Statistical Investigation of First-Order Algebraic ODEs and their Rational General Solutions

In [3] we gave an algorithm for deciding the existence of a rational general solution of strongly parametrization first-order AODEs (SP1AODEs). This report shall give, on the one hand a list of strongly parametrizable AODEs and their solutions and on the other hand a statistical investigation on the relative number of such AODEs in well known collections such as Kamke [1] and Polyanin and Sajzew [2]. For detailed explanations on the algorithm in use, we refer to [3]. To get some insight we also give some intermediate computations of the algorithm here. The given first-order AODE is considered as an algebraic curve over K(x). We need a strong rational parametrization of the curve. Then we compute an associated ODE, which, in order to allow a rational solutions, has to be of a specific form. A rational general solution of the associated ODE leads to a rational general solution of the original AODE. In the following we write NoRatGenSol , if an AODE does not have a rational general solution. If it just does not have a strong rational general solution we write NoStrongRatGenSol.