closed integral formula for a non-zero solution of a homogeneous linear ODE of order 2

Question

Let $$ (\star) \;\;\;\;\;\;\;\;\;\;\;\; y''+p(x)y'+q(x)y=0, $$ be a homogeneous linear ODE of order $2$ with $p(x)$ and $q(x)$ complex valued analytic functions in a small neighbourhood of $0$.

Q: Is there a "closed formula" involving only

(1) elementary functions

(2) elementary arithmetic operations

(3) Integrals (line integrals)

(4) the functions $p(x)$ and $q(x)$

which gives always a non-zero solution to the ODE $(\star)$ in a neighborhood of $0$?

P.S. Of course if $(\star)$ has (at most) a regular singularity at $0$, then using power series (which is not part of the toolkit allowed in my question) it is possible (in principle) to write a non-zero solution.

added: For example, if $y_1$ is a non-zero solution to $\star$, one may obtain another solution $y_2$ of $(\star)$ (linearly independent of $y_1$) using Lagrange's method (the "variation" of the constant) which gives using indefinite integrals $$ y_2(x)=y_1(x)\int (y_1(x))^{-2}e^{-\int p(x)dx} dx $$

Note that $q(x)$ does not appear explicitly in the formula but it is hidden in $y_1(x)$. So one may find a second solution if one has an "input" solution $y_1$

Answer 1

The answer is basically 'no', there is no 'elementary method' involving elementary operations and quadrature (i.e., finding antiderivatives of known holomorphic functions) that will give you a solution to the general second order linear equation with variable coefficients.

For a glimpse at why (an explanation is too complicated to summarize here), you might consult the delightful book by Michio Kuga, Galois' Dream: Group Theory and Differential Equations.

Answer 2

The answer is "no" in a very strong sense to (1), (2). For example $y"+zy=0$ defines Airy functions, and Liouville proved that they cannot be expressed in terms of elementary functions or their (indefinite) integrals. (See, for example Kaplanski, Introduction to differential algebra).

The rest of the questions are ill-defined. What does it mean "in terms of integrals"? Integrals of what and with respect to what? For example, solutions of the Airy equation (above) can be expressed as Laplace transforms of elementary functions. Is this counted as "in terms of integrals" or not?

Even less clear is the last question. What sort of "expressions in terms of $p$, $q$" is allowed? If there is at most one regular singularity, there are power series solutions which converge everywhere. If you do not care about everywhere and want to represent your solution somewhere, then choose a regular point and write a power series at this point "in terms of $p$ and $q$".

EDIT. from the edited question I conclude that a convergent series (for example when $0$ is a regular singularity) is considered a closed form'' solution. When the singularity at $0$ is irregular, there is a divergent series, which nevertheless defines a solution uniquely, and can besummed'' via Borel's summation method. Which means that there is an integral representation (Lalace transform) with some explicit convergent series under the integral. Coefficients of this series can be computed explicitly in terms of coefficients of $p$ and $q$.

Answer 3

In the case where $p, q$ are rational functions, there is the Kovacic algorithm. (J. Symb. Comp, 1986), which is an extension of Risch's algorithm (which can be thought of as an algorithm to get solution of first order homogeneous ODE). The solutions are usually hypergeometric functions, but one does get lucky sometimes and gets elementary functions.