in the us you can leave anytime after giving birth, you just have to ask to sign an AMA discharge form (Against Medical Advice)... However, due to the pku test babies get at 24hrs , the baby generally has to stay that entire time, so mom's usually do to. Also, with insurance you're "entitled" to 48hrs after a vaginal birth, so most dont even think to question it. (I work in a hospital
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Another difference I've observed is internal exams in late pregnancy... Because of our medical intervention and OB-led care (mostly, not saying everyone has an OB..) internal exams to determin cervix and dialation generally start at 36 weeks here.