He didn't change his mind properly speaking. When the whole birther conspiracy theory popped up in 28 he released his birth certificate to refute it despite the fact that to my knowledge no presidential candidate has ever done so or been expected to do so before. After he did so the birther movement barely even broke its stride rapidly claiming either that the birth certificate was a fake or that it didn't really italic prove Obama was born in the country. At that point it was obvious that nothing he could do would satisfy the conspiracists all reasonable people dropped it and the Obama campaign ignored the rest knowing full well that arguing about it was a fool game and would only draw more attention to the issue. Then around 211 the movement started coalescing around a new narrative the long-form birth certificate. Donald Trump was the most prominent voice in this movement but by no means the only one. They claimed that if Obama would only release his long-form certificate then we know for sure. The thing about this is the long-form certificate wasn't his to release. It's not like he just had to dig it out of his closet. He almost certainly never seen his long-form most people haven't. It wasn't in his possession and he had no innate right to it. He would have to petition the Hawaiian government to release it and they would have to decide whether or not to. Obama never refused to do that but also had little reason to. It was obvious from the start (as subsequent events have proven) that birthers would immediately denounce it as a fraud. It would give the issue more publicity and lower the President to squabbling with dilattantes. But eventually this small ring of very loud people managed to shout it up into a visible issue. I suspect that Obama deliberately decided to find a time to release it that would do the most political damage to the movement. That opportunity presented itself when they were planning the raid on Bin Ladenpound. By releasing it then he could both distract the media from potentially catching wind of the raid in advance and he could point out that it was a silly issue that needed to be dropped because he had more important things to do. Then when he had the most wanted man on earth killed immediately afterward he made that point in the strongest possible way. Trump and hispatriots still tried to keep the issue alive after that but they been smacked down in the most dramatic way and never got their traction back. Honestly I don't think the President could have timed it better.