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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).

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