  We have been working in many aspects of the problem of analyzing,
understanding and solving ordinary differential equations (first and second
order). As we have extensively mentioned, while working in the Darboux type
methods, the most costly step of our methods and algorithms of solution is the
determination of Darboux polynomials for the associated differential operators.
Here, we are going to present some algorithms to greatly reduce the time
expenditure in determining these needed Darboux polynomials. Some of them are
based on a detailed analysis of the general structure of second order
differential equations regarding the associated differential invariants. In
order to perform this analysis, we produce a theorem concerning the general
form for the differential invariants in terms of the Darboux polynomials.
