Vectorial ODEs and vectorial integration constants

In physics, it is traditional to work with vectors, as in analytic geometry, i.e. symbolic vectors, abstract as in , or projected into orthonormal basis such that the unit vectors appear explicitly. In Maple, that is implemented by the Physics:-Vectors  package. The underlying idea is the extension of the Maple computational domain to include a new category of objects: vectors, and related unit vectors and vectorial differential operators all based on .

But then, with paper and pencil, we frequently write vectorial differential equations, that when solved imply on vectorial integration constants, none of which were implemented; now they are, within the Maplesoft Physics Updates v.1341. As with everything new, there is more work to be done, mainly additional checks for consistency here and there, but the work is advanced; time to tell the story and we are grateful in advance for the always useful opinions / corrections if any.

The input/output below illustrate the new features, which by the way compose on top of the new subscripted arbitrary constants by dsolve; this time extended to also be vectorial. The presentation has for context typical material of a first undergrad course in Mechanics. The purpose, anyway, is only to illustrate the new solving of vectorial differential equations and vectorial integration constants.