Detecting degenerate behaviors in first order algebraic differential equations

AbstractDifferential geometric and algebraic ideas are exposed and transposed into algorithms to detect and classify degenerate behaviors appearing in first order ordinary differential equations. Differential geometry is considered for a local analysis. It indeed gives insight on the behavior of the solutions at singular points. A sequence of algebraic operations will be put forth to split the locus of singular points according to their properties. When part of the locus of singular points turns out to be a singular solution, the set of solutions splits and shall be considered from a global standpoint. To this aim, differential algebra allows to define rigorously the general and particular solutions. Investigating further leads to an original algorithm to compute a differential basis of the general solution. This basis give means to determine which singular solutions are particular solutions