I think she did a smart thing. She literally couldn't keep him from doing serious harm to others, or from stealing and vandalizing all over the neighborhood, without doing things that would get her arrested and result in BOTH her kids ending up in foster care. Used to be that parents could keep kids like this locked up 24/7 if need be, and give them good sound whuppings too. Can't do that anymore.
A few years back there was a family (two parents) in New Jersey who had begged the juvenile justice system and mental health system to take their son and lock him up, because they just couldn't handle him. He was about 12 IIRC, and his behavior had gone sour after he'd been molested by an adult man. But the authorities refused to take the kid. Of course AFTER he murdered a much younger boy, THEN the authorities took him.
This mother wants to avoid that. Good for her. This kid may be hopelessly mentally ill, but if there's any hope for him, he urgently needs to learn that he cannot be part of a family or free community if he behaves the way he has been.
And she's right to protect herself since she has another son depending on her. She has to show sequence of events and that she's made every attempt to do the right thing. I just worry about the boy being placed in foster care or going home after his month is up. Either way, I'd be surprised if he suddenly stops acting out.