Elementary first integrals of differential equations 论文

摘要

We show that if a system of differential equations has an elementary first integral (i.e. a first integral expressible in terms of exponentials, logarithms and algebraic functions) then it must have a first integral of a very simple form.This unifies and extends results of Mordukhai-Boltovski, Ritt and others and leads to a partial algorithm for finding such integrals.1. Introduction.It is not always possible and sometimes not even advantageous to write the solutions of a system of differential equations explicitly in terms of elementary functions.Sometimes, though, it is possible to find elementary functions that are constant on solution curves, that is, elementary first integrals.These first integrals allow one to occasionally deduce properties that an explicit solution would not necessarily reveal.Consider the following example:Example 1.The preditor-prey equations -= ax -bxy, -= -cy + dxy, a,b,c,d positive real numbers.Although these cannot be solved explicitly in finite terms, one can show that F(x, y) = dx + by -clog x -alog y is constant on solution curves (x(t), y(t)).Using the function F(x, y), one can furthermore show that all solution curves in the positive quadrant are closed, that is, all such solutions are periodic.Note that in this example the first integral is of the form w0(x,y) + '2c,logwi(x,y),where the c, are constants and the w¡ are algebraic (in this case, even rational) functions of x and y.Roughly speaking, the main result of this paper is that if a system of differential equations has an elementary first integral, it will then have one of this form.Corollaries of the main result will show that the theory presented here unifies and generalizes a number of results originally due to Mordukhai-Boltovski, Ritt and others.An attempt to do this was made in [SING: 77] but the results presented here are more general and the techniques more to the point.Some of these results also appear in [PRELLE: 82].In the following, Z stands for the integers, Q the rationals and C the complex numbers.