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f(x) = Aexp(ikx) + Bexp(-ikx)

with the periodic boundary condition f(x+L)=f(x). This condition leads to:

Aexp(ikx)exp(ikL) + Bexp(-ikx)exp(-ikL) = Aexp(ikx) + Bexp(-ikx) (1)

Now the way I figured out the constants A and B was that I said, that (1) must be true in particular for x=0 and x=π/2k which gave some equations to solve for A and B.

But something appears weird to me with this method of using two arbitrary values for x in (1) to determine A and B. Namely how do I know that I should also get this value for A and B if I use two different values of x? I do realize you could maybe show this by a calculation letting x1 and x2 be arbitrary but from a bigger perspective: Is it always true that the solution of a differential equation is uniquely specified by a periodic boundary condition? Maybe I'm just confused about something simple - it's late at night..