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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?

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    Do you assume that shifts are only integer multiplied by $\pi$? Why not use *any* shift? Anyway, these different functions solve different Cauchy problems. – 2017-01-09
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    Ah, I was also considering a given initial value, and also it should have been $2\pi n$. – 2017-01-10

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$\sin(t+\pi n)$ is equal to $\sin t$ only if $n$ is even, and in that case it is not a different solution because it is the same function. Every function can be expressed in many different ways; that doesn't make it many different functions.

If $n$ is an odd integer, or a non-integer, then $\sin(t+\pi n)$ is not $\sin t$. If $n$ is odd then $\sin(t+\pi n)=-\sin t.$

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    Sorry, the $n$ should have been $2n$, since I was looking at a given initial value. This was helpful. – 2017-01-10