
A practical solution method for 2parameter Abel ODE classesAustin Roche, CECM
The most general solvable classes of Abel ordinary differential equations (ODEs) that we know of are 2parameter classes, two of which have been found recently: 2 2 y  1 4 y  4 y y'(x) = , y'(x) = . 2 2 x + a x y + b (x  1) y + c x + d However, the standard method for matching the complete Abel ODE class associated to a given ODE is impractical when the ODE has more than one parameter. I will present an alternative approach, with two differences: it takes advantage of the factored structure of the invariants; and it uses the much simpler "pseudoinvariants". The resulting algorithm is already working to match the class of the second ODE given above, except for a few yet to be implemented 1parameter subclasses. The method appears to be applicable to the other 2parameter Abel ODE classes as well. 