Aha! Now we have it.
The motion of the pendulum is chaotic.
That second degree of freedom did it. The best thing about
chaotic motion is that it is not amenable to analysis in
closed form, at least if we are talking about physics. The
student should not be left with the idea that he is doing
physical modeling in his investigation of ideal nonlinear
differential equations.
The spreadsheet approach I suggested earlier now seems
even more appropriate.