In an attempt to actually grok sine, I came across the y''= -y definition.
This is incredibly cool, but it leads me to a whole new series of questions. Sine seems pretty prevalent everywhere in life (springs, sound, circles...) and I have to wonder, what's so special about the second derivative in this scenario?
In other words, why does nature / math seem to care more about the scenario where y'' = -y instead of, say, y' = -y or y''' = y?
Why is acceleration equal to negative the magnitude such a recurring theme in math and nature, while velocity equal to negative the magnitude ($y'=-y$) or jerk equal to negative the magnitude ($y'''=-y$) are seemingly unimportant?
In other words, what makes sine so special?
Note that this question also sort of applies to $e$, which satisfies y'' = y.
(Edit: Yes, I understand that $e$ and $\sin$ are closely related. I'm not looking for a relationship between $e$ and $\sin$.
Rather, I'm wondering why these functions in particular, which both arise from a relationship between a function and its own second derivative, are so prevalent. For example, do functions satisfying y'''=-y also recur frequently, and I just haven't noticed them? Or is the second derivative in some way 'important'?)