Let $a \in C^\infty(\Bbb R)$. Show that if $g \in D' (\Bbb R)$ satisfies $g' − ag = 0$, then $g \in C^\infty(\Bbb R)$, i.e. $g$ is a regular distribution associated with $C^\infty$ function.
I have done a similar question where if $g'=0$ then $g=\operatorname{constant}$.
P.S. can anyone think of any variations of this question?
Use the standard method for solving the differential equation: There exists $A$ with $A'=a$. Now if $u=e^{-A} g$ then $u'=0$; you say you know how to show this implies $u=c$, and that implies $g=ce^{A}$.