A practical solution method for 2-parameter Abel ODE classes

Austin Roche, CECM

The most general solvable classes of Abel ordinary differential equations (ODEs)
that we know of are 2-parameter 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 "pseudo-invariants". The resulting algorithm is already working to 
match the class of the second ODE given above, except for a few yet to be
implemented 1-parameter subclasses. The method appears to be applicable to
the other 2-parameter Abel ODE classes as well.