Is there a theory of "elementary closed form solution" at the operator level for differential equations?

Question

We begin by considering the usual general first order linear equation of the form

$$ a_0 y' + a_1 y + a_2 = 0 $$

Where $a_i,y \in \mathbb{C} \rightarrow \mathbb{C}$. Now it's well known from everyone's favorite undergrad ODE class that this has a general solution in terms of integration factors.

$$ y = - e^{-\int \frac{a_1}{a_0}} \int \left[ e^{\int \frac{a_1}{a_0}} \frac{a_2}{a_0} \right] dx$$

Now theres at least two integrals going on here and for almost all choices of $a_0, a_1, a_2$ these integrals CANNOT be expressed in terms of elementary functions. BUT we can declare that this ENTIRE abstract solution is an "elementary" general solution, i.e. it involves only finitely many elementary functions, and "elementary" operators which for now just includes integrals, derivatives, and function inverses.

So the question is kind of natural, what if we look at second order equations?

$$ a_0 y'' + a_1 y' + a_2 y + a_3 = 0 $$

I expect that this has NO such elementary general solution (since we would've found it by now) BUT as far as I recall I have never seen an explicit proof of such.

So to be clear I want to know, at the OPERATOR-level why is there no elementary general solution to that equation? And more generally, what machinery/theory is required to settle such problems?

Some hints of what this theory could encompass

We have the trick of matrix exponentials i.e. if the matrix

$$ \begin{bmatrix} 0 & -1 \\ \frac{a_1}{a_0} & \frac{a_2}{a_0} \end{bmatrix} $$

commutes with its own derivative then the matrix exponential trick must work and so this equation is "solvable". So our gadget for gauging solvability must somehow be connected to these matrices (but it's not even clear how to define that object for non linear equations).

Answer 1

There are several theories which deal with this question. One is the differential algebra, see, for example the little book

I. Kaplansky, Introduction to differential algebra, Publ. de l'Institut de Mathématique de l'Université de Nancago, No. V. Actualités Scientifiques et Industrielles No. 1251. Hermann, Paris, 1976.

Another is a generalization of Galois theory which is due to Picard and Vessiot:

MR1199112 Kuga, Michio Galois' dream: group theory and differential equations. Birkhäuser Boston, Inc., Boston, MA, 1993,

and

MR3289210 Khovanskii, Askold Topological Galois theory. Solvability and unsolvability of equations in finite terms. Springer, Heidelberg, 2014.