Periodic nature of ODEs Generally speaking without solving, how are periodic function solutions a priori 
recognized by any inspection or computation of terms contained in their ODEs?
(Prior knowledge is a not  a mathematical criterion!)
If and when it is recognized belonging to a class satisfying some rationale how is the 
frequency or time period next determined ?

The short answer is Floquet Theory, which studies solutions to $\dot{x}=A(t)x$ where the matrix $A(t)$ is periodic: $A(t+T)=A(t)$, where $T$ is the period. Loosely speaking, the achievement of this theory is that any solution $x$ to this equation is of the form $e^{iat}P(t)$, where $P(t)$ is a periodic function.
