Exact Differential Equations of Order n via Pfaffian Differential Equations?

Question

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.

Answer 1

I'll assume all along you're referring to ODE with real-analytic/holomorphic coefficients. You're looking for something called "non-linear differential Galois theory". This is related to this question Solution of linear ODE in the linear setting. Since this theory was developped after the book you cite you won't find any reference on it and one may reasonnably deem it an outdated source on the subject.

As for the non-linear setting the general framework as been established by Lie: the structure of the Lie algebra of infinitesimal symmetries of the equation accounts for the problem of integrability "in closed form". Yet this theory cannot be done at the singular locus of the differential equation, which is the natural context for many ODE arising in the "real world". There are two modern approaches to deal with the singular case:

1. (MR1425592) Umemura, H. "Differential Galois theory of infinite dimension" Nagoya Math. J. 144 (1996), 59–135
2. (MR1924138) Malgrange, B. "On nonlinear differential Galois theory. (English summary)" Chinese Ann. Math. Ser. B 23 (2002), no. 2, 219–226

I don't know much about the first reference (a little bit too "hardcore" for me ;) ). The second one is more geometric in nature and has been subsquently studied by e.g. the "French school", especially Casale and Malgrange. Although the theory is deemed technical it is more accessible with some decent background in finite-dimensional differential geometry and algebra. Some results are definitively available for foliations of codimension 1. Yet the problem your start with leads to a foliation of dimension 1, and the Galois-Malgrange groupoid for vector fields (save for the easier hamiltonian case) is not a very developped topic.

I don't know if these informations will be of much help to you, but this is (to the best of my knowledge) the state of the art regarding the singular setting for ODE.