TITLE: Are $\sin$ and $\cos$ the only functions whose derivatives are equal to each other up to a sign?
QUESTION [32 upvotes]: Are $\sin$ and $\cos$   the only functions that satisfy the following relationship: 
$$  x'(t) = -y(t)$$
 and
$$ y'(t) = x(t)
$$

REPLY [7 votes]: I suppose you can set:
$$
  x\left(t\right) = \exp\left(-i \cdot t\right)\\
  y\left(t\right) = i\cdot\exp\left(-i \cdot t\right)\\
  i = \sqrt{-1}
$$