Improving a family of Darboux methods for rational second order ordinary differential equations

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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 apply a procedure to greatly reduce the time expenditure in determining these needed Darboux polynomials for a class of second order differential equations.