THE COMPUTATIONAL APPROACH TO COMMUTING ORDINARY DIFFERENTIAL-OPERATORS OF ORDERS 6 AND 9

Entirely elementary methods are employed to determine explicit formulae for the coefficients of commuting ordinary differential operators of orders six and nine which correspond to an elliptic curve. These formulae come from solving the nonlinear ordinary differential equations which are equivalent to the commutativity condition. Most solutions turn out to be rational expressions in one or two arbitrary functions and their derivatives. The corresponding Burchnall-Chaundy curves are computed.