Hilbert's classes of differential equations: omitted solutions, discriminating solutions, and the classification of underdetermined equations into classes

In 1912 Hilbert introduced a classification of underdetermined ordinary differential equations (equations imposing fewer conditions than there are unknown functions, so that the general solution depends on arbitrary functions) under invertibly integral-free transformations, i.e. changes of variables in which each side is expressed through the other side's variables and finitely many of their derivatives, invertible by differentiation and elimination alone, without quadrature. He proved that all equations admitting an integral-free solution, meaning a parametrization of the general solution by one arbitrary function and finitely many of its derivatives, form a single class, while specific second-order equations, notably the equation identified in the later literature as $y'=(z'')^2$, admit no such solution at all; he established the existence of three distinct classes and indicated how the same argument yields arbitrarily many. For a Monge example $z'=y'^2$ he showed that the solutions omitted by the integral-free parametrization coincide with the discriminating solutions, on which the first variation of an associated integral vanishes, and he proposed the calculus of variations as the proper framework for the class concept. These notions have modern descendants: integral-free solvability is now called differential flatness, the transformation concept underlies Lie–Bäcklund, dynamic, and absolute equivalence of control systems, and omitted solutions reappear as flat singularities. The surviving open core is threefold: a general characterization of omitted solutions and of their coincidence with discriminating solutions beyond the first-order Monge case; development of the variational connection, including the role of the non-invariance of discriminating solutions under the class transformations; and a classification of underdetermined equations into classes, for which no complete system of invariants is known.

Background

An ordinary differential equation is called underdetermined when it imposes fewer independent conditions than there are unknown functions, so that its general solution depends on arbitrary functions of the independent variable rather than on constants alone. Hilbert's 1912 paper studies this situation in its prototypical case: a single differential equation of his type (1) for two unknown functions $y$ and $z$ of one independent variable $x$, involving $x$, $y$, $z$ and finitely many of their derivatives, and assumed not to be obtainable from lower-order equations by differentiation and linear combination. Two notions are central. An integral-free solution (German: integrallose Auflösung, 'solution without integrals') of such an equation is a parametrization of its general solution in which the variables — as a rule including the independent variable, reparametrized by a new parameter — are expressed through one arbitrary function of that parameter and finitely many of its derivatives, using only differentiation and elimination, so that no quadrature is needed to obtain any solution. An invertibly integral-free transformation (umkehrbar integrallose Transformation) is a change of variables in which the old variables $x, y, z$ and a new independent variable together with new dependent functions are mutually expressed through each other and finitely many of their derivatives, the inversion being obtained by differentiation and elimination alone, again without quadrature; two equations are then said to lie in the same class when one is carried into the other by such a transformation. Hilbert proved that all integral-freely solvable equations form a single class, and that conversely certain equations, among them the second-order equation identified in the later literature as $y'=(z'')^2$ (the Hilbert–Cartan equation), admit no integral-free solution whatever; he established the existence of three distinct classes and indicated how the same line of argument yields arbitrarily many, so that there are in fact infinitely many distinct classes. As tools for a 'deeper and more systematic study' of the class concept, the same paper introduces the omitted solutions (ausgelassene Lösungen), meaning solutions of the equation that are not captured by a given integral-free parametrization, and the discriminating solutions (diskriminierende Lösungen), meaning solutions for which the first variation of the integral associated with the equation vanishes — the variational notion Hilbert introduces for this purpose (1912, Math. Ann. 73, pp. 105–106; reference 4) — in analogy with singular solutions of variational problems. For the Monge example $z'=y'^2$ (a first-order underdetermined equation in the two unknown functions $y$ and $z$, the first-order case whose integral-free solvability Hilbert attributes to Monge) he exhibited the integral-free parametrization $x=t^2w_{tt}-2tw_t+2w$, $y=tw_{tt}-w_t$, $z=w_{tt}$ in the arbitrary function $w$ and parameter $t$, and proved that its omitted solutions are exactly the discriminating solutions $y=ax+b$, $z=a^2x+c$. He further noted that discriminating solutions are not invariant under invertibly integral-free transformations and that this non-invariance stands in the closest connection with the omission phenomenon, leaving that connection undeveloped. In modern language, integral-free solvability is differential flatness: a control system or underdetermined equation is flat when all its trajectory variables can be recovered from a fictitious output (the flat output) and finitely many of its derivatives, which makes motion planning and feedback design straightforward; the flatness school of Fliess, Lévine, Martin, and Rouchon, which introduced this terminology in the 1990s, credits Hilbert's invertibly integral-free transformations and Cartan's absolute equivalence (equivalence of systems of differential equations decided through geometric invariants) as the origins of the notion. The companion transformation concepts in current use are Lie–Bäcklund equivalence (a correspondence between the solution sets of two systems in which each variable of one is expressed through the variables of the other and finitely many of their derivatives) and dynamic equivalence of control systems (the same idea, allowing the comparison to pass through auxiliary variables added by dynamic feedback), together with absolute equivalence; points where a flat parametrization degenerates are now studied as flat singularities, the direct descendant of Hilbert's omitted solutions. Separately, the geometry of Monge equations $y'=F(x,y,z,z',z'')$ is equivalent to the study of rank-2 distributions (two-dimensional subbundles of the tangent bundle whose successive Lie brackets — commutators of vector fields — grow in a controlled way), analyzed in Cartan's 1910 five-variable paper; within that theory the equation $y'=(z'')^2$ reappears as the exceptional Hilbert–Cartan equation, with symmetry algebra the exceptional simple Lie algebra $g_2$ of dimension 14, and it serves as the universal building block of all flat Monge equations in the sense of Anderson–Kruglikov. The differential-algebraic side of the class question is the differential Lüroth problem posed by Ritt, asking whether intermediate differential fields (fields equipped with a derivation) of a purely differentially transcendental extension are themselves purely differentially transcendental (generated by one element over the base), answered negatively in two variables by Ollivier.

Problem Statement

The problem, in Hilbert's own framework (1912), is the deeper and systematic study of the class concept for underdetermined ordinary differential equations — equations imposing fewer independent conditions than there are unknown functions, so that the general solution depends on arbitrary functions rather than constants — with Hilbert's paper working in the prototypical case of a single equation for two unknown functions $y$ and $z$ of one independent variable $x$ (his equations of type (1), not obtainable from lower-order equations by differentiation and linear combination), two equations belonging to the same class when one is carried into the other by an invertibly integral-free transformation (a change of variables in which the old and the new variables are mutually expressed through each other and finitely many of their derivatives, invertible by differentiation and elimination alone, without quadrature). Concretely, three linked objectives, all of which are required: (1) Characterize the omitted solutions — solutions of the equation that are not captured by a given integral-free parametrization of it — for arbitrary equations that admit an integral-free solution; Hilbert proved only for the Monge example $z'=y'^2$ that the omitted solutions are exactly the discriminating solutions, i.e. the solutions on which the first variation of an associated integral vanishes, and the question is for which equations of type (1) this coincidence between the analytic omission phenomenon and the variational discriminating condition holds, and what replaces it when it fails. Objective (1) asks, for every equation of type (1) that admits an integral-free solution, whether this coincidence holds and what replaces it when it fails. (2) Develop the proposed variational framework: determine and settle, by proof or by counterexample, the precise connection Hilbert noted but left undeveloped between the non-invariance of discriminating solutions under invertibly integral-free transformations and the omission phenomenon. The logical form of that connection and the quantities it links are not prescribed, but the connection proposed must be admissible: (i) it must be stated for equations of type (1) in general and not only for the Monge example $z'=y'^2$; (ii) it must yield at least one concrete, checkable prediction for a named equation of type (1) other than that example; and (iii) it must be consistent with Hilbert's Monge example, in which the omitted solutions are exactly the discriminating solutions although an invertibly integral-free transformation can remove discriminating solutions altogether. The objective is to prove an admissible connection in the stated generality, or to prove that no connection meeting (i)–(iii) exists — equivalently, that those requirements are jointly unsatisfiable; a proof of the latter kind is a general argument, not a refutation of one self-chosen candidate connection, which refutes only that candidate. (3) Classify underdetermined equations of type (1) into classes under invertibly integral-free transformations: Hilbert proved that the integral-freely solvable equations form one class and that some equations, such as the second-order equation identified in the later literature as $y'=(z'')^2$, admit no integral-free solution at all and hence lie outside it; he established the existence of three distinct classes and indicated how the argument extends to arbitrarily many, but the general proof is only sketched — for transformations involving derivatives of order $r \geq 3$ the argument is direct, while the cases $r=1$ and $r=2$ are said to require a separate special treatment whose details the paper does not give. A complete classification, with an explicit system of class invariants constant within each class and separating distinct classes, determining for a given equation whether it is integral-freely solvable and, if not, which class it belongs to, remains to be given. The modern control-theoretic reformulation — characterize differentially flat systems and the singularities of their flat parametrizations up to Lie–Bäcklund, dynamic, or absolute equivalence, allowing change of the independent variable — is the direct descendant of objectives (1) and (3) for general underdetermined systems and is open in general; objective (2) is untouched in the modern literature.

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