Looking for the order of a system of arbitrary ordinary differential equations De investigando ordine systematis æquationum differentialium vulgarium cujuscunque

Hanc commentationem in medium protulerunt S. Cohn et C.W. Borchardt e manuscriptis posthumis Caroli G. J. Jacobi. Variæ formæ canonicæ quas datum systema æquationum differentialium vulgarium inducere potest considerantur. Investigatio ordinis systematis, sine formæ canonicæ auxilio, ad solvendum problema inæqualitatum reducitur: affectationum problema. Novum genus formularum, determinantia manca, introductum est. Cuiusmodi quantitas non evanescens indicio est, ordinem equalem esse solutioni H problematis inæqualitatum, quæ per algorithmum, Haroldi Kuhn methodi hungariæ similem, invenitur. This paper was edited by S. Cohn and C.W. Borchardt from posthumous manuscripts of C.G.J. Jacobi. The various canonical forms that a given system ordinary differential equations may take are considered. Looking for the order of the system, without using a normal form, is reduced to a problem of inequalities: the affectation problem. A new type of formulas, the truncated determinants, is introduced. The non vanishing of this quantity means that the order will be equal to the value H, solution of this inequalities problem, which is obtained by an algorithm similar to Harold Kuhn’s Hungarian method.