I want to solve a pair of ODEs using the FEM solver of Abaqus. Is it possible for the user to supply the equation and ask Abaqus solver to solve the equations?
No, ABAQUS is not a general-purpose PDE solver. It solves the PDE for a number of pre-defined applications such as structural analysis or heat transfer in solids. Of course many physical phenomena are governed by the same equation as heat transfer so you can solve those problems by treating them as heat transfer problems in ABAQUS.
I write this as an answer, since I cannot comment. As already explained, Abaqus is not well suited for your problem. You should have a look at Fenics (http://fenicsproject.org/), which is a general purpose PDE solver.
Update: I learned that Comsol also provides an interface for user-defined PDE and weak forms.