> compare the explanation of integrating factor and exact equations

those are fringe subjects, completely irrelevant for the modern usage of differential equations. They are useful only in computer algebra when you want to implement differential galois theory. In practice you want to understand the overall behavior of your system (qualitative theory) or compute particular solutions numerically (using numerical methods, which are more precise than evaluating the expression of the exact solution).

You'd do much better with a qualitative book about differential equations (e.g., Arnold), about numerical analysis, or about dynamical systems (e.g. Strogatz).