0

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.

Zuriel
  • 5,331
  • If by "expressible in terms of elementary functions" you disallow functions that are expressible only as integrals of elementary functions, then this appears to me a severe restriction on the coefficients.. After all, even an equation as simple as $y'' - xy = 0$ has linearly independent solutions (the Airy functions) that (to my knowledge) are not expressible in terms of elementary functions without integration. – Gyu Eun Lee Oct 10 '19 at 03:46

1 Answers1

0

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.

Robert Israel
  • 448,999