My 10 year old son has had two bouts of vile smelling burps followed by vomiting. The vomit doesn t smell like vomit. It smells like the burps. Like rotten egg. The vomiting is at night. The burps stop after the vomiting. The first bout he had a fever and loose stools and abdominal pain but the second was just burps all day then vomited at 4 am.
Hi...this looks like a stomach flu. Usually this will start with fever and then vomiting and then diarrhea (loose stools).  Once it starts it will take 5-7 days to completely get better. Unless the kids having low urine output or very dull or excessively sleepy or blood in motion or green bilious vomiting...you need not worry. There is no need to use antibiotics unless there is blood in the motion. Antibiotics might worsen if unnecessarily used causing antibiotic associated diarrhea. I suggest you use zinc supplements & ORS (Each small packet mixed in 200ml of potable water and keep giving sip by sip) as HY Chat Doctor.  If there is vomiting you can use Syrup Ondansetron (as prescribed by your pediatrician). Regards -