Consider a differential equation,
$$L(T)=f,$$
where $L$ is a linear ordinary differential operator with smooth coefficients and $f$ is a $L_{loc}^1$ function.
Does there always exist a distribution T such that above equation holds (if not what conditions on $f$ are sufficient to guarantee that)?
What is an example s.t. distribution solution exists but any function is not a solution in the sense of distribution? I know examples of ODE's for which there are distributional solutions that are not functions (e.g. $x^2y’=0$), but I want an example in which no function is a solution but distributional solution exist.
I am learning this stuff on my own so questions might be trivial. Thanks for any help!