A new method for functional decomposition of rational invariants, and the solution of Abel's differential equation via the equivalence method

The equivalence method for ordinary differential equations (ODEs) involves finding a transformation mapping a given equation into a target equation with known solution. Such equivalence transformations are found by solving systems relating the differential invariants of the input equation to those of the target equation. Standard solution techniques, applicable when the invariants are rational functions, use algebraic elimination and can solve complete families of equations characterized by many parameters. However, the complexity of such methods increases exponentially with the number of parameters, and can become impractical in interesting cases. A new technique is presented for overdetermined rational function decomposition, specifically tailored to systems of invariants. Its complexity is effectively independent of the number of parameters defining the system. The use of this technique in the ODE equivalence method is described, including the use of minimal invariants which ensures the solution of all rational coefficient equations in the target class. Recognizing that related invariants tend to be composed as products of powers of a set of common polynomials, and furthermore that the pattern of these polynomials is invariant under composition by rational functions, we can compute these component polynomials invariantly. Using the target system as a model, a sequence of invariant computations is built that successively simplifies the system, leading eventually to the determination of the parameters and transformation function. The resulting algorithm mimics a formula in its specificity but lacks the associated expression growth. Additionally, necessary conditions are checked after each step, minimizing time spent testing invalid classes. For certain parameter combinations, the structure of the component polynomials changes and the general algorithm can fail. Such cases are analysed in advance and a hierarchy of sub-algorithms is built to handle them, resulting in one super-algorithm to match the full super-class. The new equivalence method is demonstrated by the implementation in Maple of a first complete algorithm for the solution of Abel differential equations of the inverse-linear or inverse-Riccati classes. Together these two super-classes, depending on two and three parameters respectively, comprise the bulk of the solvable Abel classes described in the literature.