Sine and Cosine parametrize the unit circle with constant speed 1

The solution set of any equation y^2 = F(x),
e.g. y^2 = 1 - x^2 for the unit circle,
can be parametrized as x = f(t), y = f '(t), obviously
if f satisfies the1st order ODE f'(t)^2 = F(f(t)),
circle case: f'(t)^2 = 1 - f(t)^2.
This ODE has constant solutions f(t) = c, if F(c) = 0,
and along such solutions uniqueness of solutions fails.
To eliminate these singular solutions, differentiate the ODE
and cancel f' to get the Lipschitz-ODE: 2f'' = F'(f(t)),
in case of the circle f'' = -f, the well known ODE of sine and cosine. Solutions of this 2nd order ODE satisfy the 1st order ODE, if the initial values do.