I am trying to understand how to solve differential equations of distributions.
For example if one consider
$ u' + u = \delta_{0}$, where $ u \in \mathcal{D}'$,
this would correspond to
$<u, -\phi' + \phi> = <\delta_0, \phi> = \phi(0)$.
How would one in general go on to solve this? And can one see if it is solvable? I just made this up so I don't know if a solution exists.
Would it be possble to use Fourier Transforms?