Contributions to the development of differential systems exactly solved by multistep finite-difference schemes

The motivation underlying this contribution has been to complete some of the topics concerning exact schemes for numerically solving ordinary differential equations. A procedure for obtaining differential systems exactly solved by a given finite-difference method is described. Examples illustrating the application of the procedure for obtaining first-order, second-order and systems of differential equations exactly solved by different numerical methods are given. Among the numerical methods considered there are the Trapezoidal Rule, the two-step Adams-Bashforth method and the Numerov method. Some numerical examples are presented to provide evidence that the procedure works properly.