For b=0 you have a zero solution. That's correct. Casting it in the more general framework you want to find something that holds for all b. So b defines her a space of functions.
More general you would look at
dx/dt= ax/t + g(x,t)
with g in a specific space. So you would want to find results for all g in that specific space.
These problems are motivated by classical problems of the type:
dx/dt = Ax + f(x)
with f(x) in O(x^2)
As always, I focus too much on the pathological cases. Thanks for the clarifications ;)