software for obtaining differential equations for circuit simulation
As far as I remember the SPICE algorithms you do not want its internal matrices dumped on you. Also I believe for transient analysis it converts all capacitors to voltage sources (and inductors to current sources) for every time step and solves the non-linear circuit like a DC one. So there are no true differential equations in SPICE.

That said, SPICE could output the transfer function of the circuit; IIRC as a list of polynomial coefficients. This sometimes suffers heavily from rounding errors but for simple circuits it may be what you are looking for.