You might be confusing straightforward linear systems of differential equations (some exceptions when you have a nasty heterogenous part, but which are 'always

You might be confusing straightforward linear systems of differential equations (some exceptions when you have a nasty heterogenous part, but which are 'always' solvable by a standard method) with so-called Differential Algebraic Equations (DAEs). These beasts often pop up when trying to solve larger electrical circuits with time-dependent elements (ie. RLC). The only thing you can really do to solve them usually is linearizing them approximate them by some integration scheme (this is what Spice, the electronics simulator is based on).