## Error Bounds: Real Variables

The integral (2.01) will be called the error-controlfinction for the solutions (2.03). It suffices to establish the theorem for the case j = 1 ; the corresponding result for j = 2 then follows on replacing x in (2.02) by - x . 2.2 We begin the proof of Theorem 2.1 by applying the transformations (1.05) and w = f -'I4(x) W. Equation (2.02) becomes

d2W/d(Z = (1 +I)(()) W, (2.05) where

and obtain h''(() + 2h'(() - $ (oh(() = $(o. (2.08)

To solve this inhomogeneous differential equation for h((), the term I(/(()h(() is regarded as a correction and transferred to the right. Applying the method of variation of parameters (or constants), we find that

h (() = 3 {I - e2(v-C') I) (v) (1 + h (0)) do. I' (2.09) Conversely, it is easily verified by differentiation that any twice-differentiable solution of this Volterra integral equation satisfies (2.08).