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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 $y_1$

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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.

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  • $\begingroup$ @HugoChapdelaine: You're welcome. By the way, the method of getting a complete solution starting from a particular solution that you mention in your added comment is also explained in Kuga's little book. The problem you raise is one of the first ones treated by differential Galois theory. It's a much more extensive subject; current research in the area probably seems somewhat forbidding, but Kuga provides a very nice introduction at the undergraduate level. $\endgroup$ – Robert Bryant Nov 6 '14 at 9:10
  • $\begingroup$ Dear Robert, I just had a look at Kuga's book. Basically, the answer to my question is positive if and only if the monodromy representation is triangulizable. Unfortunately, the book only proves the if direction (which is quite easy) and says that the only if direction is beyond the scope of the book. Do you know a reference off-hand where I could find a proof for the only if direction? $\endgroup$ – Hugo Chapdelaine Nov 6 '14 at 12:58
  • $\begingroup$ There's a really nice little book by Andy Magid, Lectures on Differential Galois Theory (University Lecture Series, Volume 7, AMS, 1994) that has a thorough treatment and has a good sample of references to the literature for further developments and details. Have a look at that and see whether that does what you want. $\endgroup$ – Robert Bryant Nov 6 '14 at 13:12
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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 $q$.

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  • $\begingroup$ Dear Alexandre, see my added comment $\endgroup$ – Hugo Chapdelaine Nov 6 '14 at 2:57
  • $\begingroup$ Your added comment is insufficient: what is a closed formula? Convergent power series is a closed formula? How about divergent series, which is summable and defines a function uniquely? $\endgroup$ – Alexandre Eremenko Nov 6 '14 at 12:39
  • $\begingroup$ Dear Alexandre, I agree that my problem setup is vague and as you pointed out, ill-defined. So I just had a look at Kuga's book pages 109-115 (mentioned by Robert) and it seems that one may formulate precisely what I was intending to in my question. Any way, thanks a lot for pointing out the important distinction that one should make between a formal solution versus a convergent one; I had not thought about that. $\endgroup$ – Hugo Chapdelaine Nov 6 '14 at 12:55
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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.

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  • $\begingroup$ Thanks a lot Igor, I did not know about this algorithm. $\endgroup$ – Hugo Chapdelaine Nov 6 '14 at 3:14

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