If your forcing function is step-like, then integrate step-wise through the system. That is a bit more tedious, but this is actually the preferred solution for the case of step-function inputs.
I rediscovered this the hard way, but I’m glad to hear it’s something that’s already known. Is it documented anywhere? I haven’t found any other reference to it. If it’s not already documented, I would suggest giving it some prominence, given that it has the potential to cause very mysterious behavior.
All the best,
Robert Dodier