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.