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and so on. The initial conditions, (4.4), become

which give

et cetera. It is now a very straightforward exercise to solve each of these problems, in
sequence.
The general solution of (4.7a) is

although it is rather more convenient to write this as

where and are arbitrary functions; we require, from (4.8),

which we satisfy by selecting

The equation for (4.7b), can now be written

where, for simplicity, we have written But for to be periodic
in T, all terms on the right-hand side, in and must vanish (for otherwise
we will generate particular integrals for like this removal of
secular (i.e. non-periodic) terms requires

The solution of (4.11a) and then (4.11b), with (4.10), gives directly