Ordinary Differential Equations: An Elementary Text-Book: With an Introduction to Lie's Theory of the Group of One Parameter

Description

Mentioned in Ince's book, Page's is the earliest book that introduces Lie's theory in English (even earlier than Cohen's).

James Morris Page (cf. his entry on the Math Genealogy Project)—an American student of Lie, Ph.D., University of Leipzig, fellow by courtesy Johns Hopkins University (1895-1896), adjunct professor of pure mathematics, University of Virginia—attended Lie/Scheffers's Leipzig 1886-87 lectures and appears to have been the first to bring Lie's discoveries to the U.S.

He reviews the basics (e.g,. Cauchy (et al.)'s method of characteristics relating ODEs and quani-linear PDEs; cf. "simultaneous equations of an ODE" in ch. 1, esp. DjVu p. 38) and proceeds very systematically to more advanced topics, such as "To find all differential equations of the first/second order".

ch. 11, DjVu pp. 214 ff. are a good overview of how Lie's symmetry methods apply to systems of equations (system of ODEs → 1 PDE).

Page's work is even older than An Introduction to the Lie Theory of One-Parameter Groups: With Applications to the Solution of Differential Equations by Cohen (also of JHU)—which, unlike Page's, jumps right into Lie without any introduction to characteristics, etc.

AMS review and Nature review

cf. Page's short intro to Lie theory:

• Page, James M. “Transformation Groups Applied to Ordinary Differential Equations.” Annals of Mathematics 9, no. 1/6 (1894): 59–69.