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Is there a theory of "Elementary Closed Form Solution" at the Operator Level for Differential Equations?

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 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?