Whether you go to jail depends on many circumstances. You state this is your first offense...which is helpful. What was the nature of the domestic violence, since there are varying degrees of domestic violence offenses? The most common seems to be DV/Assault 4th degree. I have had success getting this charge deferred (i.e., deferred prosecution or deferred sentencing) pending good behavior by the defendant for a year or two at which time the case is dropped and no jail time had to be served. I guess a lot depends on the Court since the Court has discretion in these kinds of matters. Whether your boyfriend presses charges or not may not matter. Once an incident is reported to the police and passed along to the prosecutor, the prosecutor can proceed with or without the boyfriend's cooperation as it's now "the people" versus you. Practically speaking, however, if your boyfriend doesn't show up for trial, the Court may dismiss the case....or the prosecutor can ask for a continuance and then personally subpoena your boyfriend and if he doesn't show up again, he could get in trouble himself. Again, it largely depends on the Court and prosecutor.
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If your boyfriend is willing to drop the charges, its likely the DA will go along, but not always, so be careful and if that happens, hire a good lawyer immediately.