# Exact Differential Equations of Order n?

1. Sep 5, 2013

### 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:

Thanks!

2. Sep 6, 2013

### Simon Bridge

... 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

3. Sep 6, 2013

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...

Last edited: Sep 6, 2013
4. Sep 6, 2013

### Simon Bridge

Sorry - no insult intended.
Please note that I have not provided any definitions.
If the text works for you then that is great and I'm sure you'll figure out what the author is talking about in due course.

However, I don't think I can provide the answer in the form you are looking for.
I'll see if I can attract someone else.
Cheers.