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202 5. Some worked examples arising from physical problems

Equation (5.8), with becomes

and we seek a solution in the form

which is periodic in T. In this problem, we have yet to choose the frequency is
what the child can control in order to increase the amplitude of the swing. Without
any damping in the model, we must anticipate that we can find an which allows the
amplitude to grow without bound. First, with (5.10) in (5.9), we obtain

and so on. Equation (5.11a) has the general solution

for arbitrary functions and which leads to equation (5.11b) in the form

In order to make clear the forcing terms in equation (5.12), which may lead to
secularities in T, we expand the last term to give

We see immediately that periodicity in T requires and so we
will, for simplicity, choose (We are not particularly concerned about how
the motion is initiated, which would serve to select a particular value of Now if
(all sign combinations allowed), then we require for period-
icity in T, and the amplitude remains constant: the arc of the swing is not increased.
However, we are seeking that condition which will allow the amplitude to increase on
the time scale thus we expect that will increase as increases. This is possi-
ble only if and this can arise here if (all sign combinations to be
considered), and so we may choose (and the sign of is immaterial).
Thus with periodicity in T requires that