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## The Research about Invariants of Ordinary Differential Equations 开放存取 Deposited

Abstract. Several basic relative invariants for homogeneous linear differential

equations were discovered during the years shortly after 1878. Also, a basic

relative invariant was found by Paul Appell in 1889 for a type of nonlinear

differential equation. There was little progress during the years 1892--1988 as

researchers who worked with homogeneous linear differential equations were

unknowingly handicapped by the standard practice of introducing binomial

coefficients in the writing of their equations. They thereby failed to develop

adequate formulas for the coefficients of equations resulting from a change of

the independent variable. Consequently, for relative invariants as the most

important kind of invariant, progress was stymied.

The notation was simplified in 1989, adequate transformation formulas

were developed, and explicit expressions were deduced in 2002 for all of the

basic relative invariants of homogeneous linear differential equations. In 2007,

explicit formulas were obtained for all of the basic relative invariants of a

type of ordinary differential equation involving two parameters m and n that

represent positive integers. When n = 1 and m >= 3, the formulas specialize to

provide all of the basic relative invariants for homogeneous linear differential

equations of order m; and, when m = n = 2, they yield all three of the basic

relative invariants for the equations of Paul Appell.

A general method developed in 2014 combines two relative invariants of

weights p and q for the same type of equation to explicitly obtain a relative

invariant of weight p+q +r, for any r >= 0. With that, the principal problems

about relative invariants have now been solved.

This monograph provides clear perspective about the reformulation begun

after 1988 and recently completed. Chapters 15 and 18 show how the major

difficulties confronting earlier researchers have been overcome.

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