I'm wondering if somebody could both shed some light on, & offer references for more details about, this interesting quote:
The derivation of the conditions of exact integrability of an ordinary differential equation of the nth. order (or of a differential expression involving derivatives of a single dependent variable with regard to a single independent variable) is sometimes made to depend upon the theory of integration of an expression, exact in the sense of the foregoing chapter. As however the connection is not immediate and this method is not the principal method, it will be sufficient here to give the following references to some of the writers on the subject, in whose memoirs references to Euler, Lagrange, Lexell, and Condorcet, will be found in ... Forsyth - Page 33
In other words, I'm interested in how one studies the question of exactness for higher order ode's in terms of pfaffian (total) differential equations & how each illuminates the other.
Thus far my only insight into this really comes from Goursat (Page 115 if necessary) who basically says that Lagrange originally came up with the idea of the adjoint & the Lagrange identity as a means to extend the theory of integrating factors to linear equations of order n. They seemed like pretty distinct approaches to the subject of ode's until I read Forsyth's quote so if the originators of this theory naturally arrived at some deep relationship then surely something interesting is going on here.
Currently this is where I stand, I can't find the material Forsyth refers to, I can't find any material on my own & I've asked professors who have never encountered these ideas, thus I'm hoping somebody on here has encountered these idea's before & that it's a good enough question for this site, thanks.