As you have written the differential equation with the given initial conditions and with x increasing from 0 to 2, there is no other "branch". The solution is uniquely determined. With a = +1 it is
and nothing else.
You would get a different solution if you took the negative of the square root or if you set a equal to minus one - that is, if your differential equation were:
and you chose the negative root for y". Or, if x was set to decrease from 0 to -2, the solution would of course be different.