B737 pilot reported being unable to start the number 2 engine. Start was successful after selecting an alternate ignition system but crew failed to consult maintenance prior to conducting flight.

Narrative

The number 2 engine did not start so I directed completion of the Aborted Engine Start QRC and QRH. Proper motoring was observed and we confirmed that all Packs Switches and Isolation Valve Switch was in the proper position. A second start was attempted and again unsuccessful. Once again; we referenced the QRC/QRH and proper motoring was accomplished per the QRH. I consulted the AOM regarding Starter Duty Cycle limits (AOM 18.10.3) and we observed all limitations. After a quick review of the Engine Start Considerations (AOM 4.2.4); I decided that we would attempt one more engine start and then call maintenance if it was unsuccessful. I coordinated with ramp and the informed the cabin that we may be a few minutes delayed; but at no time was there any external pressure to rush. Without thinking through my next action; I switched the Engine Ignition from the Right to Left. On this attempt the engine started normally. We continued uneventfully with starting number 1; after start; taxi; takeoff and climbout. When we got to cruise; we discussed the engine start difficulty and I opened up the logbook to enter an deferral item. It wasn't until I started thinking about how to convey our difficulty with the engine start; that I realized I should have consulted with maintenance before changing the ignition switch. I entered the deferral item in the logbook as simply and concisely as possible; 'Number 2 engine start unsuccessful. Switched from Right to Left Ignition and engine started normally'. After landing in ZZZ; maintenance ops checked the ignition system and found it to be working normally and returned the aircraft to service. After discussing this situation with a couple mentors and my own research I realized that I failed to consider if there was an MEL that potentially would apply.Cause: My own failure to fully consider the systems aspect of why the engine didn't start like I expected. Incorporating others (local maintenance; Maintenance Control; other crewmember) during the problem solving and decision making phase could have prevented this. In the future; I will make sure to carefully consider all possibilities and resources when dealing with aircraft anomalies. To a lesser extent; the AOM is a little vague about how many start attempts are reasonable or acceptable.