Being a physicist, I have never really paid due attention to the little details such as - is the domain on which the differential equation is given an open or closed set? Or, can it be both? Now, my research has led me back to these kinds of questions.
For concreteness, let's say we're solving a simple ODE $y' = im y$, with $m$ a real constant, on a circle. The circle is usually represented as the interval $I = \langle - \pi, +\pi ]$ and so my solution $y(x)$ should be defined for all $x \in I$. However, since the point $x = +\pi$ is a boundary point, there's no way to define the derivative $y'$ at that point (right?) and, therefore, our ODE only makes sense on an open interval $\langle -\pi, +\pi \rangle$. If all this is correct, I would need to define $y(+\pi)$ "by hand" and this definition is nothing more than a boundary condition.
Is the above reasoning correct?