# Exact Differential Equations of Order n?

bolbteppa
A second order ode $Py'' + Qy' + Ry = 0$ is exact if there exists a first order ode $Ay' + By$ such that

$$(Ay' + By)' = Ay'' + (A' + B)y' + B'y = Py'' + Qy' + Ry = 0$$

How can one cast the analysis of this question in terms of exact differential equations?

In other words, could somebody explain 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

Thanks!

1 person

Homework Helper
exact in the sense of the foregoing chapter
... see foregoing chapter.
(reads) seems the author launches into investigating "exact" equations without making a general definition.
That's a pretty nasty text btw. I'd be remiss if I didn't advise you to ditch it.

For an idea how the concept of "exactness" may apply to higher order ODEs see instead:
http://reference.wolfram.com/mathematica/tutorial/DSolveExactLinearSecondOrderODEs.html

That doesn't help, neither the insult nor the link, but thanks...

The book does Frobenius' theorem in like a page, & spends hundreds of pages on inexact equations & is filled with history as well as substance, I'm surprised anyone with any appreciation for such a subject would write something like this off so easily. Also, the definition you've provided should really be a theorem if we're trying to link higher order exactness with first order exactness, but how and ever...

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1 person