The problem of induction doesn't need to be solved because the problem refutes itself when time is brought into the picture.

Let's assume Hume's premise "there can be no demonstrative arguments to prove, that those instances, of which we have had no experience, resemble those, of which we have had experience" is true.

Ok, then how do we know this premise will stay true in the future? Because it is true right now? Because every moment in the future will eventually become the present? But how do we know that? Because it has always happened this way. Time has only ever moved forward, never backward, at least in my experience so far. Oops, this is inductive.

If the premise might not stay true in the future, then that means we've managed to acquire a piece of information about the future: the premise of the problem might not be true by then. We've just gained a bit of knowledge about the future which the premise claims we can't possibly do!