What is the name of the function space formed by solutions to algebraic linear differential equations? Where can I find a discussion of its properties?

By an algebraic linear differential equation I mean a linear partial differential equation in $n$ variables whose coefficients are polynomials in those variables over a field $k$. The solutions of these equations obviously include all the polynomials, but also non-polynomial functions, e.g., the exponential function.

These equations and their generalizations are studied in the theory of algebraic $D$-modules, but literature on $D$-modules moves quickly to sheaves of differential operators on algebraic varieties without answering this basic question for the simplest affine case. The focus seems to be more on the properties of the ring of differential operators and their modules, rather than of their solutions, but I may be wrong.

Even just for $k = \mathbb{C}$ or $\mathbb{R}$ where can I find a description of the space of solutions of all algebraic differential equations in $n$ variables including its algebraic and functional-analytic properties?