In a first part, this article presents the adaptation of Scheifele functions to forced and damped oscillators,
designing a series method based on these, which integrates the non perturbed problem with no truncation error. This
method is highly accurate, however it is difficult to adapt to each specific problem. In order to overcome this difficulty, in
a second part, we describe the transformation of the series method to a multistep scheme.
Explicit and implicit methods are formulated and combine to create a predictor-corrector method, which precisely
integrates the homogenous problem.
The computational algorithm is developed and the results obtained are contrasted by the series method and by the multistep
algorithm with other known integrators.