I'm reading Fundamentals of Differential Equations by Nagle. Given the equation $$\frac{dy}{dx}=f(x,y)$$ Nagle says at times we can rewrite as an exact differential form $$M(x,y)dx+N(x,y)dy=0$$ So it seems this is the case if it holds that $$f(x,y)= \frac{-M(x,y)}{N(x,y)}$$ and if we do cross multiplying, we go from the differential equation to the differential form. But why is this so? I think I was taught in my calculus classes that $dy/dx$ is notation for $y'$ and not literally a ratio of $dy$ and $dx$ which the professor at that time said were "deep concepts" and that sometimes you can't do "intuitive" algebraic operations with them. So why can we do it here?
As a side note, I have read Analysis on Manifolds by Munkres about 1.5 years ago, and he talks about (exact) differential forms. Sadly I forgot a lot after that and looking again at the definition of differential forms, it's pretty abstract, and takes many other ideas to "build up to" (ie alternating tensors, dual transformations, tangent spaces, etc). So maybe there's some neat connection between differential equations and differential forms that I'm missing?
Thanks a lot in advance, any guidance is greatly appreciated.