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An exact solution of equation (4.31) is

where

(The confirmation of this, using the properties given in the Appendix, is left as an
exercise.) Here, is a constant (to ensure that the period in T is a constant which,
with the chosen form (4.33), is 1). Given and equations (4.34a,b) provide
two equations for the three unknown functions: and A third equation
is obtained by imposing periodicity on
The task of solving (4.32) is not as difficult as it might appear; the important ma-
noeuvre is to write

and then to solve for Equation (4.32), with (4.35), gives

but from (4.31) we also have that

and so (4.36) becomes, after multiplication by

Now, for to be periodic with period 1, must be similarly periodic
(because is, in (4.35)). Thus we must have, for any T,

which can be written

which is our third relation. When (4.33) is used in (4.37), the integration is possible
(but this does require some additional skills with, and knowledge of, elliptic functions
and integrals; see e.g. Byrd & Friedman, 1971), to give