Gallery of integrating factors for non-linear first-order differential equations

This paper discusses a gallery of useful results in connection with integrating factors that are often left as problems for discovery learning and are generally not taught in typical Ordinary Differential Equations courses. Most often than not the approach earlier writers employ is to give a possible form for an integrating factor that may results in an integrating curve without practical prove as far as the subject matter is concerned. In this write-up, an attempt is made by solving the resulting partial differential equation emanating from an underlining general differential equation of a non-exact form, by the use of the ratio theorem to establish various intricate possibilities of integrating factors that are seldom and often relegated to the background, even though they may be equally be applied as a function of a unitary variable or a linear combination of both the dependent and independent variables under certain conditions. Granted an integrating factor is found and such a function applied, the benefit is enormous especially the non-exact differential equation reduces into a known type which may be identified as exact, homogeneous, and or separable that yields a solution.