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Rational $S$ -functions for second order ODEs with non-Liouvillian first integrals

This paper extends the $S$ -function method for rational second-order ODEs to cases with non-Liouvillian first integrals by establishing a correspondence with linear differential equations, thereby reducing the decidability of Liouvillianity to classical algorithms and successfully recovering known equations like Airy and Bessel from high-degree examples.

Original authors: L. G. S. Duarte, L. A. C. P. da Mota

Published 2026-10-26

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CC BY 4.0

Original authors: L. G. S. Duarte, L. A. C. P. da Mota

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of mathematics, there is a vast landscape of equations that describe how things change over time. Some of these are simple enough that we can write down their answers using standard functions like polynomials, exponentials, or trigonometric waves. These are the well-behaved solutions, the ones that fit neatly into our existing libraries of knowledge. However, many equations describing the real world are far more stubborn. They resist these standard forms, producing answers that are so complex they require entirely new categories of functions to describe. For centuries, mathematicians have struggled to tell the difference between an equation that has a "nice" answer and one that does not, especially when the equations involve rates of change that depend on multiple variables at once. The difficulty lies in the fact that the surface appearance of an equation can be deceiving; two equations that look almost identical might have completely different types of solutions, one solvable and the other hopelessly tangled.

This is the territory explored by researchers L. G. S. Duarte and L. A. C. P. da Mota. They focused on a specific class of these complex equations, known as second-order differential equations, which often appear in physics and engineering to model oscillations, forces, and motion. Their work centers on a tool called an S-function, which acts like a fingerprint for these equations. In previous studies, this fingerprint was useful only for equations that had "nice" solutions. The researchers asked a critical question: what happens when the fingerprint looks rational and simple, but the solution underneath is actually one of those stubborn, complex types? They discovered that the fingerprint alone cannot tell the whole story. A simple-looking S-function can belong to an equation with a standard solution, or it can belong to an equation with a solution so complex it cannot be written down using standard methods. The two types of equations can look identical from the outside, making it impossible to distinguish them just by glancing at the S-function.

To solve this puzzle, the authors developed a new method that acts as a decoder. They realized that while the S-function hides the true nature of the solution, it is connected to a hidden structure. By analyzing the relationship between the S-function and the equation itself, they found a way to strip away the complexity and reveal a simpler, underlying linear equation. This hidden equation is much easier to study. The researchers proved that the original complex equation has a "nice" solution if and only if this hidden linear equation has a "nice" solution. This connection is powerful because it allows them to use a known, reliable testing method to decide the fate of the original, much harder problem. Instead of trying to solve the complex equation directly, they can now transform it into a simpler form and run a specific test to see if a standard solution exists.

The team put their method to the test with a variety of challenging examples, including equations that had been constructed to hide their origins. In one striking case, they took an equation that looked nothing like a famous problem in physics, fed it into their procedure, and the method successfully identified it as being related to the Airy equation, a classic problem describing the diffraction of light and the behavior of particles. In another instance, they identified a connection to the Bessel equation, which governs the vibration of circular membranes like drumheads. In every case where the underlying solution was complex, their method correctly identified that no standard solution existed. Conversely, when they tested equations known to have simple solutions, the method confirmed this as well. The process was fast, taking less than a second per example on a computer, and it worked even when the equations were disguised by complicated transformations.

What makes this work particularly significant is that it closes a gap in our ability to classify these mathematical objects. Before this, if an equation had a simple-looking S-function but a complex solution, there was no guaranteed way to know for sure without trying to solve it, a task that could be impossible. The new procedure provides a definitive answer. It separates the equations into three clear categories: those that are solvable with standard functions, those that are not, and those that fall outside the scope of this specific method entirely. For the vast majority of cases they examined, the method worked perfectly, recovering the hidden linear structure and applying the test with certainty. The researchers also noted that their approach relies on established algorithms that have been trusted for decades, ensuring that the results are not just guesses but mathematically proven conclusions.

The implications of this work extend beyond pure mathematics. By providing a reliable way to determine whether a complex system has a standard solution, the method helps scientists and engineers know when to stop looking for a simple formula and when to accept that a more complex description is necessary. It turns a question that was previously a matter of trial and error into a systematic procedure. The researchers demonstrated that even when an equation is constructed to be deceptive, hiding its true nature behind layers of algebra, the underlying structure can be uncovered. This ability to see through the disguise of the S-function to the core nature of the equation represents a significant step forward in understanding the behavior of dynamic systems. The work does not claim to solve every equation in existence, but for the specific class they studied, it offers a complete and decisive answer, transforming a problem of uncertainty into one of clear, actionable knowledge.

Technical Summary: Rational S-functions for Second Order ODEs with Non-Liouvillian First Integrals

Problem Statement
The paper addresses a specific gap in the integration theory of rational second-order ordinary differential equations (2ODEs) of the form $y'' = \phi(x, y, y')$ . The existing "S-function method," implemented in the Maple package InSyDE, relies on an associated function $S = I_y / I_z$ derived from a first integral $I$ . This method was previously designed and analyzed under the hypothesis that $I$ is a Liouvillian function (constructible from rational functions, exponentials, and integrals).

The central question posed is whether the S-function can be rational even when the first integral $I$ is not Liouvillian. If so, can one algorithmically distinguish between the two cases? The authors highlight that inspection of the rational S-function alone is insufficient for this distinction, as demonstrated by a pair of equations differing by a single term ($xy$) which share the same rational $S$ but possess fundamentally different first integrals (one rational/Liouvillian, the other non-Liouvillian).

Methodology
The authors develop a framework based on the relationship between rational 2ODEs and linear second-order ODEs with polynomial coefficients.

The Generator (Forward Construction):
Starting with a rational function $g(x, y, z)$ (where $z=y'$ ) and a linear equation $A_2(x)w'' + A_1(x)w' + A_0(x)w = 0$ , the authors impose the substitution $w'/w = g(x, y, y')$ . Differentiating this relation along the flow yields a rational 2ODE.
- Key Insight: The resulting S-function is given explicitly by $S = g_y / g_z$ . Crucially, the Wronskian of the linear equation cancels out in this expression. Consequently, a single rational $S$ is shared by an entire family of 2ODEs generated by varying the linear equation coefficients. This family includes equations with Liouvillian first integrals (where the linear equation has Liouvillian solutions) and those without.
The Inverse Problem (Characterization):
Given an admissible pair $(\phi, S)$ , the paper provides a procedure to determine if it belongs to this "linear type" class and to recover the underlying linear equation.
- Step 1 (Planar Reduction): The condition $S = g_y/g_z$ implies that $g$ is an invariant of a planar vector field $W = D_S \partial_y - N_S \partial_z$ (where $S=N_S/D_S$ ). The authors integrate this field to find a rational invariant $\psi(x, y, z)$ .
- Step 2 (Riccati Test): The pair is of "linear type" if and only if the evolution of $\psi$ along the flow satisfies a Riccati equation of the form $X(\psi) = c_2(x)\psi^2 + c_1(x)\psi + c_0(x)$ .
- Step 3 (Recovery): If the test passes, the coefficients $c_i$ uniquely determine a linear second-order ODE (up to a Möbius transformation of the generator).
The Transfer Theorem:
The paper establishes a critical equivalence: The first integral $I$ of the 2ODE is Liouvillian if and only if the recovered linear second-order ODE possesses a Liouvillian solution. Since the Liouvillianity of linear ODEs with polynomial coefficients is decidable via Kovacic's algorithm, the problem of determining the nature of $I$ for this class of 2ODEs becomes decidable.

Key Contributions

Closed-Form Generator: A formula ( $S = g_y/g_z$ ) generating rational 2ODEs with rational S-functions where the first integral is built from a linear ODE. This proves that rationality of $S$ carries no information regarding the Liouvillianity of $I$ .
Characterization and Recovery: A theorem characterizing pairs $(\phi, S)$ of this type and providing a constructive method to recover the hidden linear equation.
Decision Procedure: A transfer theorem linking the Liouvillianity of the 2ODE's first integral to the solvability of the recovered linear equation. This reduces a three-variable problem (with no a priori degree bound) to two classical problems: finding a rational first integral of a planar vector field and applying Kovacic's algorithm.
Trichotomy: The authors define a classification for rational 2ODEs with rational S-functions:
- Linear type with Kovacic positive (Liouvillian $I$ ).
- Linear type with Kovacic negative (Non-Liouvillian $I$ ).
- Not of linear type (Requires Darboux-theoretic search; Liouvillianity is not guaranteed by this method).

Results and Validation
The procedure was validated on three sets of experiments using Maple and SymPy:

Worked Examples: The algorithm successfully recovered the Airy equation from a 2ODE constructed without explicit reference to it, correctly identifying the non-Liouvillian nature of the first integral. It also handled cases where the generator involved Möbius transformations.
Round Trip: 28 pairs generated from four linear equations and seven rational functions $g$ were processed. The algorithm correctly recovered the original linear equations (up to gauge) and correctly classified the Liouvillianity of the first integrals in all cases.
Pull-back Equations: Nine rational 2ODEs of degree up to 15, constructed from Airy and Bessel equations via point transformations and provided without origin information, were analyzed. The algorithm successfully identified the underlying Airy or Bessel equations and correctly determined that all nine possessed non-Liouvillian first integrals.
Controls: The method successfully distinguished between "linear type" Liouvillian pairs and "Darboux type" Liouvillian pairs (where the S-function arises from a closed rational 1-form but not a linear equation), showing the two sources are structurally distinct.

Significance and Scope
The paper claims to resolve the ambiguity of rational S-functions in the context of non-Liouvillian first integrals. It demonstrates that while the S-function alone cannot distinguish between Liouvillian and non-Liouvillian cases, the pair $(\phi, S)$ allows for a complete decision procedure for a significant class of equations.

The authors explicitly note the limitations and open problems:

The method relies on finding a rational first integral for a planar vector field, which is Poincaré's problem and lacks a priori degree bounds.
The current procedure assumes the generator $g$ is related to the invariant $\psi$ by a Möbius transformation. If the relationship is non-Möbius (rational but higher degree), the problem reduces to a different inverse problem for first-order ODEs, which is not fully solved here.
The work does not claim to solve the integration of all rational 2ODEs, but rather provides a decidable path for those admitting a rational S-function that arises from a linear second-order structure.

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