a lot has been learned from it (mainly how
not to go about doing philosophy). But ultimately it's been a giant detour that's led to no end of trouble in both philosophy and politics (particularly the megadeaths of Communism and Fascism) - and a lot of crappy 20th century art to boot.