Abstract
The Riccati transformation is used to obtain Liouville-Green approximations, along with error bounds possessing a particularly simple and explicit form, for the equation u″ = f( t) u. The resulting error bounds involve the function f and its first derivative, whereas the earlier bounds of Olver [1961, 1974] and Taylor [1978, 1982] involve also the second derivative of f. The theorems given here resemble corresponding results of Olver [1974, Chapter 6], and indeed the present work owes a great debt to Olver. The Riccati transformation used here is based on work of Y. Sibuya, K. W. Chang, and W. A. Harris, and can be used in a more general study of linear ordinary differential equations and systems of such equations (cf. Smith [1984, 1985]).
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