An example of these class of problems would be:

under what conditions on the parameter a do there exist solutions to

dx/dt= ax/t + bt such that \lim_{t \to 0} x(t)/t^2 exists (where b is a free parameter, which here acts like a perturbation term)

with some straightforward calculus you can show that if a \neq 2 the desired solution exists.

The b-term acts like a perturbation term to the ax/t part so the more general version of the above you would replace the b-term by some very general perturbation.