Consider the following linear differential equation:
$$f_n(x)y^{(n)}+f_{n-1}(x)y^{(n-1)}+\cdots+f_1(x)y'+f_0(x)y=Q(x),$$ where $f_n(x), f_{n-1}(x), \ldots,f_1(x), f_0(x), Q(x)$ are elementary functions defined on some interval $I$.
My question is:
When do we know if all solutions to the above mentioned differential equation are expressible in terms of elementary functions?
I am aware of this question but my question should be different and more generalised.
Here's one case: Kovacic's algorithm can determine "Liouvillian" solutions to a second-order linear homogeneous differential equation with rational-function coefficients. I don't think there's anything similar for $n$' th order equations.