One thing that stuck with me is not to blame yourself for the attitude of your pupils. There may be many things going on that you have no control over.
There are various differences to consider:
In the state sector, perhaps more than private, it is important to remember that the child you think is not engaging may be having a bad time at home, may not have had breakfast or perhaps is not getting enough sleep. Many children grow up in far from ideal circumstances that can impact their performance at school so they fail to reach their potential.
At private school parents have invested in education because they deem it important, so passing exams may be the most important thing happening in the child's life. In a state school, some pupils and their parents will perhaps not see the point of education resulting in much lower levels of effort.
Pupils at state school often don't have the self-confidence that is usually evident in private school pupils. This can be a self-fulfilling prophecy leading to failure. It can take time to build self-esteem. Whereas at private school you may want to pitch your lessons to stretch the pupils, if your class is struggling then it becomes more important to start with something easily achievable that will instill pride in pupils and encourage them to keep trying.
The best thing you can do is provide some stability for them, try to nurture an interest in the subject and give encouragement to build their confidence which in turn may increase engagement.
Apologies if I am telling you what you already know and stating the obvious, but when I was starting out I was too inward looking, blaming myself for behaviour issues, etc. A mentor suggested I stopped focusing on what I needed to do and focused more on what the children needed from me. I gained more awareness of the issues that may impact their behaviour and stopped execting too much from them. I appreciate that this can be difficult given the confines of a set curriculum but anything you can do will help.