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Lets say I have some inexact differential equation, is there anyway to know if an integrating factor exists and if so how to find it?

So far I only know of one.

Let $\mu dx+\nu dy=0 $ if $({\frac{\partial \mu}{\partial y}-\frac{\partial \nu}{\partial x}})/\nu$ or $({\frac{\partial \nu}{\partial y}-\frac{\partial \mu}{\partial x}})/\mu=I$ which is a function of 1 variable then the integrating factor is $e^{F}$ where $F$ is the indefinite integral of $I$ wrt the variable it's a function of.

What other common ones are there?

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    $\begingroup$ If an equation of the form $M dx + N dy = 0$ is not exact, the equation always has integrating factors. There is no general methods to find integrating factors. It should be remember that there are an infinite number of integrating factors for an equation of the form $M dx + N dy = 0$ . There are so many rules available in the literature. For those you can follow "Introductory Course in Differential Equations for Students in Classical and Engineering Colleges" by D. A. Murray. $\endgroup$ – nmasanta May 4 at 9:40

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