The Diffiety Institute Preprint Series

Preprint DIPS-2/97 April 30, 1997

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The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Contact classification of 3-order linear ODEs

by

Valeriy A. Yumaguzhin

Program System Institute of Russian Academy of Sciences

April 30, 1997

Abstract. A local classification of regular 3-order linear ordinary differential equations with respect to contact transformations is given.

0. Introduction

It is well known that any two 2–order linear ordinary differential equations (ODEs) are locally equivalent. The similar statement for 3-order linear ODEs is not correct. The point is that a necessary condition for one differential equation to be locally equivalent to another one is that dimensions of the classical symmetry algebras of these equations are the same; but dimension of the classical symmetry algebra of an arbitrary 3–order linear ODE can be equal to one of the numbers 10, 5, or 4 (see the corollary of Theorem 2.1).

The aim of this paper is to obtain a local classification of 3–order linear ODEs at a regular point with respect to contact transformations.

Our approach to this problem is the following.

It is easy to prove that any 3–order linear ODE can be transformed to the form

y′′′ = v(x) · y′ + w(x) · y.  (0.1)

Therefore our problem is reduced to the classification of equations (0.1).

Every ODE of the form (0.1) is identified in the intrinsic way with the geometric structure γ on R^{1}. The coefficients v(x) and w(x) of this equation are the components of the coordinate expression of γ in the coordinate system x on R^{1}. Now our problem is reduced to a local classification of those geometric structures with respect to diffeomorphisms of R^{1}.

To this end we calculate differential invariants of these structures. The invariant differential form ω and the scalar differential invariant I obtained here make possible to solve this problem completely.

As a result we obtain the following local classification of 3–order linear ODEs at a regular point with respect to contact transformations. Let E be an ODE of the form (0.1) considered as the submanifold in the corresponding 3–jet bundle J^{3}π and let Sym E be the algebra of classical symmetries for E; then

(1) If dim Sym E = 10, then E is locally equivalent to the equation

y′′′ = 0  (0.2)

at every point x_{3} ∈ E.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 34A26, 53A55, 34A30, 34C20, 58F35.

Key words and phrases. Linear ordinary differential equation, jet bundle, geometric structure, contact transformation, point transformation, equivalence problem, differential invariant..

(2) If dim Sym E = 5, then E is locally equivalent to one of the following equations

y′′′ = K · y′ + y, K ∈ R^{1}  (0.3)

at every point x_{3} ∈ E such that ω|_{π(x_{3})} ≠ 0.

(3) If dim Sym E = 4, then E is locally equivalent to an equation of the form (0.4) at every point x_{3} ∈ E such that ω|_{π(x_{3})} ≠ 0 and dI|_{π(x_{3})} ≠ 0.

y′′′ = [xu^{2} − 2 (\frac{u′}{u} )′ + (\frac{u′}{u} )^{2}] y′

+ \frac{1}{2} [(xu^{2} − 2 (\frac{u′}{u} )′ + (\frac{u′}{u} )^{2})′ − u^{3}] y, (0.4)

here u = u(x) is a nowhere vanishing smooth function. Two equations of the form (0.4) are locally equivalent iff the corresponding functions u coincide in a suitable neighborhood.

Thus the complete list of ”simplest” locally nonequivalent regular 3–order linear ODEs consists of equation (0.2), the infinite family of equations (0.3) and the infinite family of equations (0.4).

Note that this list of ”simplest” ODEs is a complete list of ”simplest” locally nonequivalent 3–order linear ODEs with respect to point transformations (see [8]).

Recall that the general method to investigate local equivalence problem based on the theory of differential invariants was originated by classics of the end of XIX century S.Lie, A.Tresse, G.–H.Halphen, E.Laguerre, R.Liouville and others. In particular, first the abovementioned invariant differential form ω was used by E.Laguerre in [4] to solve the local equivalence problem of 3–order linear ODEs with respect to point transformations of the form

X = f(x), Y = g(x) · y.

In [3], G.–H.Halphen used scalar differential invariants to obtain a criterion for a 3–order linear ODE to be locally equivalent to a linear ODE with constant coefficients with respect to those transformations.

This paper is organized as follows. In section 1 we recall preliminary definitions and facts. In section 2 we calculate the classical symmetry algebra for ODEs of the form (0.1), for ODEs admitting a 5 or 4 – dimensional classical symmetry algebra we reduce the problem of local classification with respect to contact transformations to the problem of local classification with respect to point transformations. In section 3 we solve the local equivalence problem for ODEs of the form (0.1). In section 4 we calculate differential invariants and give the local classification of 3–order linear ODEs with respect to contact transformations.

Below everything is supposed to be smooth.

By R^{n} denote the n – dimensional arithmetical space, by [f]_{x}^{k} denote the k–jet of a map

f at the point x, and by definition, put A_{x_{1}...x_{k}} = \frac{∂^{k}A}{∂x_{1} … ∂x_{k}}.

1. Preliminaries

In this section we recall necessary notations and results of the geometry of differential equations (see [6]) and the theory of geometric structures and differential invariants (see [1],[2]).

1.1. Lie transformations. Let λ : E → M be a smooth bundle, let dim M = n, and let dim E = n + m. By J^{k}λ k = 0, 1, 2,…, denote the bundle of k–jets [s]_{x}^{k} of all crosssections s of λ. Let λ_{k} : J^{k}λ → M be the projection that takes each [s]_{m}^{k} to m and let λ_{k,r} : J^{k}λ → J^{r}λ, k > r, be the projection that takes each k-jet [s]_{m}^{k} to the r-jet [s]_{m}^{r}.

Every cross-section s : U → E, U ⊂ M, of λ generates the cross-section j_{k}s : U → J^{k}λ of the jet bundle λ_{k} by the formula j_{k}s : x ↦ [s]_{m}^{k}. By L_{s}^{(k)} denote the image of the cross-section j_{k}s.

By T_{x_{k}}(J^{k}λ) denote the tangent space to J^{k}λ at x_{k} ∈ J^{k}λ, by T_{x_{k}}(L_{s}^{(k)}) denote the tangent space to L_{s}^{(k)} at x_{k} ∈ L_{s}^{(k)}.

The Cartan plane C_{x_{k}} ⊂ T_{x_{k}}(J^{k}λ) at x_{k} ∈ J^{k}λ is the subspace spanned on the union of all T_{x_{k}}(L_{s}^{(k)}) such that x_{k} ∈ L_{s}^{(k)}.

The Cartan distribution C on J^{k}λ is the distribution defined by the formula

C : x_{k} ⟼ C_{x_{k}}.

A (local) diffeomorphism of J^{k}λ conserving the Cartan distribution is called a Lie transformation.

Every Lie transformation f : U → U′ of J^{k}λ is lifted canonically up to the Lie transformation f^{(r)} : λ_{k+r,k}^{−1}(U) → λ_{k+r,k}^{−1}(U′) of J^{k+r}λ, r = 1, 2,…, such that the diagram

λ_{k+r,k}^{−1}(U) \underrightarrow{f^{(r)}} λ_{k+r,k}^{−1}(U′)
λ_{k+r,k}↓  ↓λ_{k+r,k}
U  \overrightarrow{f}  U′

is commutative.

Recall that f^{(r)} is defined in the following way. A point x_{k+1} ∈ J^{k+1}λ is identified with K_{x_{k+1}} = T_{x_{k}}(L_{s}^{(k)}), where x_{k} = λ_{k+1,k}(x_{k+1}). The differential f_{∗} maps K_{x_{k+1}} onto the subspace f_{∗}( K_{x_{k+1}} ). If f_{∗}( K_{x_{k+1}} ) is projected on M without a degeneration, then there is x′_{k+1} ∈ J^{k+1}λ such that K_{x′_{k+1}} = f_{∗}( K_{x_{k+1}} ) and we set f^{(1)}(x_{k+1}) = x′_{k+1}. It is obvious that f^{(1)} is a Lie transformation of J^{k+1}λ defined almost everywhere in λ_{k+1,k}^{−1}(U). Setting f^{(r+1)} = (f^{(r)})^{(1)}, we define the Lie transformation f^{(r)} for all r = 1, 2,…. Clearly, that f^{(r)} is defined almost everywhere in λ_{k+r,k}^{−1}(U). (We shall say for brevity that f^{(r)} is defined in λ_{k+r,k}^{−1}(U).)

A Lie transformation of J^{1}λ is called a contact transformation.

A Lie transformation of J^{0}λ (that is an arbitrary diffeomorphism of J^{0}λ) is called a point transformation.

It is well known (see [6]) that if m = 1, then every Lie transformation of J^{k}λ, where k > 1, is the lifting of some contact transformation.

A vector field ξ in J^{k}λ is called a Lie field if flow of ξ is generated by Lie transformations. Obviously, the lifting of Lie transformations defines the lifting of every Lie field ξ in

J^{k}λ up to the Lie field ξ^{(r)} in J^{k+r}λ for all r = 1, 2,….

Let λ : R^{1} × R^{1} → R^{1} be the product bundle and let x, y, p_{1}, …, p_{k} be the standard coordinates on J^{k}λ. Then it is easily shown that a contact transformation is defined in the standard coordinates by the formulas

\begin{aligned}X & = X(x, y, p_{1}), \\ Y & = Y (x, y, p_{1}), \\ P_{1} & = \frac{Y_{x} + p_{1}Y_{y}}{X_{x} + p_{1}X_{y}} , \end{aligned}  (1.1)

where the functions X(x, y, p_{1}) and Y (x, y, p_{1}) are connected by the relation

Y_{p_{1}}(X_{x} + p_{1}X_{y}) − X_{p_{1}}(Y_{x} + p_{1}Y_{y}) = 0.

Obviously, a point transformation is defined in the standard coordinates x, y of J^{0}λ by the formulas

\begin{aligned}X & = X(x, y), \\ Y & = Y (x, y). \end{aligned}  (1.2)

1.2. Geometric structures. Let G_{k}^{n}, k = 1, 2,…, be the group of k-jets at the point 0 ∈ R^{n} of all diffeomorphisms R^{n} → R^{n} conserving the point 0.

Let M be a smooth n – dimensional manifold and let S_{k}(M) be the manifold of k-jets at 0 ∈ R^{n} of all diffeomorphisms s : R^{n} → M. The projection ρ_{k} : S_{k}(M) → M is defined by ρ_{k}( [s]_{0}^{k}) = s(0). The right action S_{k}(M)×G_{k}^{n} → S_{k}(M) of the group G_{k}^{n} on S_{k}(M) is defined by

( [s]_{0}^{k}, [g]_{0}^{k}) ⟼ [s ∘ g]_{0}^{k} ∀ [s]_{0}^{k} ∈ S_{k}(M), ∀ [g]_{0}^{k} ∈ G_{k}^{n}.

It is easy to verify that ( S_{k}(M), ρ_{k}, M, G_{k}^{n} ) is a principal bundle with structural group G_{k}^{n}.

Let Q be a smooth N – dimensional manifold and let μ : G_{k}^{n} × Q → Q be a left action of the group G_{k}^{n} on Q. We often will write g_{k} · q instead μ( g_{k}, q) g_{k} ∈ G_{k}^{n}, q ∈ Q.

Let ρ_{μ} : E → M be the Q – associated bundle of the principal bundle S_{k}(M) generated by μ.

We recall the definition of this bundle. The actions G_{k}^{n} on S_{k}(M) and on Q generate the equivalence relation on S_{k}(M) × Q by the formula

( s_{k}, g_{k} · q) ∼ ( s_{k} · g_{k}, q) ∀ g_{k} ∈ G_{k}^{n}, ∀ ( s_{k}, q) ∈ S_{k}(M) × Q.

By [s_{k}, q] denote the equivalence class of the element (s_{k}, q), by E denote the manifold of all those equivalence classes, and the projection ρ_{μ} : E → M is defined by the formula ρ_{μ}( [s_{k}, q] ) = ρ_{k}(s_{k}).

A cross-section γ of ρ_{μ} is called a geometric structure of type μ on the manifold M. The bundle ρ_{μ} is called a bundle of geometric structures of type μ too.

Every coordinate system (local chart) ( U, h = (x^{1},…, x^{n}) ) of M generates the local trivialisation h_{E} : ρ_{μ}^{−1}(U) → U × Q of the bundle ρ_{μ} in the following way. The coordinate system ( U, h = (x^{1},…, x^{n}) ) generates the cross-section h_{k} : U → S_{k}(M) by the formula

h_{k}(m) = [h^{−1} ∘ t_{h(m)}]_{0}^{k},

where t_{h(m)} is the translation of R^{n} defined by t_{h(m)}(x) = x + h(m). Let e ∈ ρ_{μ}^{−1}(U) and m = ρ_{μ}(e); then there is g_{k} ∈ G_{k}^{n} such that

e = [s_{k}, q] = [h_{k}(m) · g_{k}, q] = [h_{k}(m), g_{k}^{−1} · q].

By definition, put

h_{E} : e ↦ (m, g_{k}^{−1} · q).

Let γ be a geometric structure of type μ on M, let pr_{2} : U ×Q → Q be the projection on the second factor; then for every coordinate system ( U, h = (x^{1},…, x^{n}) ) of M the map γ_{h} = pr_{2} ∘ h_{E} ∘ (γ | U) : U → Q is called the expression of γ in the coordinate system ( U, h = (x^{1},…, x^{n}) ).

Let ( U, h = (x^{1},…, x^{n}) ) and ( U′, h′ = (y^{1},…, y^{n}) ) be coordinate systems of M such that U ∩U′ ≠ ∅ and let (y^{1}(x^{1},…, x^{n}),…, y^{n}(x^{1},…, x^{n})) be the transformation of these coordinate systems. Then it follows easily that the expressions of γ in these coordinate systems are connected by the following formula.

γ_{h′}( y(m) ) = μ( \frac{∂y^{i}}{∂x^{j}} (x(m)),…, \frac{∂^{k}y^{i}}{∂x^{j_{1}} … ∂x^{j_{k}}} (x(m)), γ_{h}( x(m) ),  (1.3)

where m ∈ U ∩ U′ and ( \frac{∂y^{i}}{∂x^{j}} (x(m)),…, \frac{∂^{k}y^{i}}{∂x^{j_{1}} … ∂x^{j_{k}}} (x(m) ) ∈ G_{k}^{n}. Relation (1.3) is

called the transformation law of coordinate expressions.

Obviously, every geometric structure is defined completely by the collection of all its coordinate expressions and the transformation law of these expressions.

Example. A function, an arbitrary tensor field, and a linear connection are examples of geometric structures of different types.

1.3. Differential invariants. Every diffeomorphism f of the manifold M generates the Lie transformation f_{E} of E by the formula

f_{E} : [ [s]_{0}^{k}, q] ⟼ [ [f ∘ s]_{s(0)}^{k}, q].  (1.4)

Obviously, the following diagram is commutative

E \underrightarrow{f_{E}} E
ρ^{μ}↓  ↓ρ^{μ}
M \overrightarrow{f} M

Therefore every diffeomorphism f of M generates the transformation of geometric structures by the formula

γ ⟼ f_{E} ∘ γ ∘ f^{−1}.  (1.5)

Let (y^{1} = f^{1}(x^{1},…, x^{n}),…, y^{n} = f^{n}(x^{1},…, x^{n}) be the expression of diffeomorphism f in coordinate systems ( U, h = (x^{1},…, x^{n}) ) and ( U′, h′ = (y^{1},…, y^{n}) ) of M and let γ′ = f_{E} ∘ γ ∘ f^{−1}; then it follows easily that

γ′_{h′}(m) = μ( \frac{∂f^{i}}{∂x^{j}} ( f^{−1}(m) ),…, \frac{∂^{k}f^{i}}{∂x^{j_{1}} … ∂x^{j_{k}}} ( f^{−1}(m) ), γ_{h}( f^{−1}(m) ) ).

Let (ρ_{μ})_{r} : J^{r}ρ_{μ} → M be the r–jet bundle of ρ_{μ}, r = 0, 1, 2,…. By f_{E}^{(r)} denote the lifting of f_{E} up to the Lie transformation of J^{r}ρ_{μ}.

Obviously, this lifting defines the lifting of every vector field ξ on M up to the Lie vector field ξ_{E}^{(r)} on J^{r}ρ_{μ}, r = 0, 1, 2,….

A diffeomorphism f of M is called a symmetry of the geometric structure γ if γ = f_{E} ∘ γ ∘ f^{−1}.

A vector field ξ on M is called an infinitesimal symmetry of the geometric structure γ if the flow of ξ consists of symmetries for γ.

It is obvious that a vector field ξ is an infinitesimal symmetry for γ iff the vector field ξ_{E} tangents to L_{γ}^{(0)}.

A geometric structure on the manifold J^{r}E is called a differential invariant of order r if it is invariant with respect to all diffeomorphisms of the form f_{E}^{(r)}. A differential invariant is called a scalar differential invariant if it is a function.

Let Γ be the Lie pseudogroup of all diffeomorphisms from M to itself. The Lie pseudogroup Γ_{E}^{(r)} = { f_{E}^{(r)} | f ∈ Γ } acts on the J^{r}E. As a result J^{r}E is divided into orbits. It is clear that a function I on J^{r}E is a scalar differential invariant iff I is constant on each orbit.

The coordinate definition of a scalar differential invariant is the following. Let q^{1},…, q^{N} be a coordinate system on the manifold Q; then for every geometric structure γ we have γ_{h} = (γ_{h}^{1},…, γ_{h}^{N}). A smooth function I of variables q^{α}; q_{j_{1}}^{α}; …; q_{j_{1}...j_{r}}^{α}, α = 1, 2,…, N, j_{l} = 1, 2,…, n, is said to be a scalar differential invariant of order r if for every geometrical structure γ of type μ and for every transformation of coordinate systems y = y(x), the following condition holds

I( γ^{α}(x), \frac{∂γ^{α}}{∂x^{j}} (x),…, \frac{∂^{r}γ^{α}}{∂x^{j_{1}} … ∂x^{j_{r}}} (x) )

= I( γ̃^{α}(y(x)), \frac{∂γ̃^{α}}{∂y^{j}} (y(x)),…, \frac{∂^{r}γ̃^{α}}{∂y^{j_{1}} … ∂y^{j_{l}}} (y(x)) ),

where γ^{α} are the components of the expression of γ in the coordinates x^{1},… x^{n} and γ̃^{α} are the components of the expression of γ in the transformed coordinates y^{1},… y^{n}.

Let I be a differential invariant of order r and let γ be a geometric structure of type μ on M; then by I(γ) denote the restriction I | L_{γ}^{r} of I on image of j_{r}γ. One can assume that I(γ) is defined on M because the restriction (ρ_{μ})_{r} | L_{γ}^{r} : L_{γ}^{r} → M is a diffeomorphism. I(γ) is said to be the value of I on the geometric structure γ.

Let ξ be infinitesimal symmetry of a geometric structure γ and let I be a scalar differential invariant. Then it is easy to prove that I(γ) is an integral of the vector field ξ.

1.4. Equivalence problem for ODEs. Let π : R^{1} × R^{1} → R^{1} be the product bundle. We identify every ODE of order k

F (x, y(x), \frac{dy}{dx},…, \frac{d^{k}y}{dx^{k}}) = 0

with the submanifold E ⊂ J^{k}π defined by the equation

F (x, y, p_{1},…, p_{k}) = 0.

Let E and E′ be ODEs of order k. We say that E and E′ are locally equivalent if there exist a local contact transformation f : U → U′, U, U′ ⊂ J^{1}π, transforming E to E′ that is f^{(k−1)}( E ∩ π_{k,1}^{−1}(U) ) = E′ ∩ π_{k,1}^{−1}(U′); in this case for every point x_{k} ∈ E ∩ π_{k,1}^{−1}(U) we say that the equation E is locally equivalent to the equation E′ at x_{k}.

The local equivalence problem for ODEs consists in obtaining of a criterion for local equivalence of two equations.

1.5. Classical symmetries of ODEs. A Lie field on J^{1}π is called a contact vector field. It is easily shown that a contact vector field ξ can be represented in the standard coordinates x, y, p_{1} by the formula

ξ = ξ_{φ} = −φ_{p_{1}} \frac{∂}{∂x} + (φ − p_{1}φ_{p_{1}})\frac{∂}{∂y} + (φ_{x} + p_{1}φ_{y})\frac{∂}{∂p_{1}} ,

where the function φ = φ(x, y, p_{1}) is called the generating function of ξ. Obviously, the generating function defines the contact vector field completely.

Subjecting a contact vector field to an arbitrary contact transformation (1.4), we get a contact vector field. It is easy to verify that the generating functions Φ of the obtained vector field and the generating function φ of the initial one are connected by the formula

Φ( X(x, y, p_{1}), Y (x, y, p_{1}), P_{1}(x, y, p_{1}) ) = \frac{X_{x}Y_{y} − X_{y}Y_{x}}{X_{x} + p_{1}X_{y}} φ(x, y, p_{1}) .  (1.6)

An arbitrary vector field ζ in J^{0}π is called a point vector field. In the standard coordinates ζ is of the form

ζ = a(x, y)\frac{∂}{∂x} + b(x, y)\frac{∂}{∂y} .  (1.7)

The lifting ζ^{(1)} of ζ is a contact vector field. It is easy to verify that the generating function of ζ^{(1)} is of the form

−a(x, y) · p_{1} + b(x, y).  (1.8)

Conversely, if the generating function of a contact vector field is of the form (1.8), then this contact vector field is the lifting of some point vector field.

We shall say that function (1.8) is the generating function of point vector field (1.7).

A contact transformation f : U → U′ in J^{1}π is called a symmetry of a differential equation E ⊂ J^{k}π if E ∩ π_{k,1}^{−1}(U′) = f^{(k−1)}( E ∩ π_{k,1}^{−1}(U) ).

A contact vector field ξ_{φ} in J^{1}π is called a classical symmetry of a differential equation E ⊂ J^{k}π if the lifting ξ_{φ}^{(k−1)} is tangent to the submanifold E.

Let Sym E be the set of all classical symmetries of a differential equation E. It is easy to verify that Sym E is a Lie algebra over R^{1} with respect to the Lie bracket.

It follows from the general theory of symmetries of differential equations ( see [6] ) that the space of generating functions of classical symmetries for the equation

p_{k} − F (x, y, p_{1},…, p_{k−1}) = 0.

coincides with the space of solutions of the form φ = φ(x, y, p_{1}) for the following linear PDE

( D^{k} − \frac{∂F}{∂p_{k−1}} D^{k−1} −… − \frac{∂F}{∂p_{1}} D − \frac{∂F}{∂y} )(φ) = 0,  (1.9)

where D = \frac{∂}{∂x} + p_{1} \frac{∂}{∂y} + p_{2} \frac{∂}{∂p_{1}} + … + p_{k−1} \frac{∂}{∂p_{k−2}} + F \frac{∂}{∂p_{k−1}} .

2. Classical symmetries of 3–order linear ODEs

2.1. Let E be an ODE of the form p_{3} = v(x) · p_{1} + w(x) · y.

Theorem 2.1. Suppose coefficients v and u of E are defined in a domain U and ξ_{φ} ∈ Sym E; then the following statements hold:

(1) The generating function φ is of the form

φ = a(x)p_{1}^{2} + (b(x) − 2a_{x}y)p_{1} + (2a_{xx} − va(x))y^{2} + (K − b_{x})y + c(x)

with K ∈ R^{1} and a(x), b(x), c(x) are solutions of E iff v_{x} − 2w ≡ 0 in U.

(2) The generating function φ is of the form

φ = K^{1}(v_{x} − 2w)^{−1/3}p_{1} + [K^{2} − K^{1}((v_{x} − 2w)^{−1/3})_{x}]y + c(x)

with K^{1}, K^{2} ∈ R^{1} and c(x) is a solution of E iff (v_{x}−2w)(x) ≠ 0 almost everywhere in U and (v_{x} − 2w)^{−1/3} is a solution of the equation p_{3} = v · p_{1} + (1/2) · v_{x} · y.

(3) The generating function φ is of the form

φ = Ky + c(x)

with K ∈ R^{1} and c(x) is a solution of E iff (v_{x}−2w)(x) ≠ 0 almost everywhere in U and (v_{x} − 2w)^{−1/3} is not a solution of equation p_{3} = v · p_{1} + (1/2) · v_{x} · y.

Proof. Equation (1.9) corresponding to an equation p_{3} = v(x)·p_{1} +w(x)·y is the following one

( D^{3} − v(x) · D − w(x) )(φ) = 0,  (2.1)

where D = \frac{∂}{∂x} + p_{1} \frac{∂}{∂y} + p_{2} \frac{∂}{∂p_{1}} + ( v(x) · p_{1} + w(x) · y)\frac{∂}{∂p_{2}} .

Taking into account that φ depends on x, y, p_{1}, we see that the left hand side of (2.1) is a polynomial of degree 3 on p_{2}. Therefore (2.1) holds iff coefficients of this polynomial are equal to zero. Therefore equation (2.1) is equivalent to the following system of four PDEs

φ_{p_{1}p_{1}p_{1}} = 0,  (2.2)
φ_{xp_{1}p_{1}} + p_{1}φ_{yp_{1}p_{1}} + φ_{yp_{1}} = 0,  (2.3)
(v · p_{1} + w · y)φ_{p_{1}p_{1}} + φ_{xy} + p_{1}φ_{yy}
+ φ_{xxp_{1}} + 2p_{1}φ_{xyp_{1}} + p_{1}^{2}φ_{yyp_{1}} = 0,  (2.4)
(v_{x} · p_{1} + w_{x} · y)φ_{p_{1}} − w(φ − p_{1}φ_{p_{1}})
− v(φ_{x} + p_{1}φ_{y}) + (v · p_{1} + w · y)(3φ_{xp_{1}} + 3p_{1}φ_{yp_{1}} + φ_{y})
+ φ_{xxx} + 3p_{1}φ_{xxy} + 3p_{1}^{2}φ_{xyy} + p_{1}^{3}φ_{yyy} = 0.  (2.5)

From equation (2.2) we get φ(x, y, p_{1}) = a(x, y)p_{1}^{2} + b^{1}(x, y)p_{1} + c^{1}(x, y).

Substituting this expression for φ in equation (2.3), we get the equation 4a_{y}p_{1} + 2a_{x} + b_{y}^{1} = 0. Taking into account that the functions a, b depend on x, y, we see that this equation is equivalent to the following system of PDEs

a_{y} = 0, 2a_{x} + b_{y}^{1} = 0

. From this system we get φ(x, y, p_{1}) = a(x)p_{1}^{2} + (b(x) − 2a_{x}y)p_{1} + c^{1}(x, y).

In the same way, substituting the obtained expression for φ in equation (2.4), we get the following system

c_{yy}^{1} + 2va − 4a_{xx} = 0, c_{xy}^{1} + 2way − 2a_{xxx}y + b_{xx} = 0.

From the first equation we get c^{1}(x, y) = (2a_{xx} − va)y^{2} + c^{11}(x)y + c(x). Substituting this expression for c in the second equation, we get the following system

b_{xx} + c_{x}^{11} = 0  (2.6)
a_{xxx} − va_{x} − wa − (v_{x} − 2w)a = 0  (2.7)

From equation (2.6) we obtain c^{11} = K − b_{x}, where K ∈ R^{1}. Therefore we have

φ(x, y, p_{1}) = a(x)p_{1}^{2} + (b(x) − 2a_{x}y)p_{1} + (2a_{xx} − va)y^{2} + (K − b_{x})y + c(x).  (2.8)

Substituting the obtained expression for φ in equation (2.5), we see that left hand side of the obtained equation is a polynomial of degree 2 on p_{1}. Therefore this equation is equivalent to a system of three equations. The first one corresponding to the coefficient of p_{1}^{2} is the following equation

7( a_{xxx} − va_{x} − wa) − 4(v_{x} − 2w)a = 0.  (2.9)

Comparing equations (2.7) and (2.9), we get

(v_{x} − 2w)a = 0.  (2.10)

Let v_{x} − 2w ≡ 0 in U.

Then it follows from (2.7) or (2.9) that a is a solution of E. Now substituting expression (2.8) for φ in equation (2.5), we get

[ −5( a_{xxx} − va_{x} − wa)_{x}y + 2( b_{xxx} − vb_{x} − wb) ]p_{1}

− ( a_{xxx} − va_{x} − wa)_{xx}y^{2} + ( b_{xxx} − vb_{x} − wb)_{x}y
− ( c_{xxx} − vc_{x} − wc) = 0.

From this equation we see that b and c are solutions of E. This completes the proof of the first statement of Theorem 2.1.

Now let (v_{x} − 2w)(x) ≠ 0 almost everywhere in U.

Then it follows from (2.10) that a ≡ 0 in U. Now from (2.8) we see that

φ(x, y, p_{1}) = b(x)p_{1} + (K − b_{x})y + c(x).

Substituting this expression for φ in equation (2.5), we get the following system

b_{xxx} − vb_{x} − \frac{1}{2}v_{x}b = 0  (2.11)
2(b_{xxx} − vb_{x} − \frac{1}{2}v_{x}b)_{x} − 3(v_{x} − 2w)b_{x} − (v_{x} − 2w)_{x}b = 0  (2.12)
c_{xxx} − vc_{x} − wc = 0.  (2.13)

Equation (2.13) means that c is a solution of E. From equations (2.11) and (2.12) we obtain 3(v_{x} − 2w)b_{x} + (v_{x} − 2w)_{x}b = 0. From this equation we see that b = K^{1}(v_{x} − 2w)^{−1/3}, where K^{1} ∈ R^{1}.

Now we see that if b = (v_{x} − 2w)^{−1/3} is a solution of equation (2.11), then from (2.8) we have φ = K^{1}(v_{x} − 2w)^{−1/3}p_{1} + [K^{2} − K^{1}((v_{x} − 2w)^{−1/3})_{x}]y + c(x) else b ≡ 0 and from (2.8) we have φ = Ky + c(x). ■

Corollary. dim Sym E is equal to one of the numbers 10, 5, 4, moreover (1) dim Sym E = 10 iff v_{x} − 2w ≡ 0 in U.

(2) dim Sym E = 5 iff (v_{x} − 2w)(x) ≠ 0 almost everywhere in U and (v_{x} − 2w)^{−1/3} is

a solution of the equation p_{3} = v · p_{1} + \frac{1}{2}v_{x} · y.

(3) dim Sym E = 4 iff (v_{x} − 2w)(x) ≠ 0 almost everywhere in U and (v_{x} − 2w)^{−1/3} is

not a solution of the equation p_{3} = v · p_{1} + \frac{1}{2}v_{x} · y.

Proof. Let v_{x} − 2w ≡ 0 in U and let { a^{1}(x), a^{2}(x), a^{3}(x) }, { b^{1}(x), b^{2}(x), b^{3}(x) }, { c^{1}(x), c^{2}(x), c^{3}(x) } be three collections of linear independent (over R^{1}) solutions of E. Then it follows from Theorem 2.1 that the collection of the following functions

A^{i} = a^{i}(x)p_{1}^{2} − 2a_{x}^{i}yp_{1} + (2a_{xx}^{i} − va^{i}(x))y^{2}, i = 1, 2, 3;
B^{j} = b^{j}(x)p_{1} − b_{x}^{j}y, j = 1, 2, 3;
C^{k} = c^{k}(x), k = 1, 2, 3;
D = y

is a basis for the space (over R^{1}) of all generating functions of classical symmetries for E. Therefore the collection of corresponding classical symmetries ξ_{A^{1}}, ξ_{A^{2}}, ξ_{A^{3}}, ξ_{B^{1}}, ξ_{B^{2}}, ξ_{B^{3}}, ξ_{C^{1}}, ξ_{C^{2}}, ξ_{C^{3}}, ξ_{D} is a basis for Sym E. It follows that dim Sym E = 10.

A proof of statements (2) and (3) are analogous. ■

2.2.

Theorem 2.2. Let E and E′ be ODEs of the form p_{3} = v(x) · p_{1} + w(x) · y and let both dim Sym E and dim Sym E′ be equal to 4 or 5; then if f is a contact transformation transforming E to E′, then f is the lifting of some point transformation.

Proof. Suppose E is the equation P_{3} = V (X) · P_{1} + W (X) · Y, E′ is the equation p_{3} = v(x)·p_{1}+w(x)·y, f is defined by (1.1), and f transforms E to E′. Suppose dim Sym E = 5, then obviously, dim Sym E′ = 5 . Let the collection Φ_{1}(X), Φ_{2}(X), Φ_{3}(X) be a fundamental system of solutions for E. It follows from Theorem 2.1 that they are generating functions for some classical symmetries of E. Therefore transformation (1.1) connects each of these functions with a generating function of some classical symmetry of E′ by formula (1.6). Taking into account the form of last generating functions (see Theorem 2.1), we obtain from formula (1.6):

Φ_{1}(X(x, y, p_{1})) = F · { K_{1}^{1}(v_{x} − 2w)^{−1/3}p_{1} + [K_{1}^{2} − K_{1}^{1}((v_{x} − 2w)^{−1/3})′]y + c_{1} },
Φ_{2}(X(x, y, p_{1})) = F · { K_{2}^{1}(v_{x} − 2w)^{−1/3}p_{1} + [K_{2}^{2} − K_{2}^{1}((v_{x} − 2w)^{−1/3})′]y + c_{2} },
Φ_{3}(X(x, y, p_{1})) = F · { K_{3}^{1}(v_{x} − 2w)^{−1/3}p_{1} + [K_{3}^{2} − K_{3}^{1}((v_{x} − 2w)^{−1/3})′]y + c_{3} },

where F = \frac{X_{x}Y_{y} − X_{y}Y_{x}}{X_{x} + p_{1}X_{y}} and c_{1}, c_{2}, c_{3} are some solutions of E′ and K_{j}^{i} ∈ R^{1}, i =

1, 2, j = 1, 2, 3. If one of the numbers K_{1}^{1}, K_{2}^{1}, K_{3}^{1} is not equal to zero, say K_{1}^{1} ≠ 0, then the following relations hold

Φ_{2} − \frac{K_{2}^{1}}{K_{1}^{1}} Φ_{1} = F · \{ [K_{2}^{2} − \frac{K_{2}^{1}}{K_{1}^{1}} K_{1}^{2}]y + c_{2}(x) − \frac{K_{2}^{1}}{K_{1}^{1}} c_{1}(x) \},
Φ_{3} − \frac{K_{3}^{1}}{K_{1}^{1}} Φ_{1} = F · \{ [K_{3}^{2} − \frac{K_{3}^{1}}{K_{1}^{1}} K_{1}^{2}]y + c_{3}(x) − \frac{K_{3}^{1}}{K_{1}^{1}} c_{1}(x) \}.

It follows that \frac{K_{1}^{1}Φ_{2} − K_{2}^{1}Φ_{1}}{K_{1}^{1}Φ_{3} − K_{3}^{1}Φ_{1}} does not depend on p_{1}. Therefore

\frac{∂}{∂p_{1}} (\frac{K_{1}^{1}Φ_{2} − K_{2}^{1}Φ_{1}}{K_{1}^{1}Φ_{3} − K_{3}^{1}Φ_{1}} ) = \frac{d}{dX} (\frac{K_{1}^{1}Φ_{2} − K_{2}^{1}Φ_{1}}{K_{1}^{1}Φ_{3} − K_{3}^{1}Φ_{1}} ) · X_{p_{1}} = 0 .

If X_{p_{1}} ≠ 0 in some neighborhood, then

K_{1}^{1}Φ_{2} − K_{2}^{1}Φ_{1} = K( K_{1}^{1}Φ_{3} − K_{3}^{1}Φ_{1} ),

where K ∈ R^{1}, in this neighborhood. But it is impossible, because the collection Φ_{1}, Φ_{2}, Φ_{3} is the fundamental system of solutions. Hence X_{p_{1}} ≡ 0. This means that f is the lifting of some point transformation.

The above-mentioned arguments are valid if all numbers K_{1}^{1}, K_{2}^{1}, K_{3}^{1} are equal to zero. Obviously, a proof for the case dim Sym E = 4 is analogous. ■

Below we prove that for any 3–order linear ODE E with dim Sym E = 10 there is a point transformation transforming E to the equation y′′′ = 0. Taking into account this fact and the last theorem, we obtain that the problem of local classification of 3-order linear ODEs with respect to contact transformations is reduced to the problem of local classification of these equations with respect to point transformations.

3. The equivalence problem for 3–order linear ODEs

3.1. Let E be the equation P_{3} = V (X) · P_{1} + W (X) · Y, let E′ be the equation p_{3} = v(x) · p_{1} + w(x) · y, and let f be an arbitrary point transformation defined by (1.2). It is not hard to prove that the transformation f transforms E to the following equation

p_{3} = u^{1} · (p_{2})^{2} + u^{2} · p_{2} + u^{3},

where the coefficients u^{1}, u^{2}, u^{3} are defined by the formulas

u^{1} = \frac{X_{x} + p_{1}X_{y}}{X_{x}Y_{y} − Y_{x}X_{y}} [ X_{y}Z_{p_{1}} − (X_{x} + p_{1}X_{y})Z_{p_{1}p_{1}} ],

u^{2} = \frac{X_{x} + p_{1}X_{y}}{X_{x}Y_{y} − Y_{x}X_{y}} [ Z_{p_{1}}(X_{xx} + 2p_{1}X_{xy} + (p_{1})^{2}X_{yy})
− 2(Z_{xp_{1}} + p_{1}Z_{yp_{1}})(X_{x} + p_{1}X_{y})
− Z_{y}(X_{x} + p_{1}X_{y}) + (Z_{x} + p_{1}Z_{y})X_{y}],
u^{3} = \frac{X_{x} + p_{1}X_{y}}{X_{x}Y_{y} − Y_{x}X_{y}} [ (X_{xx} + 2p_{1}X_{xy} + (p_{1})^{2}X_{yy})(Z_{x} + p_{1}Z_{y})
− (Z_{xx} + 2p_{1}Z_{xy} + (p_{1})^{2}Z_{yy})(X_{x} + p_{1}X_{y})
+ (V Z + W Y)(X_{x} + p_{1}X_{y})^{3} ],

where Z = \frac{Y_{x} + p_{1}Y_{y}}{X_{x} + p_{1}X_{y}}.

It is clear that f transforms E to E′ iff the collection of functions X(x, y), Y (x, y) defining f is a solution of the following system of PDEs

X_{y}Z_{p_{1}} − (X_{x} + p_{1}X_{y})Z_{p_{1}p_{1}} = 0,  (3.1)

Z_{p_{1}}(X_{xx} + 2p_{1}X_{xy} + (p_{1})^{2}X_{yy})

− 2(Z_{xp_{1}} + p_{1}Z_{yp_{1}})(X_{x} + p_{1}X_{y})
− Z_{y}(X_{x} + p_{1}X_{y}) + (Z_{x} + p_{1}Z_{y})X_{y} = 0,  (3.2)

\frac{X_{x} + p_{1}X_{y}}{X_{x}Y_{y} − Y_{x}X_{y}} [(X_{xx} + 2p_{1}X_{xy} + (p_{1})^{2}X_{yy})(Z_{x} + p_{1}Z_{y})

− (Z_{xx} + 2p_{1}Z_{xy} + (p_{1})^{2}Z_{yy})(X_{x} + p_{1}X_{y})
+ (V Z + W Y)(X_{x} + p_{1}X_{y})^{3}] = vp_{1} + wy.  (3.3)

Therefore the local equivalence problem for E and E′ with respect to point transformations consists in obtaining of an existence criterion of a point transformation ( X(x, y), Y (x, y) ) satisfying to system (3.1) – (3.3).

Let us find this criterion. To this end we consider equation (3.1). Solving this equation with respect to Z_{p_{1}} and taking into account that ( X(x, y), Y (x, y) ) define a point transformation, we get

Z_{p_{1}} = A(x, y) · (X_{x} + p_{1}X_{y}) and A(x, y) ≠ 0 everywhere.

It follows that,

Z = \frac{1}{2}AX_{y}p_{1}^{2} + AX_{x}p_{1} + B(x, y).

Taking into account that Z = \frac{Y_{x} + p_{1}Y_{y}}{X_{x} + p_{1}X_{y}}, we get the following equation

\frac{1}{2}A(X_{y})^{2}p_{1}^{3} + \frac{1}{2}AX_{x}X_{y}p_{1}^{2} + (AX_{x}^{2} + BX_{y} − Y_{y})p_{1} + BX_{x} − Y_{x} = 0.

The functions X, Y, A, B do not depend on p_{1}. Therefore the last equation is equivalent to the following system of PDEs

A(X_{y})^{2} = 0, AX_{x}X_{y} = 0, AX_{x}^{2} + BX_{y} − Y_{y} = 0, BX_{x} − Y_{x} = 0.

From this system we easily get the following results:

X = X(x), Z = A(x, y)X_{x}p_{1} + B(x, y), ∀x, y A(x, y) ≠ 0,  (3.4)
Y_{x} = BX_{x}, Y_{y} = AX_{x}^{2},  (3.5)
B_{y} = 2AX_{xx} + A_{x}X_{x}.  (3.6)

Now substituting obtained expressions for X and Z in equation (3.2), we get

3A_{y}X_{x}^{2}p_{1} + ( AX_{xx} + 2A_{x}X_{x} + B_{y})X_{x} = 0.

Obviously, this equation is equivalent to the following system of PDEs

A_{y} = 0, AX_{xx} + 2A_{x}X_{x} + b_{y} = 0.

It follows from the first equation of this system that A = A(x). Comparing the second equation of this system with equation (3.6), we get AX_{xx} + A_{x}X_{x} = 0. Whence

A = \frac{K}{X_{x}} , K ∈ R^{1} \ \{0\}.

Substituting this expression for A in the second equation of (3.5), we get Y_{y} = KX_{x}. Whence Y = KX_{x}y + C(x). Thus we have

X = X(x), Y = KX_{x}y + C(x), K ∈ R^{1} \ {0}.  (3.7)

Now substituting the obtained expressions for X and Y in equation (3.3), we get

[ −2 (\frac{X_{xx}}{X_{x}} )_{x} + (\frac{X_{xx}}{X_{x}} )^{2} + X_{x}^{2}V − v ] p_{1}

+ [ − (\frac{X_{xx}}{X_{x}} )_{xx} + \frac{X_{xx}}{X_{x}} (\frac{X_{xx}}{X_{x}} )_{x} + X_{xx}X_{x}V + X_{x}^{3}W − w ] y

+ \frac{1}{K} [ − (\frac{C_{x}}{X_{x}} )_{xx} + (\frac{C_{x}}{X_{x}} )_{x} \frac{X_{xx}}{X_{x}} + X_{x}V C_{x} + X_{x}^{2}W C ] = 0

Obviously, this equation is equivalent to the following system of PDEs

−2 (\frac{X_{xx}}{X_{x}} )_{x} + (\frac{X_{xx}}{X_{x}} )^{2} + X_{x}^{2}V − v = 0,  (3.8)
− (\frac{X_{xx}}{X_{x}} )_{xx} + \frac{X_{xx}}{X_{x}} (\frac{X_{xx}}{X_{x}} )_{x} + X_{xx}X_{x}V + X_{x}^{3}W − w = 0 ,  (3.9)
− (\frac{C_{x}}{X_{x}} )_{xx} + (\frac{C_{x}}{X_{x}} )_{x} \frac{X_{xx}}{X_{x}} + X_{x}V C_{x} + X_{x}^{2}W C = 0.  (3.10)

Now it is clear that if there exist a point transformation satisfying system (3.1) – (3.3), then it is defined by formulas (3.7), where X(x) satisfies to system (3.8) – (3.9) and C(x) is a solution of the ODE obtained by substitution the solution X(x) in (3.10). Therefore our local equivalence problem for E and E′ is reduced to obtaining of an existence criterion of a local diffeomorphism X(x) satisfying to system (3.8) – (3.9).

Let us obtain this criterion. To this end we differentiate equation (3.8) with respect to x and subtract the obtained equation from doubled equation (3.9). As a result we get the following equation

(V_{X} − 2W)X_{x}^{3} − (v_{x} − 2w) = 0.  (3.11)

It is clear that system of ODEs (3.8) – (3.9) is equivalent to system of ODEs (3.8), (3.11).

Suppose both dim Sym E and dim Sym E′ are equal to 10. Then it follows from Theorem 2.1 that V_{X} − 2W ≡ 0 and v_{x} − 2w ≡ 0. Therefore system (3.8), (3.11) is reduced to ODE (3.8). Obviously, there is local diffeomorphisms satisfying to this equation.

Thus in particular we obtain the following statements:

Theorem 3.1. Let E and E′ be 3–order linear ODEs admitting a 10–dimensional algebras of classical symmetries; then there exist a point transformation transforming E to E′.

Corollary. (The local contact classification of 3–order linear ODEs admitting a 10– dimensional classical symmetry algebra.)

Let E be a 3–order linear ODE with dim Sym E = 10; then the equation E is locally equivalent to the equation p_{3} = 0 at every point x_{3} ∈ E.

Now let dim Sym E ≠ 10 and dim Sym E′ ≠ 10. Then it follows from Theorem 2.1 that (V_{X} − 2W)(X) ≠ 0 and (v_{x} − 2w)(x) ≠ 0 almost everywhere. Therefore every solution X(x) of (3.11) is a local diffeomorphism.

It follows from (3.11) that X_{x} = (\frac{v_{x} − 2w}{V_{X} − 2W} )^{1/3}. Substituting this expression for X_{x}

in (3.8), we obtain the following relation

[\frac{2}{3} (\frac{(V_{X} − 2W)_{X}}{V_{X} − 2W} )_{X} − \frac{1}{9} (\frac{(V_{X} − 2W)_{X}}{V_{X} − 2W} )^{2} + V ] (V_{X} − 2W)^{−2/3}( X(x) ) = (3.12)

= [\frac{2}{3} (\frac{(v_{x} − 2w)_{x}}{v_{x} − 2w} )_{x} − \frac{1}{9} (\frac{(v_{x} − 2w)_{x}}{v_{x} − 2w} )^{2} + v] (v_{x} − 2w)^{−2/3}(x).

Thus we obtain the following statement

Theorem 3.2. Let E be the equation P_{3} = V (X) · P_{1} + W (X) · Y, let E′ be the equation p_{3} = v(x) · p_{1} + w(x) · y, and let both dim Sym E and dim Sym E′ be equal to 5 or 4. Then E and E′ are locally equivalent iff there is a solution of (3.11) satisfying to equation (3.12).

The following statement follows from above-mentioned arguments.

Theorem 3.3. If the equations P_{3} = V (X) · P_{1} + W (X) · Y and p_{3} = v(x) · p_{1} + w(x) · y are locally equivalent, then a transformation transforming the first equation to the second one can be chosen in the form

\begin{aligned}X & = X(x), \\ Y & = X_{x} · y. \end{aligned}  (3.13)

Thus we have reduced the problem of local classification of 3–order linear ODEs with respect to contact transformation to the problem of local classification of equations p_{3} = v(x) · p_{1} + w(x) · y with respect to transformations of the form (3.13).

4. Differential invariants of 3-order linear ODEs

4.1. Geometric structures of equations p_{3} = v(x) · p_{1} + w(x) · y. Let E be an arbitrary equation P_{3} = V (X) · P_{1} + W (X) · Y. It is easy to verify that every transformation (3.13) transforms this equation to the equation p_{3} = v(x) · p_{1} + w(x) · y of the same form. The coefficients v and w of the last equation are expressed through the coefficients V and W of the initial equation by the formulas

v = −2 (\frac{X_{xx}}{X_{x}} )_{x} + (\frac{X_{xx}}{X_{x}} )^{2} + X_{x}^{2}V,  (4.1)
w = − (\frac{X_{xx}}{X_{x}} )_{xx} + \frac{X_{xx}}{X_{x}} (\frac{X_{xx}}{X_{x}} )_{x} + X_{xx}X_{x}V + X_{x}^{3}W.  (4.2)

Note that these formulas are the same as (3.8), (3.9).

Obviously, the family of all transformation of the form (3.13) is a Lie pseudogroup. It follows that formulas (4.1), (4.2) define the action μ : G_{4}^{1} × R^{2} → R^{2}, μ : (X_{x}, X_{xx}, X_{xxx}, X_{xxxx}) × (V, W) ⟼ (v, w).

Therefore the collections V (X), W (X) and v(x), w(x) are the expressions of some geometric structure of type μ on R^{1} in the coordinate systems X and x respectively. We denote this structure by γ_{E}.

Obviously, E is identified with γ_{E}.

We see that the bundle of geometric structures of type μ in our case is the product bundle ρ_{μ} : R^{2} × R^{1} → R^{1}. We denote by x; u^{1}, u^{2}; u_{1}^{1}, u_{1}^{2}; …; u_{k}^{1}, u_{k}^{2} the standard coordinates in J^{k}ρ_{μ}.

Let Sym γ_{E} be algebra of all infinitesimal symmetries for γ_{E}.

Proposition 4.1.

(1) if dim Sym E = 10, then dim Sym γ_{E} = 3;

(2) if dim Sym E = 5, then dim Sym γ_{E} = 1;

(3) if dim Sym E = 4, then dim Sym γ_{E} = 0;

Proof. It is obvious that a transformation (3.13) is a symmetry for E iff the diffeomorphism x ↦ X(x) is a symmetry for γ_{E}.

Let ζ be a vector field on J^{0}π and let the flow of ζ be generated by transformations (3.13). Then ζ is expressed in the standard coordinates x, y on J^{0}π by the following formula

ζ = a(x)\frac{∂}{∂x} + a_{x} · y \frac{∂}{∂y} .

It is obvious that ζ is a classical symmetry for E iff the vector field a(x)\frac{∂}{∂x} is an infinitesimal symmetry for γ_{E}.

Generating function of ζ is

−a(x) · p_{1} + a_{x} · y  (4.3)

.

Let φ be the generating function of an arbitrary classical symmetry for E.

It follows from Theorem 2.1 that if dim Sym E = 10, then φ is of the form (4.3) iff φ = b(x) · p_{1} − b_{x} · y, where b is a solution of E. Taking into account that there are three linear independent solutions for E, we see that dim Sym γ_{E} = 3.

In just the same way, if dim Sym E = 5, then φ is of the form (4.3) iff φ = K · ( (v_{x} − 2w)^{−1/3} · p_{1} + ( (v_{x} − 2w)^{−1/3} )′ · y ), where K ∈ R^{1}. This means that dim Sym γ_{E} = 1.

If dim Sym E = 4, then there is no φ of the form (4.3). Therefore dim Sym γ_{E} = 0. ■

4.2. Differential invariants of equations p_{3} = v(x) · p_{1} + w(x) · y. Let geometric structure γ_{E} of type μ on R^{1} be defined by components u^{1} = V (X), u^{2} = W (X) in the coordinate system X on R^{1}. We subject γ_{E} to an arbitrary diffeomorphism x ↦ X(x). Suppose that obtained geometric structure γ_{E′} is defined by components u^{1} = v(x), u^{2} = w(x); then components of these structures are connected by formulas (4.1), (4.2).

We differentiate equation (4.1) with respect to x and subtract the obtained equation from doubled equation (4.2). As a result we get the relation (V_{X} − 2W)(X_{x})^{3} − (v_{x} − 2w) = 0. Therefore,

X_{x} · (V_{X} − 2W)^{1/3} = (v_{x} − 2w)^{1/3}.  (4.4)

This means that the differential form

ω = ( u_{1}^{1} − 2 · u^{2} )^{1/3}dx

on J^{1}ρ_{μ} is a differential invariant for geometric structures of type μ.

We say that ω is a differential invariant for ODEs of the form p_{3} = v(x) · p_{1} + w(x) · y.

We denote by ω(E) the value ω( γ_{E} ) of differential invariant ω on the geometric structure γ_{E} (see subsection 1.3). Thus if E is the equation p_{3} = v(x) · p_{1} + w(x) · y; then

ω(E) = ( v_{x} − 2 · w)^{1/3}dx.

Obviously, the following proposition is valid.

Proposition 4.2. Let E be the equation p_{3} = v(x) · p_{1} + w(x) · y; then (1) dim Sym E = 10 iff ω( E ) = 0;

(2) if ω(E) = 0, then the equation E is locally equivalent to the equation p_{3} = 0 at every point x_{3} ∈ E.

Now let ω(E) ≠ 0.

Taking into account (4.4), we have X_{x} = \frac{(v_{x} − 2w)^{1/3}}{(V_{X} − 2W)^{1/3}} . Substituting this expression

for X_{x} in (4.1), we obtain relation (3.12). This relation means that the function I defined on J^{3}ρ_{μ} by formula (4.5) is a scalar differential invariant for geometric structures of type μ.

I = [\frac{2}{3}D (\frac{D(u_{1}^{1} − 2u^{2})}{u_{1}^{1} − 2u^{2}}) − \frac{1}{9} (\frac{D(u_{1}^{1} − 2u^{2})}{u_{1}^{1} − 2u^{2}})^{2} + u^{1}] (u_{1}^{1} − 2u^{2})^{−2/3},  (4.5)

here D = ∑_{i=1}^{2} \frac{∂}{∂x} + u_{1}^{i} \frac{∂}{∂u^{i}} + u_{2}^{i} \frac{∂}{∂u_{1}^{i}} + … + u_{k}^{i} \frac{∂}{∂u_{k−1}^{i}} + … is full derivative operator with

respect to x.

We say that I is a scalar differential invariant for ODEs of the form p_{3} = v(x) · p_{1} + w(x) · y.

By definition, put I( E ) = I( γ_{E} ). Thus if E is the equation p_{3} = v(x) · p_{1} + w(x) · y; then

I(E)(x) = [\frac{2}{3} (\frac{(v_{x} − 2w)_{x}}{v_{x} − 2w} )_{x} − \frac{1}{9} (\frac{(v_{x} − 2w)_{x}}{v_{x} − 2w} )^{2} + v] (v_{x} − 2w)^{−2/3}(x).  (4.6)

Now Theorem 3.2 can be rewritten in the following way.

Proposition 4.3. Let E be the equation P_{3} = V (X) · P_{1} + W (X) · Y, let E′ be the equation p_{3} = v(x) · p_{1} + w(x) · y, and let ω(E) ≠ 0 and ω(E′) ≠ 0. Then E and E′ are locally equivalent iff there is a solution of (3.11) satisfying to the equation

I(E)(X(x)) = I(E′)(x).  (4.7)

4.3. Equations with 5–dimensional algebra of classical symmetries.

Proposition 4.4. Let E be the equation p_{3} = v(x) · p_{1} + w(x) · y; then dim Sym E = 5 iff I(E) is a constant.

Proof. Suppose dim Sym E = 5. It follows from Proposition 4.1 that γ_{E} admits a 1– dimensional algebra of infinitesimal symmetries. Hence I(E) is an integral for any ξ ∈ Sym γ_{E}. It follows that I(E) is a constant.

Conversely, suppose I(E) is a constant. By definition, put u = (v_{x} − 2w)^{−1/3}, where v, w are components of expression of γ_{E} in the coordinate system x on R^{1}. Then I(E) can

be rewritten in the form I(E) = −2u_{xx}u + (u_{x})^{2} + vu^{2}. Therefore \frac{dI(E)}{dx} = −2u(u_{xxx} −

vu_{x} − \frac{1}{2}v_{x}u). Taking into account that the set \{x | u(x) ≠ 0\} is everywhere dense, we

obtain that the condition \frac{dI(E)}{dx} = 0 is equivalent to the condition of corollary of Theorem 2.1 guaranteeing that dim Sym E = 5. ■

Proposition 4.5. Suppose E is the equation P_{3} = V (X) · P_{1} + W (X) · Y, E′ is the equation p_{3} = v(x) · p_{1} + w(x) · y, and I(E) and I(E′) are constants; then E and E′ are locally equivalent iff I(E) = I(E′).

Proof. This statement is an obvious consequence of Proposition 4.3. ■

It follows from (4.6) that for every equation E of the form p_{3} = K · p_{1} + y, K ∈ R^{1}, we have I(E) = 2^{−2/3} · K.

Now the following theorem is obvious.

Theorem 4.6. (The local contact classification of 3–order linear ODEs admitting a 5– dimensional classical symmetry algebra.)

(1) Let E be a 3-order linear ODE with dim Sym E = 5; then E is locally equivalent to the equation p_{3} = 2^{2/3} · I(E) · p_{1} + y at almost every point x_{3} ∈ E .

(2) If numbers K and K′ are not equal, then the equations p_{3} = K · p_{1} + y and p_{3} = K′ · p_{1} + y are not local equivalent.

(3) The complete list of ”simplest” locally nonequivalent regular 3–order linear ODEs with 5–dimensional algebra of classical symmetries consists of equations p_{3} = K · p_{1} + y, K ∈ R^{1}.

4.4. Equations with 4–dimensional algebra of classical symmetries. Let E be the equation p_{3} = v(x)·p_{1} +w(x)·y with dim Sym E = 4. Then I(E) is not a constant function in domain of definition of the coefficients v and w. Therefore I(E) can be considered as

new independent variable in some neighborhood of point x such that \frac{d I(E)}{dx}(x) ≠ 0. Subjecting E to the point transformation

\begin{aligned}X & = I(E)(x), \\ Y & = \frac{d I(E)}{dx} · y, \end{aligned}

we obtain the equation P_{3} = V (X) · P_{1} + W (X) · Y, where the coefficients V and W considering as a functions of I are scalar differential invariants. We say that this equation is the invariant form of E.

Let E be the equation p_{3} = v(x) · p_{1} + w(x) · y; then this equation is the invariant form of E iff the following identity holds

I(E)(x) ≡ x.  (4.8)

Proposition 4.7. Suppose E and E′ are equations of the form p_{3} = v(x) · p_{1} + w(x) · y and dim Sym E = dim Sym E′ = 4; then E and E′ are locally equivalent iff the invariant forms for E and E′ are the same.

Proof. Let P_{3} = V (X) · P_{1} + W (X) · Y be the invariant form of E and let p_{3} = v(x) · p_{1} + w(x) · y be the invariant form of E′.

Suppose E and E′ are locally equivalent. Then equality (4.7) for these equations is X = x. Hence it follows from formulas (4.1) and (4.2) that V = v and W = w.

The converse statement is obvious. ■

Let the equation p_{3} = v(x) · p_{1} + w(x) · y be the invariant form of some equation E. By definition, put u = (v_{x} − 2w)^{1/3}. Then it follows from (4.6) that the coefficients v and w are represented as the functions of u in the following way:

\begin{aligned}v & = xu^{2} − 2 (\frac{u_{x}}{u} )_{x} + (\frac{u_{x}}{u} )^{2} \\ w & = \frac{1}{2} [(xu^{2} − 2 (\frac{u_{x}}{u} )_{x} + (\frac{u_{x}}{u} )^{2})_{x} − u^{3}] . \end{aligned}

Consider an arbitrary ODE of the form

p_{3} = [xu^{2} − 2 (\frac{u_{x}}{u} )_{x} + (\frac{u_{x}}{u} )^{2}] · p_{1}

+ \frac{1}{2} [(xu^{2} − 2 (\frac{u_{x}}{u} )_{x} + (\frac{u_{x}}{u} )^{2})_{x} − u^{3}] · y, (4.9)

where u is an arbitrary nowhere vanishing smooth function of x.

Proposition 4.8. Equation (4.9), is an invariant form.

Proof. The proof is by direct verification of identity (4.8). ■

Theorem 4.9. (The local contact classification of 3–order linear ODEs admitting a 4– dimensional classical symmetry algebra.)

(1) Let E be a 3-order linear ODE with dim Sym E = 4; then E is locally equivalent to some equation (4.9) at almost every point x_{3} ∈ E.

(2) If functions u and u′ are not equal one to another in some neighborhood and they are nowhere vanishing, then the corresponding equations of the form (4.9) are not locally equivalent.

(3) The complete list of ”simplest” locally nonequivalent regular 3–order linear ODEs with 4–dimensional algebra of classical symmetries consists of equations (4.9), where functions u are not equal one to another in some neighborhood and they are nowhere vanishing.

Proof. (2) Assume the converse. Then the invariant forms of these equations are the same. Therefore u = u′.

Statements (1) and (3) are obvious . ■

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Program System Institute, m.Botik, Pereslavl-Zalesskiy, 152140, Russia E-mail address: yuma@diffiety.botik.ru