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The exact difference equation of the first order

Published online by Cambridge University Press:  24 October 2008

L. M. Milne-Thomson
L. M. Milne-Thomson
Affiliation:
Corpus Christi College
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Extract

The theory of the exact difference equation in the general linear case has been fully developed, but the corresponding theory for the non-linear equation of the first order does not appear to have been considered. In this paper necessary and sufficient conditions for the difference equation of the first order to be exact and the form of the primitive are obtained. It appears that two conditions are required for a difference equation to be exact, one of which is identically satisfied in the limiting case of the exact differential equation. These conditions are applied to determining the primitive in some cases where the conditions for exactness are not satisfied.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 29 , Issue 3 , 30 July 1933 , pp. 373 - 381
Copyright
Copyright © Cambridge Philosophical Society 1933

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References

* Wallenberg, Guldberg u., Theorie d. linearen Differenzengleichungen, 1911, p. 78Google Scholar. See also Nörlund, , Équations linéaires, 1929, ch. 1.Google Scholar

* See Nörlund, N. E., Acta Math. 44 (1923), 71–211.CrossRefGoogle Scholar If Nörlund writes,

The theory of this operator is very elaborate, but the above will serve to make the present argument intelligible.

* Nörlund, loc. cit.