Abstract
The well-known first-order nonlinear difference equation yn+1=2yn−xyn2,n∈N0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ y_{n+1}=2y_{n}-xy_{n}^{2}, \\quad n\\in {\\mathbb {N}}_{0}, $$\\end{document} naturally appeared in the problem of computing the reciprocal value of a given nonzero real number x. One of the interesting features of the difference equation is that it is solvable in closed form. We show that there is a class of theoretically solvable higher-order nonlinear difference equations that include the equation. We also show that some of these equations are also practically solvable.
Highlights
IntroductionF (x, y) can be chosen in various ways, and the choice depends on its usefulness in solving the problem of computation of f (x)
We use the following standard notation: N, Z, R, and C are the sets of natural, integer, real, and complex numbers, respectively, and N0 = N ∪ {0}
To compute the value of f (x), it is suggested that relation (1) is written in an implicit form F(x, y) = 0
Summary
F (x, y) can be chosen in various ways, and the choice depends on its usefulness in solving the problem of computation of f (x). Two concrete examples of some suitable choices of function F(x, y) are given below Such a chosen function F(x, y) is a function of two variables, and the given x in relation (1) (a fixed number therein) belongs to the domain of definition of the function. It is natural to assume that yn ≈ yn, so for computing f (x), from (2) we obtain the following recursive relation: yn+1. This is the Newton method, applied to F as a function of y (here x is fixed).
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