I had the same issue with my son and we were homeschooling. He's say he's done his best (which was true for the 5-10 minutes he put into it. I couldn't seem to convince him that I knew he could do better if he used more time efficiently.
Your boys are counting on you giving up and the whole issue going away. I personally don't recommend that. Perhaps if you set aside an hour (or appropriate time for each boy) as homework time. No getting it done early, if you finish in less than an hour on the "immediate" homework, start work on longer term projects such as reading a book for an upcoming book report, term paper, etc.
Talk with their teachers (as it sounds like you have) and get a exact list of their homework. If they fail to cooperate with this system, MAKE THEM CARRY a notebook that is written in by you and the teachers EACH DAY. I suspect the school will enforce that with you. They will HATE that.
During this "Homework hour" either you or your husband occupy the younger sibling with stories, bath, etc. You booth will benefit from the time together and that removes the boys excuse.
Forget the yelling. It is probably only upsetting you and not them anyway. It's a means of delaying homework. Just set the rules. If they are not obeyed, make them stay longer at the homework table. By the way, provide a place away from distractions that is comfortable for each boy. Some like a table and chairs, others prefer their lap. Many teens have gotten used to studying with headphones on. When I was in the school system they told me it was to block out the other noises. They had to keep them low enough so no one else could hear them. School rule. I used to try to make my son study at the table because that was how I was comfortable. Turned out for him it was generally lying on his bed or floor (when he could find it).
If possible also separate the boys during this time. It might be a good time for no TV for anyone if that is an issue. Make sure the computer is available if they need it, and make sure they stay on task. Sometimes we assume they know more about using the computer than we do. But when it comes to searching the net for a specific topic, etc. They aren't always as well versed as they think they are.
Make them responsible for your requirements. put more time into checking that the "completed paper" is for the current assignment. It will take more time on your part, but once he figures out you're going to do that, he'll smarten up.
This is extreme but it worked. An oriental friend of mine found out her grade school son was not doing his work. She got ALL UNCOMPLETED assignments from his teacher, took a weekend and sat right beside him while he did each and every one. Her husband said a weekend was too long for the boy to sit and do schoolwork. After all he's only a kid. But she stuck to her guns. After that experience her son's response was, I'm not getting behind again. I don't ever want to spend a weekend like that again!
Good luck and I'd love to hear how it all ends up.
By the way, my "lazy" homeschooled son is now 24, has two children and is an excellent worker on his job and a very responsible father and husband. He has beem tested by fire in his marriage (wife had an affair) but through it all he realized he wanted his family and her more than anything, forgave her, took her back and they are now working on becoming a couple again. So, lack of responsility now, doesn't necessarily mean a bad outcome in the future. Your boys may not see any reason why they should do this "busywork". So there is alot of hope ahead for your boys. In spite of their teen years ! Just be there for them ALL the time.
You mentioned depression. Has that been dealt with and are you comfortable with it now? If not, look into visiting a counselor. Most places have a sliding fee scale if you don't have mental health insurance. I live with depression, am caring for my terminally ill mother and have had to make numerous adjustments in my life. I find my counselor invalueable. I also find my medication invalueable also, as it allows me to live a normal live. Before I was constantly dealing with, "I'm having all I can do to maintain my cool, and you dare to breathe the same air I am!" There is help there. Use it. I've been to the ER for OD's etc. It isn't worth it. Get help before you need help and can't get it. Getting into the mental health system sometimes takes time.
Again, I'd love to hear how you make our. Feel free to reply to me privately or here on the posting.
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P. Bublat