Thanks for writing in. Isn't it interesting how two brothers can be so different in so many areas? What are some other things or activities your younger son values besides basketball? Sometimes, to motivate kids to take care of their academic requirements, it's effective to require that they cannot access the things they like until their assignments are completed. This gives them a personal reason and motivation to complete them. If you set something up like this, be sure you can enforce it.
As far as your expectation for your older son to be a friend and coach to his younger brother, it is not unrealistic. However, that probably won't happen as long as he is allowed to be critical or make fun of his brother. Those are disrespectful actions whether they are toward his brother or anyone else. It may be a good idea to address with the boys how to show respect to each other and with others. You can teach them what respect is, what it looks, what it sounds like, and exactly how to be respectful. In some families, if one family member is disrespectful to another with a comment, a sigh, or even a roll of the eyes, there is an immediate consequence that consists of an apology, including doing an act of kindness for the person they were disrespectful to. You might try that, too.
Your older son probably would not treat his friends the way he treats his brother or he wouldn't have good friends like he does. So, it is likely he already knows his behavior is not acceptable, but he is getting by with it at home. As long as this is allowed to continue, your hope that they could be friends is probably not going to happen. So, at a neutral time, we recommend you address this to both of them as a concept or a character trait that you want them both to have in your home and throughout their lives.
Let us know how this goes, and if additional suggestions are needed, don't hesitate to let us know. We will do what we can to help.