ODE trends in computer algebra: Four ODE challenges
Edgardo S. Cheb-Terrab MITACS - CECM, Theoretical Physics Department, UERJ
ODE solving algorithms typically evolve by enlarging the ODE classes we know how to solve and by introducing techniques for mapping given problems into these solvable ones. The implementation of these algorithms in computer algebra systems has given a remarkable boost in the capability for computing exact ODE solutions; in that way it has also facilitated the investigation of new algorithms, forming a positive developing cycle. Bearing the above in mind, this talk presents four current ODE challenges together with some insights about them and possible related solving strategies. Current computer algebra systems have poor performance with these four ODE types, which - from some point of view - are just "one step" out of the scope of the currently implemented algorithms. The problems are: * Linear ODEs: 1. of second order, with rational coefficients but not admitting Liouvillian solutions; 2. of third and higher order with rational coefficients, all types of solutions. * Non-Linear ODEs: 3. of first order which are "linearizable" through transformations of the form x -> P(x) y(x), y(x) -> F(x) where F(x) and P(x) are arbitrary functions; 4. Second order nonlinear ODEs of the form y'' = R(x,y,y') (rational R), not admitting ("point") symmetries or integrating factors.