How can i find out if a child has been born in the state of arkansas? I know the hospitals name, and the birth mothers name but not the childs or when the child was born.

I think it will be more than enough info you need. Hospitals has to keep all the birth records. Mother's name and date of birth will surely ease your work. Child name is usually not required. So, visit that hospital and see if they are ready to show their records or not.
Good luck.