A Constructive Example:

Construct an implicit linear multistep (two-step) method of maximal order, containing one free parameter, and find its order.

Solution: Here by hypothesis. Let be the free parameter. There remain four undetermined coefficients and we can thus set

.

Solving, we get

and the method is

(*)

Moreover,

If , then and the method is of order 3.

If , then and the method which is now Simpson's rule, is of order 4. Note that when is the two step Adam-Moultan method, while if , it is an explicit method.