Ok, First time this happens to me so I'm a bit freaked out.
Colt is 2 days old and has a persistent urachus: basically his umbilicus is not sealing up and letting some urine driblle out.
He is on an agressive antibiotic course to prevent infection.
Colt is 2 days old and has a persistent urachus...
Do they close on their own?
If it's still open at 2 days, then it's probably unlikely to close by itself. The problem is a catch-22 - as long as it's wet, it won't seal off; as long as it won't seal off, the urine will continue to leak and keep it wet. The usual treatment is use of a caustic agent such as silver nitrate to attempt to cauterize it. It is a "main line" right into the foal's system for bacteria, so you do need to do something about it, and make sure that the antibiotics you have him on are suitable, broad-spectrum and hard-hitting (Excenel is often used in this type of a situation).
FWIW, I did have a filly last year who had a nice dry umbillicus, but at about 3 weeks, when the dried up stump fell off, she started to get slightly damp on the end of the umbillicus from urine. The silver nitrate sticks helped close it off. My vet also had me put aluspray over the umbillicus after I did the silver nitrate stick.
Silver nitrate has also worked well for us to close off patent urachuses. We have had two - out of 24+ foals - one this year and one last, both colts (which I hear is more common), and both showed within 2-5 days of birth. They both received Naxcel as well and both did great. Hope it recitifies quickly for you