In the preceding chapter the solution of ordinary differential equations in terms of standard functions or numerical integrals was discussed, and methods for obtaining such solutions explained and illustrated. The present chapter is concerned with a further method of obtaining solutions of ordinary differential equations, but this time in the form of a convergent series which can be evaluated numerically [and if sufficiently commonly occurring, named and tabulated]. As previously, we will be principally concerned with second-order linear equations.

There is no distinct borderline between this and the previous chapter; for consider the equation already solved many times in that chapter

The solution in terms of standard functions is of course

but an equally valid solution can be obtained as a series. Exactly as in ▸1 of chapter 5 we could try a solution

and arrive at the conclusion that two of the an are arbitrary [a0 and a1] and that the others are given in terms of them by

Hence the solution is

It hardly needs pointing out that the series in the brackets are exactly those known as cos x and sin x and that the solution is precisely that of (6.2); it is simply that the cosine and sine functions are so familiar that they have a special name which is adequate to identify the corresponding series without further explanation.

It will also be true of most of our examples that they have a name (although their properties will be slightly less well known), but the methods we will develop can be applied to a variety of equations, both named and un-named.