# Integrating Factor Techniques for Exact ODE

As explained in this answer, an inexact ODE in the form $$M(x,\ y)\ dx + N(x,\ y)\ dy = 0$$ can be transformed into an exact ODE via multiple integrating factors which vary from each other non-trivially (by more than a factor of a constant). This implies that an exact ODE can be transformed into a non-trivially different exact ODE by some integrating factor, even though there would be less motivation to do so.

The usual method of finding integrating factors of only $$x$$ or only $$y$$ always returns $$1$$ when it eats an exact ODE, so my guess is that such a non-trivial transformation can only be a function of both $$x$$ and $$y$$ explicitly. One example of this might be multiplying an exact ODE by an integrating factor in $$x$$ and $$y$$ which separates the variables, which is guaranteed to output another exact ODE. However, this technique can just as easily be applied to inexact ODEs if the variables are separable.

What I want to know is if there are any new techniques that become available to find a non-trivial integrating factor when the given ODE is already exact. By analogy, there may be some "difficulty (at least if you don't have a calculator or patience)" in finding a particular linear combination of nickels and quarters that total $$\2,547,042,997.85$$, but once you find one, the rest follow effortlessly via the nullspace $$n*\begin{bmatrix}1\ quarter \\ -5\ nickels\end{bmatrix}$$. I'm wondering if finding one exact form of an ODE can aid in finding others (aside from the trivial case of multiplying the integrating factor by a constant).