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3

and, using the same argument as before, we must choose

where the ‘1’ in the second condition is accommodated by (If the initial
conditions were, say, then we would have to select

Thus the first approximation is represented by the problem

the general solution is

where A and B are arbitrary constants which, to satisfy the initial conditions, must
take the values A= 1, B = 0. The solution is therefore

The problem for the second term in the series becomes

The solution of this equation requires the inclusion of a particular integral, which here
is the complete general solution is therefore

where C and D are arbitrary constants. (The particular integral can be found by any
one of the standard methods e.g. variation of parameters, or simply by trial-and-error.)
The given conditions then require that and D = 0 i.e.

and so our series solution, at this stage, reads

Let us now review our results.
The original differential equation, (1.1), should be recognised as the harmonic
oscillator equation for all and, as such, it possesses bounded, periodic solutions.
The first term in our series, (1.5), certainly satisfies both these properties, whereas
the second fails on both counts. Thus the series, (1.7), also fails: our approximation