concerning work to be done: I have created only a skeleton
ode_systems.py solver. While I like the design, it is incomplete (it
raises NotImplementedError) and not at all integrated in the rest of
sympy. And all the rest of the projects suggested by you are still
possible.

The parts that are incomplete will result in only a few lines of code
_if_ we have jordan decomposition.

If you know what eigenvalue decomposition is and why it is necessary
for solving systems of linear differential equations, then the rest is
a straightforward generalization for solving systems that can not be
decomposed in this way. Check the code that I have written in
ode_systems.py (it is commented) and wikipedia, and work out a few
simple examples by hand (check the failing tests and try them step by
step with sage/mathematica/wolfram alpha)

If you wish you can cannibalize my code and provide a pull request in
whatever style you want.

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