Page 248 -

231

becomes, with

where the solution is to be

The equations for the are therefore

and so on. The most general solution of equation (5.89), with a constant period is

for arbitrary and note that, for a constant period, we will require that
The initial conditions, (5.88), now on are satisfied by
the choices

and the existence of a real, oscillatory solution (of the form (5.91)) requires that
0 > a > – ln 2.
The periodicity condition, which will define can be obtained from (5.90) by
first writing and then using the T-derivative of (5.89); this
yields

This can be integrated once directly, when multiplied by so that we now have

i.e.

For to be periodic, then so must be both F and with the period prescribed as