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This same property of the functions leads to a corresponding statement for
(again, independent of which now gives

where the upper sign applies if and the lower if (Often in
arguments of this type, we cannot incorporate both signs, and we are reduced to
working with the modulus of the function; we will see here that we can allow the
form given in (2.20).) Between the two inequalities, we have an expression associated
with a constant coefficient Riccati equation; let us therefore consider

where and for arbitrary (bounded) functions and
If an appropriate unique solution of (2.21), satisfying exists for all
and all as specified, then we will certainly have satisfied (2.20). However, we will,
in this text, give only the flavour of how the development proceeds, by considering
a restricted version of the problem with the special choice: and constant (but
satisfying the given bounds).
To solve (2.21), we introduce to obtain

which has, in our special case, the general solution

where the arbitrary constants are A and B, and the auxiliary equation for the exponents
is

The roots of this equation have been written as

where for i = 1, 2, as Finally, the solution which satisfies the
condition on x = 1 is

which is bounded for as Thus the
error is for as required. (A comprehensive discussion