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.