Hi

Concerning only

>1 . Solving a set of homogenous equations with constant coefficients:

check out https://github.com/sympy/sympy/pull/1322

There you can find a lot a discussion about how it should be done and
code that solves not only inhomogeneous linear systems but also some
nonlinear ones. It breaks on degenerate cases, which can not yet be
done, because the matrix module does not support certain
generalizations of eigenvalue decomposition.

It is quite problematic to write tests for this module because of the
mess with integration constants.

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