I'm not a parent but I've been a teenager!
I think most of the points have been covered so I won't repeat them, but one thing I wanted to say was that if they haven't done their homework for ages they are probably way behind and don't understand it. When that happens teenagers go one of two ways - they try their hardest to catch up and they might succeed, but they also run the risk of failing.
Teenagers often feel like this (I think a lot of people do) - that failing after doing your best is sometimes more humiliating than failing because you didn't try. When you try your best and fail there's no excuse other than your own inability, but when you fail because you haven't tried then the reason is because you didn't do it, not because you can't. There's always the possibility that you could have so it shows that "I could, but I don't wanna."
I'm not tying to justify their laziness, what I'm saying is that perhaps they feel way too far behind that they've given up, and that it's too scary to risk failure. Sort of a "Better to keep your mouth shut and have people think you're a fool, than open it and erase all doubt" (but obviously more eloquent than that!)
I guess what I'm having difficulty saying is that their laziness might be because they don't see you as a father, or because they don't have a stable family life, or it may be because they need lots and lots of help with their homework to get back up to speed, and that's a real scary step. Of course I'm just going by my experiences here, but perhaps the younger child is being influenced by the 15 year old. If you got the 13 year old alone and offered to help him a lot - instead of threatening him or rewarding his success - and do his homework with him, it might be less scary and he might be able to catch up. He's probably following the 15 year old's actions.
I'm not sure if anything I've said here is true but I figured I'd offer that since lots of other points have been covered. I know that in my experience the risk of failure despite hard work can often be more humiliating than failure because of apathy, or projected apathy.
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