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.