Consider the first-order differential equation
$$\dot{x}+p(t)x=q(t).$$
This can be generally solved using an integrating factor
$$a(t)=\exp\left(\int p(t)dt\right)$$
and the solution is
$$x(t)=\frac{1}{a(t)}\left(\int a(t)q(t)dt+C\right)$$
(I'm going to skip over why this works - I can more information if people don't recognize this). The point is that this is a general method - it should work for any smooth functions $p(t)$ and $q(t)$.
What I'm interested in is if a similar thing is true for the same differential equation,
$$\dot{x}+P(t)x=Q(t),$$
but where $P(t)$ and $Q(t)$ are distributions, that is, linear functions on test functions. One wouldn't normally think techniques on smooth functions would extend to distributions, but since the final solution basically just depends on integrals of the distributions, I would be optimistic that it might work for all distributions.
Are there any results in this area? I do have a specific case in mind, in which $P$ and $Q$ are made up of delta functions and it's derivatives, but the question is more general, and I would love a specific reference (which is hopefully readable :-) )