Jacobi's Problem on the Order of a System of Differential Equations

Abstract

where j1, * *, j. is a permutation of 1, * * *, n. He arrives at the conclusion that the number of arbitrary constants in the solution of (1) does not exceed the greates sum (2). Now the number of arbitrary constants in the solution of a system is not a notion which is definite in advance. A system may have many families of solutions, each with its own arbitrary constants. Not every unknown need be determined by the system to within arbitrary constants. It seems safe to assume that before one can speak with completeness on the question of arbitrary constants, one will have to possess a theory for the decomposition of a system into systems of some normal type. Apparently no such theory exists at present for systems other than algebraic systems. It is thus not surprising that the argument on which Jacobi bases his conclusion, in the first of the above mentioned papers, should be whimsical in aspect. Without justification, Jacobi replaces the given equations by their equations of variation, which are linear. He then asserts without proof that, for his purpose, a linear system may be imagined to have constant coefficients. His treatment of linear systems with constant coefficients was completed by Chrystal.2 There are indications in Jacobi's second paper that his conclusion was actually based on a speculation as to the number of times which the given equations would have to be differentiated to permit the elimination of all unknowns except one. Investigations of the problem from this point of view were given by Nan-