An efficient method for computing Liouvillian first integrals of planar polynomial vector fields

Here we present an efficient method to compute Darboux polynomials for polynomial vector fields in the plane. This approach is restricted to polynomial vector fields presenting a Liouvillian first integral (or, equivalently, to rational first order differential equations (rational 1ODEs) presenting a Liouvillian general solution). The key to obtaining this method was to separate the procedure of solving the (non-linear) algebraic system resulting from the equation that translates the condition for the existence of a Darboux polynomial (i.e., from the equation D(p) = q p) into feasible steps (procedures that require less memory consumption). We also present a brief performance analysis of the algorithms developed. (C) 2021 Elsevier Inc. All rights reserved.