This method is based on paper "Exactness of Second Order Ordinary Differential Equations and Integrating Factors", by AlAhmad, M. Al-Jararha and H. Almefleh which now I have full implementation for. We start with the ode in the form
Then, we first verify the ode is exact using the conditions
If the above are satisfied, then next we generate a first order ode using
If we are not given initial conditions for the original ode, then the above is replaced by
Next, we solve the the above first order ode. Examples below make this method more clear. Notice that when matching our equation against the template (1), it is possible to obtain different possible matches and hence different possible