This is for homework, but it's not the question that the homework is asking.
If we have a differential equation with some periodic solution, for example $(y')^2 + y^2 = 1$ can be solved by $y = \sin t$, then do we consider the solutions of the form $y = \sin(t + 2\pi n)\ \forall n \in \mathbb{Z}$ to be distinct solutions for the initial value $y(0) = 0$? In general, are shifted periodic solutions different solutions, or are they the same solution written in a different way?