With reference to _implicit differentials_, the distinction of orders is still more important; for, besides the preceding reason, the influence of which is evidently analogous in this case, and is even greater, it is easy to perceive that the higher order of the differential equations necessarily gives rise to questions of a new nature. In fact, even if we could integrate every equation of the first order relating to a single function, that would not be sufficient for obtaining the final integral of an equation of any order whatever, inasmuch as every differential equation is not reducible to that of an immediately inferior order. Thus, for example, if we have given any relation between _x_, _y_, _dx/dy_, and _d_²_y_/_dx_², to determine a function _y_ of a variable _x_, we shall not be able to deduce from it at once, after effecting a first integration, the corresponding differential relation between _x_, _y_, and _dy/dx_, from which, by a second integration, we could ascend to the primitive equations. This would not necessarily take place, at least without introducing new auxiliary functions, unless the proposed equation of the second order did not contain the required function _y_, together with its derivatives. As a general principle, differential equations will have to be regarded as presenting cases which are more and more _implicit_, as they are of a higher order, and which cannot be made to depend on one another except by special methods, the investigation of which consequently forms a new class of questions, with respect to which we as yet know scarcely any thing, even for functions of a single variable.[10]
[Footnote 10: The only important case of this class which has thus far been completely treated is the general integration of _linear_ equations of any order whatever, with constant coefficients. Even this case finally depends on the algebraic resolution of equations of a degree equal to the order of differentiation.]
_Another equivalent distinction._ Still farther, when we examine more profoundly this distinction of different orders of differential equations, we find that it can be always made to come under a final general distinction, relative to differential equations, which remains to be noticed. Differential equations with one or more independent variables may contain simply a single function, or (in a case evidently more complicated and more implicit, which corresponds to the differentiation of simultaneous implicit functions) we may have to determine at the same time several functions from the differential equations in which they are found united, together with their different derivatives. It is clear that such a state of the question necessarily presents a new special difficulty, that of separating the different functions desired, by forming for each, from the proposed differential equations, an isolated differential equation which does not contain the other functions or their derivatives. This preliminary labour, which is analogous to the elimination of algebra, is evidently indispensable before attempting any direct integration, since we cannot undertake generally (except by special artifices which are very rarely applicable) to determine directly several distinct functions at once.