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we use the prime to denote the derivative. Thus we require

with the boundary conditions

The solution for is immediately

but this result is clearly unsatisfactory: the solution for grows exponentially, whereas
the solution of equation (1.10) must decay for (because then Per-
haps the next term in the series will correct this behaviour for large enough x; we have

Thus
and we require A = 0; the series solution so far is therefore

However, this is no improvement; now, for sufficiently large x, the second term dom-
inates and the solution grows towards Let us attempt to clarify the situation by
examining the exact solution.
We write equation (1.10) as

the general solution is therefore

and, with C = 1 to satisfy the given condition at x = 0, this yields

Clearly the series, (1.12), is recovered directly by expanding the exact solution, (1.13),
in for fixed x, so that we obtain

Equally clearly, this procedure will give a very poor approximation for large x; indeed,
for x about the size of the approximation altogether fails. A neat way to see this
is to redefine x as this is called scaling and will play a crucial rôle in what