> Only in school exercises you can solve a differential equation by expanding a sine function into a Taylor series.

> In practical physics computations, the solution of differential equations requires numerical methods that do not use the Taylor series of specific functions, even if the theory used for developing the algorithms may use the Taylor series development of arbitrary functions.

I'm sorry, but you have no idea what you're talking about. Series expansions is one of the most widely used techniques in Physics. Obviously some equations require full blown numerical methods to be solved, but one can do a whole lot with analytical techniques by doing series expansions and using perturbation theory.

Saying that this is only used "in school exercises" shows that you're completely out of touch with reality.