As long as the singularities are not "too bad", the answer will be yes, $f(x)$ will represent a solution of the differential equation. The very same way that $$ \sum_{x=0}^{\infty} (-1)^{x}n!x^{n+1} $$ represents a solution of $x^2y'+y=x$. One might object that the series diverges, but resummation theory says this is irrelevant, that sum nevertheless represents a unique function (in a sector) which is a solution of the differential equation.
Balser's book, "From divergent series to analytic differential equations", would be a good place to start. Since the theory fundamentally uses Mellin transforms, you should be able to find what you want (perhaps indirectly) there.