In my ODE class, we learned how to solve first order ordinary differential equations which are not exact yet but exact after multiplying by the right integrating factor. The integrating factor we learned about take one of the five forms: f(x), f(y), f(xy), f(x/y), and f(y/x). This is to say, the integrating factors can be a function of x only, y only, x*y, x/y, and y/x.
My teacher is expecting us to solve one of these ODE among many others about 7 minutes flat. Going through all the five possible integrating factors can be very time consuming. I want to know if there are tricks to knowing which one of the five integrating factors are more likely by perhaps examining the structure of the ODE or certain characteristics that they exhibit .
If there aren't any systematically efficient way to solve ODEs that are exact by integrating factors, are there tips or tricks in general are useful in finding the right integrating factor?