The Diffiety Institute Preprint Series

Preprint DIPS–5/2001 April 12, 2001

Available via INTERNET: http://diffiety.ac.ru/

The Diffiety Institute

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

Contact classification of linear ordinary differential equations. II.

by

V.A. YUMAGUZHIN

Contact classification of linear ordinary differential equations. II.

V.A. YUMAGUZHIN

Abstract. The classification of ODEs y^{(n)} = ∑_{i=2}^{n} a_{n−i}(x)y^{(n−i)}, n ≥ 3, in a neighborhood of a regular point up to a contact transformation is given.

1. Introduction

This paper is devoted to the local classification of linear ODEs of order n ≥ 3 in neighborhoods of regular points up to a contact transformation.

In [2], E.Cartan proved that for n ≤ 2, any n-order linear ODE can be transformed to the form y^{(n)} = 0 by a point transformation. For n ≥ 3, it is incorrect. In this case, there are infinite number of different equivalence classes of n-order linear ODEs.

First the problem of local classification of linear ordinary differential equation (ODE) up to a transformation of variables was set up by classics of XIX century E. Laguerre, G.-H. Halphen and others. They obtained first results concerning the classification of linear ODEs of 3rd and 4-th orders, see [7], [3]. Essentially, this problem was forgotten after them.

It is well known that any linear ODE can be transformed by a point transformation to the form

y^{(n)} = a_{n−2}(x) y^{(n−2)} + a_{n−3}(x) y^{(n−3)} + … + a_{0}(x) y .  (1.1)

In this paper, we classify linear ODEs of this form. The case n = 3, we considered in [13], [14].

In his book [12], E.J.Wilczynski proved that any linear ODE of order n ≥ 3 can be transformed by a point transformation to the form

y^{(n)} = a_{n−3}(x) y^{(n−3)} + a_{n−4}(x) y^{(n−4)} + … + a_{0}(x) y  (1.2)

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

Key words and phrases. Linear ordinary differential equation, contact transformation, point transformation, projective transformation, Laguerre-Forsyth transformation, scalar differential invariants, equivalence problem, classical symmetry, point symmetry.

The author is grateful to Professor V.A.Konstantinov for his support and attention to this work.

(see also, [10],[4]). In [15], [16], we obtained the local classification of linear ODEs of form (1.2).

Our approach to the problem is the following.

In their paper [8], F. M. Mahomed and P. G. L. Leach proved that dimension of the algebra of point symmetries of a n-order linear ODE is equal to either n + 4 or n + 2, or n + 1.

We prove (theorem 2.2) that dimension of algebra of point symmetries of a linear ODE is an invariant of contact transformations that take the set of linear ODEs to itself.

Further, we prove (theorem 4.1) that any linear ODE with n + 4-di-mensional algebra of point symmetries is reduced by a point transformation to the form y^{(n)} = 0. For linear ODEs with n + 2 and n + 1dimensional algebras of point symmetries, we prove (theorem 2.3) that a contact transformation that takes one of these equation to another one is a point transformation. These results reduce the problem of local classification of linear ODEs w.r.t. contact transformations to the classification w.r.t. point transformations.

Further, we reduce (theorem 2.5) the problem to the classification w.r.t. point transformations of the form

X = f(x), Y = | f′(x) |^{(n−1)/2}y  (1.3)

In his thesis [9], F. M. Mahomed proved that any linear ODE with n + 2-dimensional algebra of point symmetries can be transformed to a linear ODE with constant coefficients by a point transformation of form (1.3). We find invariants of transformations of linear ODEs with constant coefficients and n + 2-dimensional algebras of point symmetries. These invariants solve the equivalence problem for these equations (theorem 4.8). Finally, we classify linear ODEs with constant coefficients up to equivalence (theorem 4.11).

Further, we calculate (theorem 4.17) the algebra of scalar differential invariants of linear ODEs. We use these invariants to classify linear ODEs with n + 1-dimensional algebras of point symmetries. Let ℰ be an arbitrary linear ODE with n + 1-dimensional algebra of point symmetries and let I(x) be its nonconstant scalar differential invariant. The transformation of form (1.3) X = I(x), Y = | I′(x) |^{(n−1)/2}y takes ℰ to the equation ℰ′. We say that ℰ′ is a canonical form of ℰ. We prove (theorem 4.23) that equivalent equations have the same canonical form. It leads to the classification (theorem 4.24) of linear ODEs with n + 1dimensional algebras of point symmetries in a neighborhood of regular point up to equivalence.

Below, all manifolds and maps are supposed to be smooth. By ℝ^{n} denote the n-dimensional arithmetical space.

2. Preliminaries

In this section, we recall necessary notations and results of the geometry of differential equations ([5], [6]) and linear ODEs ([8],[12]). We prove some necessary results concerning symmetries and transformations of linear ODEs too.

2.1. Jet bundles.

2.1.1. Cartan distribution. Let E and M be a smooth manifolds of dimensions n + m and n respectively and let π : E → M be a smooth bundle. By [S]_{x}^{k} denote the k–jet of a section S of π at the point x ∈ M. By

π_{k} : J^{k}π → M , π_{k} : [S]_{x}^{k} ↦ x , k = 0, 1, 2, …, ∞ ,

denote the bundle of k–jets of all sections of π.

The projection π_{k,r} : J^{k}π → J^{r}π , k > r, is defined by π_{k,r}( [S]_{x}^{k}) = [S]_{x}^{r}.

Every section S of π generates the section j_{k}S of the bundle π_{k} by the formula j_{k}S : x ↦ [S]_{x}^{k}. By L_{S}^{(k)} denote the image of the section j_{k}S.

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

Let x_{k} ∈ J^{k}π. Consider all submanifolds L_{S}^{(k)} passing trough x_{k} and consider their tangent spaces T_{x_{k}}(L_{S}^{(k)}) at x_{k}. The subspace C_{x_{k}} ⊂ T_{x_{k}}(J^{k}π) spanned on the union of these tangent spaces is called the Cartan plane at x_{k}. The distribution C : x_{k} ↦ C_{x_{k}} is called the Cartan distribution on J^{k}π.

2.1.2. Lie transformations. A (local) diffeomorphism of J^{k}π that takes the Cartan distribution to itself is called a Lie transformation. A Lie transformation of J^{0}π (that is an arbitrary diffeomorphism of J^{0}π) is called a point transformation. A Lie transformation of J^{1}π is called a contact transformation if m = 1.

Every Lie transformation f : U → U′ of J^{k}π can be lifted canonically to the Lie transformation f^{(r)} : π_{k+r,k}^{−1}(U) → π_{k+r,k}^{−1}(U′) of J^{k+r}π, r = 1, 2, …, such that, for r ≥ l, the diagram

π_{k+r,k}^{−1}(U) ⟶^{f^{(r)}} π_{k+r,k}^{−1}(U′)
π_{k+r,k+l}↓  ↓^{π_{k+r,k+l}}
π_{k+l,k}^{−1}(U) ⟶_{f^{(l)}} π_{k+l,k}^{−1}(U′)

is commutative. Indeed, f^{(r)} is defined in the following way. A point x_{k+1} = [S]_{x}^{k+1} ∈ J^{k+1}π is identified with K_{x_{k+1}} = T_{x_{k}}(L_{S}^{(k)}), where x_{k} = π_{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}π 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}π defined almost everywhere in π_{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 π_{k+r,k}^{−1}(U).

It is well known (see [5], [6]) that any Lie transformation is the lifting of some point transformation if m > 1 and if m = 1, then any Lie transformation is the lifting of some contact transformation.

2.1.3. Lie fields. A vector field ξ in J^{k}π is called a Lie field if its flow is generated by Lie transformations. A vector field in J^{0}π is said to be a point vector field. A Lie field in J^{1}π is called a contact vector field if m = 1.

The canonical lifting of Lie transformations generates the lifting of Lie fields. Indeed, let ξ be a Lie field in J^{k}π and let f_{t} be its flow. Then the flow f_{t}^{(r)} of the Lie transformations defines the Lie field ξ^{(r)} in J^{k+r}π , r = 1, 2, … such that

(π_{k+r,k+l})_{∗}ξ^{(r)} = ξ^{(l)} , r ≥ l .

2.2. Ordinary differential equations.

2.2.1. Contact classification. Let π : ℝ^{1} × ℝ^{1} → ℝ^{1} and let x, y, p_{1}, …, p_{k} be the standard coordinates on J^{k}π.

Any k-order ODE

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

is identified with the submanifold

ℰ = { F (x, y, p_{1}, …, p_{k}) = 0 } ⊂ J^{k}π .

An ”usual” solution S(x) of the initial ODE is identified with the submanifold L_{S}^{(k)} ⊂ ℰ generated by the section S : x ↦ S(x) of π. Obviously, L_{S}^{(k)} is a 1-dimensional integral manifold of the Cartan distribution on J^{k}π. A ”many-valued” solution of ℰ is a 1-dimensional integral manifold L of the Cartan distribution on J^{k}π such that L ⊂ ℰ. It is easy to prove that locally, almost everywhere, a ”many-valued” solution has the form L_{S}^{(k)}.

It is natural to classify k-order ODEs up to a diffeomorphism of J^{k}π that takes the set of all solutions of ODEs to itself. Obviously, this diffeomorphism is a Lie transformation. Hence, it is generated by a contact transformation. So, we come to the problem of classification of ODEs up to a contact transformation.

Let ℰ_{1} , ℰ_{2} ⊂ J^{k}π be k-order ODEs and f be a point (contact) transformation. We say that f takes (locally) ℰ_{1} to ℰ_{2} if f^{(k)} (f^{(k−1)}) takes (locally) the submanifold ℰ_{1} ⊂ J^{k}π to the submanifold ℰ_{2} ⊂ J^{k}π. We say that ODEs ℰ_{1} and ℰ_{2} are equivalent if there exists a point (contact) transformation that takes (locally) ℰ_{1} to ℰ_{2}.

2.2.2. Point and contact transformations. Any point transformation f is defined in the standard coordinates by the formulae

X = X(x, y) , Y = Y (x, y) .  (2.1)

Obviously, the lifting f^{(k)} is defined in the standard coordinates by the formulae

X = X(x, y) , Y = Y (x, y) , P_{1} = \frac{DY}{DX} , …, P_{k} = \frac{DP_{k−1}}{DX} , (2.2)

where

D = \frac{∂}{∂x} + p_{1} \frac{∂}{∂y} + p_{2} \frac{∂}{∂p_{1}} + … + p_{k+1} \frac{∂}{∂p_{k}} + …  (2.3)

is the operator of total derivation w.r.t. x.

It is easy to show that a contact transformation is defined in the standard coordinates by the formulae

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}} ,  (2.4)

where functions X(x, y, p_{1}), 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 .

2.2.3. Point and contact vector fields. Let ξ be a contact vector field. It is easy to prove that ξ can be represented in the standard coordinates as

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}} ,  (2.5)

where the function φ = φ(x, y, p_{1}) is called the generating function of ξ.

Let ζ be an arbitrary point vector field. It has the form

ζ = a(x, y)\frac{∂}{∂x} + b(x, y)\frac{∂}{∂y}  (2.6)

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

b(x, y) − a(x, y) · p_{1} .  (2.7)

Conversely, if the generating function of a contact vector field has form (2.7), then this vector field is the lifting of some point vector field. We shall say that function (2.7) is the generating function of point vector field (2.6).

Let us transform a contact vector field by an arbitrary contact transformation (2.4). As a result, we obtain a new 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

\frac{X_{x} + p_{1}X_{y}}{X_{x}Y_{y} − X_{y}Y_{x}} Φ( X, Y, P_{1} ) = φ(x, y, p_{1}) .  (2.8)

2.2.4. Classical symmetries. A point vector field ζ in J^{0}π is called a point symmetry of a differential equation ℰ ⊂ J^{n}π if ζ^{(n)} is tangent to the submanifold ℰ. By Pnt ℰ we denote the set of all point symmetries of ℰ.

A contact vector field ξ in J^{1}π is called a contact symmetry of a differential equation ℰ ⊂ J^{n}π if ξ^{(n−1)} is tangent to the submanifold ℰ and there is no a point vector field ζ with ξ = ζ^{(1)}.

The point and contact symmetries are called classical symmetries. By Sym ℰ we denote the set of all classical symmetries of ℰ.

The space of generating functions of classical symmetries of an ODE

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

coincide with the space of solutions of the form φ = φ(x, y, p_{1}) of the linear PDE (see [5], [6])

( D̄^{n} − \frac{∂F}{∂p_{n−1}} D̄^{n−1} − \frac{∂F}{∂p_{n−2}} D̄^{n−2} − … − \frac{∂F}{∂p_{1}} D̄^{1} − \frac{∂F}{∂y} )(φ) = 0 , (2.9)

where D̄ = ∂/∂x + p_{1}∂/∂y + p_{2}∂/∂p_{1} + … + p_{n−1}∂/∂p_{n−2} + F ∂/∂p_{n−1}.

2.3. Linear ordinary differential equations.

2.3.1. Classical symmetries. Let ℰ be an arbitrary n-order linear ODE.

In [13],[14], we proved for n = 3 the following results: dim Sym ℰ can be equal to either 10 or 5, or 4; if dim Sym ℰ = 10, then Sym ℰ is generated by three contact and seven point symmetries; if dim Sym ℰ = 5 or 4, then Sym ℰ = Pnt ℰ.

Proposition 2.1. If n > 3, then Sym ℰ = Pnt ℰ.

Proof. We can assume without loss of generality that ℰ has form (1.1).

Let φ(x, y, p_{1}) be the generating function of an arbitrary classical symmetry of ℰ. The generating function of a point symmetry has the form α(x, y)p_{1} + β(x, y). Hence we should be check that φ_{y^{(1)}y^{(1)}} ≡ 0 to prove the theorem. To this end, let us consider equation (2.9) for ℰ:

( D̄^{n} − a_{n−2}D̄^{n−2} − a_{n−3}D̄^{n−3} − … − a_{1}D̄^{1} − a_{0} )φ = 0 . (2.10)

Obviously,

D̄^{n}(φ) = D̄^{n−2}( φ_{p_{1}}p_{3} + φ_{p_{1}p_{1}}(p_{2})^{2} + low degree terms ) =

= \{ \begin{matrix}D̄( 3φ_{p_{1}p_{1}}p_{2}p_{3} + l. d. t. ), if n = 4 \\ D̄^{n−3}( φ_{p_{1}}p_{4} + 3φ_{p_{1}p_{1}}p_{2}p_{3} + l. d. t. ), if n > 4 \end{matrix}

= \{ \begin{matrix}3φ_{p_{1}p_{1}}(p_{3})^{2} + l. d. t., if n = 4 \\ (\binom{n−3}{n−5} + 3\binom{n−3}{n−4} + 3)φ_{p_{1}p_{1}}p_{3}p_{n−1} + l. d. t., if n > 4 \end{matrix}

It now follows from (2.10) that φ_{p_{1}p_{1}} ≡ 0. □

Obviously, dimension of an algebra of classical symmetries is an invariant of contact transformations of ODEs. From above mentioned results of [13], [14] and this proposition, we get

Theorem 2.2. Dimension of the algebra of point symmetries of a linear ODE is an invariant of contact transformations that take the set of all linear ODEs to itself.

2.3.2. Point symmetries. Let ℰ be an arbitrary ODE (1.1). In their paper [8], F. M. Mahomed and P. G. L. Leach proved that a point symmetry of ℰ has the form

( φ(x)\frac{∂}{∂x} + \frac{n − 1}{2} φ′ y \frac{∂}{∂y} ) + C y \frac{∂}{∂y} + γ(x)\frac{∂}{∂y} ,

where γ(x) is a solution of ℰ , C ∈ ℝ, and φ(x) is a solution of the system of ODEs

\{ \begin{matrix}R_{n}^{3}φ^{(3)} − 2a_{n−2}φ^{(1)} − a_{n−2}^{(1)}φ = 0 \\ R_{n}^{k+1}φ^{(k+1)} − ∑_{s=0}^{k−3} R_{n−2−s}^{k−1−s}a_{n−2−s}φ^{(k−1−s)} \\ − ka_{n−k}φ^{(1)} − a_{n−k}^{(1)}φ = 0 , k = 3, 4 …, n , \end{matrix} (2.11)

where R_{p}^{q} = \binom{p}{q−1}(n − 1)/2 − \binom{p}{q}. Dimension of the space of solutions of system (2.11) can be equal to either 3 or 1, or 0. It follows that dim Pnt ℰ can be equal to either n + 4 or n + 2, or n + 1.

Thus the set of all linear ODEs ℰ is divided into the three nonintersecting families according to dim Pnt ℰ. From theorem 2.2, we have that these families are invariant w.r.t. contact transformations.

2.3.3. Contact transformations. Linear ODEs of order n ≥ 3 with n+4dimensional algebras of point symmetries have contact symmetries if n = 3 ( [13], [14] ). It follows that there exist contact transformations of 3-order linear ODEs with 7-dimensional algebras of point symmetries. For n = 3, the following theorem is proved in [14].

Theorem 2.3. Let ℰ_{1}, ℰ_{2} be linear ODEs with n+2 or n+1 – dimensional algebras of point symmetries and let f be a contact transformation that takes ℰ_{1} to ℰ_{2}. Then f is the lifting of a point transformation.

Proof. We can assume without loss of generality that ℰ_{1} and ℰ_{2} have form (1.1).

Suppose dim Pnt ℰ_{1} = n + 2. The contact transformation f is defined in the standard coordinates by (2.4). Let Γ_{1}(X), Γ_{2}(X), Γ_{3}(X) be linear independent solutions of ℰ_{1}. We can consider these solutions as generating functions of point symmetry of ℰ_{1} (see subsection 2.3.2). Every that function is connected with the correspondence generating function of point symmetries of ℰ_{2} by formula (2.8). Taking into account the form of generating functions of the n+2 – dimensional algebra Pnt ℰ_{2} (see subsection 2.3.2), we obtain:

ΔΓ_{1}(X(x, y, p_{1})) = K_{1}( φ(x)p_{1} − \frac{n − 1}{2}φ′y ) + C_{1}y + γ_{1}(x)

ΔΓ_{2}(X(x, y, p_{1})) = K_{2}( φ(x)p_{1} − \frac{n − 1}{2}φ′y ) + C_{2}y + γ_{2}(x)

ΔΓ_{3}(X(x, y, p_{1})) = K_{3}( φ(x)p_{1} − \frac{n − 1}{2}φ′y ) + C_{3}y + γ_{3}(x)

where Δ = ( X_{x} + p_{1}X_{y})/( X_{x}Y_{y} − X_{y}Y_{x}); K_{j}, C_{j} ∈ ℝ, j = 1, 2, 3. If one of the numbers K_{1}, K_{2}, K_{3} is not equal to zero, say K_{1} ≠ 0, then

Δ( Γ_{2} − \frac{K_{2}}{K_{1}} Γ_{1} ) = ( C_{2} − \frac{K_{2}}{K_{1}} C_{1} )y + γ_{2} − \frac{K_{2}}{K_{1}} γ_{1}

Δ( Γ_{3} − \frac{K_{3}}{K_{1}} Γ_{1} ) = ( C_{3} − \frac{K_{3}}{K_{1}} C_{1} )y + γ_{3} − \frac{K_{2}}{K_{1}} γ_{1}

It follows that ( K_{1}Γ_{2} − K_{2}Γ_{1} )/( K_{1}Γ_{3} − K_{3}Γ_{1} ) is independent on p_{1}. Therefore

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

Suppose that X_{p_{1}} ≠ 0, then

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

where K ∈ ℝ. This means that the solutions Γ_{1}, Γ_{2}, Γ_{3} of ℰ_{1} are linear dependent. From this contradiction, we have X_{p_{1}} ≡ 0. Hence f is the lifting of some point transformation.

Obviously, the proofs for the case K_{1} = K_{2} = K_{3} = 0 and for the case dim Pnt ℰ = n + 1 are analogous. □

2.3.4. Reduction to transformations (1.3). It is well known that any linear ODE can be reduced to the form (1.1) by a point transformation.

The following proposition holds ([12]).

Proposition 2.4. Let ℰ be an ODE of form (1.1). Then a point transformation takes ℰ to an ODE of the same form iff this transformation has the form

X = X(x), Y = C · | X′(x) |^{(n−1)/2} · y + β(x), C ∈ ℝ . (2.12)

It is follows from (2.2) that the lifting of point transformation (2.12) to the Lie transformation of J^{n}π is defined by

X = X(x) , Y = C |X′|^{(n−1)/2}y + β(x) ,
P_{k} = C ∇^{k}( |X′|^{(n−1)/2}y) + ∇^{k}(β) , k = 1, 2, …, n ,

where ∇ = D/DX and D is operator (2.3).

Let ℰ_{1} = { P_{n} = A_{n−2}(X) P_{n−2} + A_{n−3}(X) P_{n−3} + … + A_{0}(X) Y } and ℰ_{2} = { p_{n} = a_{n−2}(x) p_{n−2} + a_{n−3}(x) p_{n−3} + … + a_{0}(x) y }. Suppose transformation (2.12) takes ℰ_{1} to ℰ_{2}. This means

C ∇^{n}( |X′|^{(n−1)/2}y) + ∇^{n}( β(x) ) =

= ∑_{i=2}^{n}[C A_{n−i}(X(x))∇^{n−i}( |X′|^{(n−1)/2}y) + A_{n−i}(X(x))∇^{n−i}( β(x) )] .

It follows:

(1) the function β(x) is a solution of the homogeneous linear ODE

∇^{n}(β(x)) = ∑_{i=2}^{n} A_{n−i}(X(x))∇^{n−i}(β(x)).

(2) Transformation (2.12) takes ℰ_{1} to ℰ_{2} for an arbitrary nonzero

constant C ∈ ℝ. As a result, we obtain the following statement.

Theorem 2.5. Let ℰ_{1} and ℰ_{2} be ODEs of form (1.1). Then if there exist a point transformation that takes ℰ_{1} to ℰ_{2}, then there exist a point transformation of the form (1.3) that takes ℰ_{1} to ℰ_{2}.

From proposition (2.4), we have that transformations of form (1.3) take the set of all ODEs of form (1.1) to itself.

2.3.5. Laguerre-Forsyth transformations. The following proposition holds ([12], [10], [4]).

Proposition 2.6. Let ℰ = { P_{n} = A_{n−2}(X) P_{n−2} + A_{n−3}(X) P_{n−3} + … + A_{0}(X) Y } be an ODE of form (1.1). A point transformation

X = f(x), Y = | f′ |^{(n−1)/2}y ,

where f is a solution of the ODE

2f′′′f′ − 3(f′′)^{2} − 24( (n + 1)n(n − 1) )^{−1} (f′)^{4}A_{n−2}(f) = 0 ,

takes ℰ to an equation ℰ′ of form (1.2).

This transformation is called a Laguerre-Forsyth transformation of the equation ℰ. The equation ℰ′ is called a Laguerre-Forsyth form of the equation ℰ.

It follows from this proposition that the problem of local classification of linear ODEs is reduced to classification of ODEs of form (1.2).

2.3.6. Reduction to transformations (2.13). The following proposition holds (see [12]).

Proposition 2.7. Let ℰ be an ODE of form (1.2). Then a point transformation takes ℰ to an ODE of the same form iff this transformation has the form

X = \frac{α · x + β}{γ · x + δ} , Y = C · | X′ |^{(n−1)/2} · y , α, β, γ, δ, C ∈ ℝ .

Taking into account proposition 2.7, we can prove the following statement by the same way as theorem 2.5.

Theorem 2.8. Let ℰ_{1} and ℰ_{2} be ODEs of form (1.2). Then if there exist a point transformation that takes ℰ_{1} to ℰ_{2}, then there exist a point transformation of the form

f(x) = \frac{α x + β}{γ x + δ} , f̂(x, y) = | f′ |^{(n−1)/2} · y , α, β, γ, δ ∈ ℝ , (2.13)

that takes ℰ_{1} to ℰ_{2}.

From proposition (2.7), we have that transformations of form (2.13) take the set of all ODEs of form (1.2) to itself.

3. Bundles of linear ODEs

3.1. Bundle of linear ODEs of form (1.1).

3.1.1. Here, we reduce the problem of local classification of linear ODEs of form (1.1) to the classification of germs of section of the bundle of these ODEs.

Let π : E = ℝ^{1} × ℝ^{n−1} → ℝ^{1} be a product bundle. By x denote the standard coordinate on the base ℝ^{1}, by a_{n−2}, a_{n−4}, …, a_{0} denote the standard coordinates on the fiber ℝ^{n−1}.

We identify any section

S : x ↦ ( x, a_{n−2}(x), a_{n−3}(x), …, a_{0}(x) )

of π with the linear ODE

ℰ_{S} = { p_{n} = a_{n−2}(x) p_{n−2} + a_{n−3}(x) p_{n−3} + … + a_{0}(x) y } .

Clearly, this identification is a bijection from the set of all sections of π to the set of all linear ODEs of form (1.1). By S_{ℰ} we denote the section of π corresponding to the equation ℰ under this identification.

By Γ we denote the Lie pseudogroup of all local diffeomorphisms of ℝ^{1}. By Φ we denote the Lie pseudogroup of all point transformations of form (1.3). Formula (1.3) defines the isomorphism

Γ → Φ , f ↦ ( f, f̂) ,  (3.1)

where f̂(x, y) = | f′(x) |^{(n−1)/2}y.

Let

ℰ_{2} = { P_{n} = A_{n−2}(X) P_{n−2} + A_{n−3}(X) P_{n−3} + … + A_{0}(X) Y }

be an arbitrary ODE of form (1.1). Subjecting ℰ_{2} to an arbitrary transformation ( f , f̂) of form (2.13), we obtain linear ODE

ℰ_{1} = { p_{n} = a_{n−2}(x) p_{n−2} + a_{n−3}(x) p_{n−3} + … + a_{0}(x) y } .

The coefficients of ℰ_{1} are expressed in terms of the coefficients of ℰ_{2} and the transformation f by equations of the following form

a_{n−j} = F_{n−j}( A_{n−2}, …, A_{n−j}; \frac{d f}{dx} , …, \frac{d^{j+1}f}{dx^{j+1}} ) , j = 2, 3, …, n .

Obviously, the coefficients of ℰ_{2} are expressed in terms of the coefficients of ℰ_{1} and the transformation f^{−1} by the same equations

A_{n−j} = F_{n−j}( a_{n−2}, …, a_{n−j}; \frac{d f^{−1}}{dX} , …, \frac{d^{j+1}f^{−1}}{dX^{j+1}} )  (3.2)
j = 2, 3, …, n .

Equations (3.2) define the lifting of any f ∈ Γ to the diffeomorphism f^{(0)} of the bundle π such that the diagram

\begin{matrix}E & ⟶^{f^{(0)}} & E \\ ^{π}↓ & & ↓^{π} \\ ℝ & ⟶_{f} & ℝ \end{matrix}

is commutative (in the domain of f^{(0)}).

For any f ∈ Γ, we define the transformation of sections of π by the formula

S ↦ f( S) = f^{(0)} ∘ S ∘ f^{−1} .  (3.3)

Now equations (3.2) can be represented as

S_{ℰ_{2}} = f( S_{ℰ_{1}} ) .

Obviously, the following statement holds.

Proposition 3.1. Let ℰ_{1} , ℰ_{2} be equations of form (1.1). Then a transformation ( f , f̂) ∈ Φ takes ℰ_{1} to ℰ_{2} iff S_{ℰ_{2}} = f( S_{ℰ_{1}} ).

Let S be a section of π and let p be a point from the domain of S. By {S}_{p} we denote the germ of S at the point p. We say that germs {S_{1}}_{p_{1}} and {S_{2}}_{p_{2}} are equivalent if there exist f ∈ Γ with

f( {S_{1}}_{p_{1}} ) =^{def} {f(S_{1})}_{f(p_{1})} = {S_{2}}_{p_{2}} .

3.1.2. Symmetries of sections. Let S be a section of π and let ξ be a vector field in the base ℝ^{1} of π. By f_{t} we denote the flow of ξ. We say that ξ is a symmetry of S if one of the following equivalent conditions is fulfilled:

(1) the vector field ξ^{(0)} is tangent to the image L_{S}^{(0)} of S;

(2) ∀t f_{t}( S) = S;

(3) df_{t}( S)/dt = 0 . By Sym S we denote the Lie algebra of all symmetries of the section S.

Proposition 3.2. Let S : x → ( x, a_{n−2}(x), …, a_{0}(x) ) be a section of π and let ξ = φ(x)∂/∂x. Then:

(1) ξ is a symmetry of S iff φ(x) is a solution of system (2.11) ; (2) φ(x)∂/∂x is a symmetry of S iff φ(x)∂/∂x+((n−1)/2 )φ′ y ∂/∂y

is a point symmetry of the equation ℰ_{S};

(3) dim Sym S is equal to either 3 or 1, or 0 ;

(4) dim Pnt ℰ_{S} = dim Sym S_{ℰ} + n + 1 .

Proof. All statements of the proposition follows immediately from proposition 3.1 and the results of work [8] of F. M. Mahomed and P. G. L. Leach mentioned in section 2.3.2. □

The following statement is needed for sequel

Proposition 3.3. Let ξ be a symmetry of a section S of π and let f ∈ Γ. Then f_{∗}(ξ) is a symmetry of the section f(S).

Proof. Let f_{t} be the flow of ξ. Then f ∘ f_{t} ∘ f^{−1} is the flow of f_{∗}ξ and (f ∘ f_{t} ∘ f^{−1})( f(S) ) = (f ∘ f_{t})(S) = f(S). □

3.2. Laguerre-Forsyth bundles. Let E^{n−3} be the subspace of the total space E of π defined by

E^{n−3} = { ( x, a_{n−2}, a_{n−3}, …, a_{0} ) ∈ E | a_{n−2} = 0 } .

Then τ = π|_{E^{n−3}} : E^{n−3} → ℝ is a subbundle of the bundle π. Obviously, the set of sections of τ is identified with the set of linear ODEs of form (1.2).

By G denote the Lie group of all projective transformations of ℝ^{1}, that is

G = \{ f(x) = \frac{α x + β}{γ x + δ} | α, β, γ, δ ∈ ℝ and det (\begin{matrix}α & β \\ δ & γ \end{matrix}) ≠ 0 \} .

It is easy to check that the set of nonconstant solutions of the equation

2 f′′′ f′ − 3 (f′′)^{2} = 0  (3.4)

coincides with G.

From proposition (2.7), theorem (2.7), and proposition (3.1), we have that, for any f ∈ G, the diffeomorphism f^{(0)} takes the bundle τ to itself.

By 𝔤 we denote the Lie algebra of G. It can easily be checked that 𝔤 as a vector space over ℝ is generated by the vector fields

ξ_{0} = \frac{∂}{∂x} , ξ_{1} = x\frac{∂}{∂x} , ξ_{2} = x^{2} \frac{∂}{∂x} .  (3.5)

It now is obvious that any symmetry of a section of τ is an element of 𝔤.

Let S be a section of π and let f ∈ Γ. We say that a transformation f is a Lagguere-Forsyth transformation for S if the transformation (f, f̂) ∈ Φ is a Lagguere-Forsyth transformation for the ODE ℰ_{S}.

4. Classification of linear ODEs

4.1. Classification of linear ODEs with n + 4 – dimensional algebra of point symmetries.

Theorem 4.1. Let S be a section of π with dim Sym S = 3 and let 0 be the zero section of π. Then for any p from the domain of S, the germs {S}_{p} and {0}_{0} are equivalent.

Proof. Let S be a section of π with dim Sym S = 3 and let p be a point from the domain of S.

Let us choose a Laguerre-Forsyth transformation f ∈ Γ for S such that f(p) = 0 ( proposition 2.6 ). Let S′ = f(S) : x → ( x, a_{n−3}(x), …, a_{0}(x) ). From dim Sym S = 3, we have dim Sym S′ = 3. From proposition 3.2, we have that Sym S′ as a vector space over ℝ is generated by linear independent vector fields φ_{i}(x)∂/∂x , i = 1, 2, 3, where functions φ_{i} are solutions of the system (2.11). In the considered case, this system has the form

\{ \begin{matrix}φ′′′ = 0 \\ 3 a_{n−3}φ′ + a′_{n−3}φ = 0 \\ \frac{(k − 1)(n − (k − 1))}{2} a_{n−k+1}φ′′ + ka_{n−k}φ′ + a′_{n−k}φ = 0 , \\ k = 4, 5, …, n . \end{matrix} (4.1)

From the second equation of this system, we have a_{n−3} ≡ 0; from the third equation, we have a_{n−4} ≡ 0 and so on; from the last equation, we obtain a_{0} ≡ 0. □

Corollary 4.2.

(1) Let S_{1} and S_{2} be sections of π with 3-dimensional algebras of symmetries. Then any their germs {S_{1}}_{p_{1}} and {S_{2}}_{p_{2}} are equivalent.

(2) The zero section of the bundle τ is a unique section of τ that has 3-dimensional algebra of symmetries.

(3) The zero section of τ is invariant w.r.t. the group G.

4.2. Reductions of the classification problem. From theorem 2.3, proposition 3.1, and theorem 4.1, we have that the problem of local classification of linear ODEs w.r.t. contact transformations is reduced to the problem of classification of section germs of the bundle π w.r.t. the pseudogroup Γ.

Taking into account that Γ is a transitive pseudogroup, we obtain that the last problem is reduced to the classification of germs at 0 ∈ ℝ^{1} of sections of π up to a diffeomorphism from the isotropy group Γ_{0} = { f ∈ Γ | f(0) = 0 } ⊂ Γ of the point 0 ∈ ℝ^{1}.

From proposition 2.6 and theorem 2.8, we now have that the problem of classification of section germs of the bundle π w.r.t. the pseudogroup Γ can be reduced to the problem of classification of section germs of the bundle τ w.r.t. the group G. A solution of the last problem is given in [15], [16].

4.3. Classification of linear ODEs with n + 2 – dimensional algebra of point symmetries.

4.3.1. Linear ODEs with constant coefficients. Let S be a section of the bundle π with dim Sym S = 1. The algebra Sym S as a vector space over ℝ is spanned on the vector field φ(x)∂/∂x, where φ(x) is a unique solution of (2.11).

Let p be a point from the domain of S. We say that p is a regular point of S ( ℰ_{S}) if φ(p) ≠ 0. We say that a germ of S at a regular point is a regular germ.

Theorem 4.3. Let S be a section of π with dim Sym S = 1 and let {S}_{p} be a regular germ. Then there exist a constant section S′ of π such that {S}_{p} and {S′}_{0} are equivalent.

Proof. In some neighborhood of p, there exist a diffeomorphism f of ℝ^{1} straightening the vector field φ(x)∂/∂x (that is the diffeomorphism f : x ↦ X takes φ(x)∂/∂x to ∂/∂X) and f(p) = 0. It follows that the section f(S) has the symmetry ∂/∂X. Hence, f(S) is a constant section in a neighborhood of 0 ∈ ℝ^{1}. By S′ we denote the constant section of π coinciding with f(S) in some neighborhood of 0 ∈ ℝ^{1}. □

Corollary 4.4. Let ℰ be a linear ODE of form (1.1) with dim Pnt ℰ = n + 2 and p is a regular point for ℰ. Then, in some neighborhood of p, ℰ is equivalent to an ODE of form (1.1) with constant coefficients.

Note, that apparently first this statement was proved by F.M.Mahomed in [9].

From theorem 4.3, we have that the problem of classification of sections of π with 1-dimensional algebras of point symmetries in a neighborhood of a regular points is reduced to the classification of constant sections of π up to equivalence.

Note that any constant section S of π has the symmetry of the form

\frac{∂}{∂x}. It follows that dim Sym S is equal to either 1 or 3.

The following proposition makes possible to separate all constant sections with 3 – dimensional algebras of symmetries.

Proposition 4.5. Let S : x → ( x, a_{n−2}, …, a_{0} ) be a constant section of π. Then dim Sym S = 3 iff the components a_{n−k} satisfy to the following conditions:

(1) if k is odd, then a_{n−k} = 0, (2) if k is even, then the coefficients a_{n−k} are expressed in terms of

a_{n−2} by the recurrence formula

a_{n−2m} = \frac{1}{2m}( R_{n}^{2m−1}λ^{2m} − ∑_{s=0}^{2m−4} R_{n−2−2s}^{2m−1−2s}a_{n−2−2s}λ^{2m−2−2s} ) ,

where R_{p}^{q} = \binom{p}{q−1}(n − 1)/2 − \binom{p}{q} , λ^{2} = \frac{2}{R_{n}^{3}} a_{n−2} , m = 2, 4, …

In particular, if a_{n−2} = 0, then all coefficients are equal to 0.

Proof. Let S : x → ( x, a_{n−2}, …, a_{0} ) be a constant section of π. Then dim Sym S = 3 iff system (2.11) for S has three linear independent solutions. This system has the following form in this case:

\{ \begin{matrix}R_{n}^{3}φ^{(3)} − 2a_{n−2}φ^{(1)} = 0 \\ R_{n}^{k+1}φ^{(k+1)} − ∑_{s=0}^{k−3} R_{n−2−s}^{k−1−s}a_{n−2−s}φ^{(k−1−s)} − ka_{n−k}φ^{(1)} = 0 , \\ k = 3, 4, …, n . \end{matrix}

From the first equation, we obtain that the system has three linear independent solutions: 1, e^{λx}, e^{−λx}, where λ^{2} = (2/R_{n}^{3})a_{n−2}. It now is clear that the system has three linear independent solutions iff

a_{n−k} = \frac{1}{k}( R_{n}^{k+1}λ^{k} − ∑_{s=0}^{k−3} R_{n−2−s}^{k−1−s}a_{n−2−s}λ^{k−2−s} ) , k = 3, 4, …, n

λ^{2} = \frac{2}{R_{n}^{3}} a_{n−2}

and these identities are fulfilled for ±λ. This completes the proof. □

4.3.2. Classification of constant sections with 1-dimensional algebras of symmetries.

Proposition 4.6. Let S be a constant section of π with 1-dimensional algebra of symmetries and let f be a diffeomorphism of ℝ^{1}. Then f(S) is a constant section iff f(x) = λx + ν , λ, ν ∈ ℝ , λ ≠ 0.

Proof. The section S has the symmetry ∂/∂x. It follows that f_{∗} takes this symmetry to the symmetry f′( f^{−1}(X) )∂/∂X of the section f(S). It now is obvious that f(S) is a constant iff f′ = λ ∈ ℝ , λ ≠ 0. The last is equivalent to f(x) = λx + ν , ν ∈ ℝ. □

It now is clear that the problem of classification of sections of π with 1-dimensional algebras of symmetries in a neighborhood of regular point is reduced to the classification of constant sections with 1-dimensional algebras of symmetries w.r.t. the group of all linear transformations x ↦ λx , λ ≠ 0, of ℝ^{1}.

Proposition 4.7. Let S_{1} : x → ( x, a_{n−2}, a_{n−3}, …, a_{0} ) and S_{2} : X → ( X, A_{n−2}, A_{n−3}, …, A_{0} ) be constant sections of π with 1-dimensional algebras of symmetries. Then S_{1} and S_{2} are equivalent iff there exist λ ∈ ℝ , λ ≠ 0, such that

a_{n−2} = λ^{2}A_{n−2}, a_{n−3} = λ^{3}A_{n−3}, …, a_{0} = λ^{n}A_{0}.  (4.2)

Proof. Suppose S_{1} and S_{2} are equivalent. This means that there exist a linear transformation f(x) = λx that takes S_{1} to S_{2}. The point transformation ( f, f̂) of form (1.3) generated by f is defined by X = λx , Y = |λ|^{(n−1)/2}y. From (2.2), we have that the lifting ( f, f̂)^{(n)} is defined by

X = λx , Y = |λ|^{(n−1)/2}y , P_{1} = |λ|^{(n−1)/2}λ^{−1}p_{1} , …, P_{n} = |λ|^{(n−1)/2}λ^{−n}p_{n} .

It follows that p_{n} = λ^{2}A_{n−2}p_{n−2} + λ^{3}A_{n−3}p_{n−3} + … + λ^{n}A_{0}y is the equation ℰ_{S_{1}} obtained from ℰ_{S_{2}} by point transformation ( f, f̂). The statement of the proposition now is obvious. □

Let ℐ ⊂ { 2, 3, …, n } , n ≥ 3, be a nonempty subset. If ℐ contains odd numbers, then by m we denote the minimal odd number of ℐ. If ℐ consists of even numbers, then by m we denote the minimal number of ℐ.

By S^{ℐ} we denote a constant section with 1-dimensional algebra of symmetries S : x → ( x, a_{n−2}, a_{n−3}, …, a_{0} ) that satisfies a_{n−i} ≠ 0 if i ∈ ℐ and a_{n−i} = 0 otherwise. Put by definition

I_{k}(S^{ℐ}) = \{ \begin{matrix}a_{n−k}/a_{n−m}^{k/m} , k ∈ ℐ , whenever ℐ contains odd numbers \\ a_{n−k}/|a_{n−m}|^{k/m} , k ∈ ℐ , whenever ℐ consists of even numbers . \end{matrix}

Theorem 4.8.

(1) If ℐ_{1} ≠ ℐ_{2}, then any sections S^{ℐ_{1}} , S^{ℐ_{2}} are not equivalent.

(2) Sections S_{1}^{ℐ} , S_{2}^{ℐ} are equivalent iff

∀k ∈ ℐ I_{k}(S_{1}^{ℐ}) = I_{k}(S_{2}^{ℐ}).  (4.3)

Proof. 1) The first statement follows from relations (4.2).

2) Suppose the sections S_{1}^{ℐ} , S_{2}^{ℐ} are equivalent. Then, it follows from relations (4.2) that λ = a_{n−m}^{1/m}/A_{n−m}^{1/m} whenever ℐ contains odd numbers and λ = |a_{n−m}|^{1/m}/|A_{n−m}|^{1/m} whenever ℐ consists of even numbers. Substituting this expression for λ ( respectively for λ^{2} ) in (4.2), we obtain equalities (4.3).

Conversely, suppose equalities (4.3) hold. Suppose ℐ contains odd numbers. Then we define λ by formula λ = a_{n−m}^{1/m}/A_{n−m}^{1/m}. It follows that (a_{n−m})^{1/m} = λ(A_{n−m})^{1/m}. Substituting this expression for (a_{n−m})^{1/m} in (4.3), we obtain relations (4.2). If ℐ consists of even numbers, then setting λ = |a_{n−m}|^{1/m}/|A_{n−m}|^{1/m}, we obtain (4.2) by analogue arguments. □

Corollary 4.9. The collection of numbers I_{k}(S^{ℐ}), k ∈ ℐ, is a complete collection of invariants to solve the equivalence problem for sections of the form S^{ℐ}.

Below, by ℐ_{1} we denote nonempty subsets of { 2, 3, …, n } , n ≥ 3, consisting odd numbers; by ℐ_{2} we denote nonempty subsets of { 2, 3, …, n }, n ≥ 3, consisting of even numbers.

By S_{+1}^{ℐ_{1}} we denote a section of the form S^{ℐ_{1}} with a_{n−m} = 1, by S_{+1}^{ℐ_{2}} we denote a section of the form S^{ℐ_{2}} with a_{n−m} = 1, and at last by S_{−1}^{ℐ_{2}} we denote a section of the form S^{ℐ_{2}} with a_{n−m} = −1.

By ℱ we denote the family of all sections of the forms: S_{+1}^{ℐ_{1}}, S_{+1}^{ℐ_{2}}, and S_{−1}^{ℐ_{2}} for all ℐ_{1} , ℐ_{2}.

From stated above, we have the following two theorems.

Theorem 4.10. (Classification constant sections of π with 1-dimensional algebras of symmetries up to equivalence.)

(1) Any two sections from ℱ are not equivalent.

(2) Any constant section of π with 1-dimensional algebra of symmetries is equivalent to some section from ℱ.

Theorem 4.11. (Classification of linear ODEs order n ≥ 3 with n+2dimensional algebras of point symmetries in a neighborhood of a regular point up to equivalence.)

(1) Let S_{1}, S_{2} ∈ ℱ and S_{1} ≠ S_{2}. Then any their germs {S_{1}}_{p_{1}} , {S_{2}}_{p_{2}} are not equivalent.

(2) Let S be a section of π with 1-dimensional algebra of symmetries and let p be a regular point of its domain. Then {S}_{p} is equivalent to a germ of some section from ℱ.

Thus ℱ is a complete family of nonequivalent sections with 1-dimensional algebras of symmetries.

As examples, consider ℱ for n = 3, 4.

Let n = 3. Then ℐ_{1} = {1, 0}, {0} and ℐ_{2} = {1}. From (4.5), we have dim Sym S^{ℐ_{2}} = 3. Thus the family ℱ consists of the following sections:

p_{3} = a_{1}p_{1} + y , a_{1} ∈ ℝ \{0} ,
p_{3} = y .

Let n = 4. Then ℐ_{1} = {2, 1, 0}, {2, 1}, {1, 0}, {1} and ℐ_{2} = {2, 0}, {0}. From (4.5), we have that the sections S : x ↦ ( x, ± p_{2}, 0, (1/20) y ) of the form S^{ℐ_{2}} have 3-dimensional algebras of symmetries. All others sections of this form have 1-dimensional algebras of point symmetries.

Thus the family ℱ consists of the following sections:

p_{4} = a_{2}p_{2} + p_{1} + a_{0}y , a_{2}, a_{0} ∈ ℝ \{0} ,
p_{4} = b_{2}p_{2} + p_{1} , b_{2} ∈ ℝ \{0} ,
p_{4} = ±p_{2} + c_{0}y , c_{0} ∈ ℝ \{0} , c_{0} ≠ 1/20 ,
p_{4} = p_{1} + d_{0}y , d_{0} ∈ ℝ \{0} ,
p_{4} = p_{1} ,
p_{4} = ±y .

4.4. Differential invariants of linear ODEs of form (1.2). In this subsection, we calculate the algebra of scalar differential invariants of linear ODEs of form (1.2) and we solve the euivalence problem for these equations.

4.4.1. Jet bundles. The standard coordinates x, a_{n−i}, i = 3, 4, …, n, on the bundle τ define in the obvious way the standard coordinates x, a_{n−j}^{(r)}, j = 3, 4, …, n , r = 0, 1, 2, …, k, on the jet bundle τ_{k} : J^{k}τ → ℝ , k = 0, 1, 2, …, ∞.

Any diffeomorphism f^{(0)} , f ∈ G, can be lifted to the Lie transformations f^{(k)} of the jet bundles J^{k}τ , k = 1, 2, …, ∞, by the formula

f^{(k)}( [S]_{p}^{k}) = [ f^{(0)} ∘ S ∘ f^{−1} ]_{f(p)}^{k} .  (4.4)

Obviously, for any l > m, the diagram

J^{l}τ ⟶^{f^{(l)}} J^{l}τ
τ_{l,m}↓  ↓^{τ_{l,m}}
J^{m}τ ⟶_{f^{(m)}} J^{m}τ

is commutative (in the domains of f^{(l)}).

Let

G^{(k)} = { f^{(k)} | f ∈ G } , k = 0, 1, 2, …, ∞ .

Let

G_{+} = { f ∈ G | f′ > 0 } , G_{−} = { f ∈ G | f′ < 0 } .

Obviously, G_{+} is the connected component of the unit of G. By μ we denote the element of G_{−} defined by

μ(x) = −x ∀x ∈ ℝ^{1} .

Obviously, we have

G = G_{+} ∪ G_{−} , G_{−} = μ ∘ G_{+} ,

The lifting μ^{(k)} is defined by

μ^{(k)}( ( x, a_{n−j}^{(r)} ) ) = ( −x, (−1)^{j+r}a_{n−j}^{(r)} ) ,  (4.5)
j = 3, 4, …, n , r = 0, 1, …, k.

Let

G_{+}^{(k)} = { f^{(k)} | f ∈ G_{+} } , G_{−}^{(k)} = { f^{(k)} | f ∈ G_{−} } .

Obviously, G_{+}^{(k)} is the connected component of the unit of G^{(k)} and

G^{(k)} = G_{+}^{(k)} ∪ G_{−}^{(k)} , G_{−}^{(k)} = μ^{(k)} ∘ G_{+}^{(k)} .

The lifting of projective transformations of the base ℝ^{1} to diffeomorphisms of J^{k}τ generates the lifting of any vector field ξ ∈ 𝔤 to the vector field ξ^{(k)} on J^{k}τ. By definition, ξ^{(k)} is the vector field defined by the flow f_{t}^{(k)}, where f_{t} is the flow of ξ. Obviously

( τ_{l,m})_{∗}( ξ^{(l)} ) = ξ^{(m)} , l > m .

Let ξ = φ(x)∂/∂x be an arbitrary element of 𝔤. The vector field ξ^{(∞)} is defined by the formula (see [5])^{1}

ξ^{(∞)} = φD_{x} + �_{ψ} ,  (4.6)

where D_{x} = ∂/∂x + ∑_{k=0}^{∞} ∑_{j=3}^{n} a_{n−j}^{(k+1)}∂/∂a_{n−j}^{(k)} is the operator of total derivation w.r.t. x; �_{ψ} = ∑_{k=0}^{∞} ∑_{j=3}^{n} D_{x}^{k}( ψ_{n−j})∂/∂a_{n−j}^{(k)} is the operator of evolution differentiation with generating function ψ = ( ψ_{n−3}, …, ψ_{0} )^{t}. This function is defined in the following way. Let x_{1} = [S]_{x}^{1} ∈ J^{1}τ , x = τ_{1}(x_{1}); then

ψ(x_{1}) = (\begin{matrix}ψ_{n−3}(x_{1}) \\ ⋯ \\ ψ_{0}(x_{1}) \end{matrix}) = \frac{d}{dt}( f_{t}^{(0)} ∘ S ∘ f_{t}^{−1} )(x)|_{t=0}  (4.7)

Let S(x) = ( x, a_{n−3}(x), …, a_{0}(x) ). Then, taking into account that df_{t}/dt|_{t=0} = φ and φ′′′ = 0, we can calculate that

ψ = (\begin{matrix}−3 a_{n−3}φ′ − a_{n−3}^{(1)}φ \\ −\frac{3(n−3)}{2} a_{n−3}φ′′ − 4 a_{n−4}φ′ − a_{n−4}^{(1)}φ \\ … \\ −\frac{(k−1)(n−(k−1))}{2} a_{n−k+1}φ′′ − k a_{n−k}φ′ − a_{n−k}^{(1)}φ \\ … \\ −\frac{n−1}{2} · 1 · a_{1}φ′′ − n a_{0}φ′ − a_{0}^{(1)}φ \end{matrix})  (4.8)

1 Here we use the Cyrillic letter �, which is pronounced like “e” in “ten”.

It now follows from (4.6) and (4.8) that for any k = 0, 1, 2, …, ∞,

ξ_{0}^{(k)} = \frac{∂}{∂x} .  (4.9)
ξ_{1}^{(k)} = x\frac{∂}{∂x} − ∑_{r=0}^{k} ∑_{j=3}^{n}(j + r)a_{n−j}^{(r)} \frac{∂}{∂a_{n−j}^{(r)}} .  (4.10)

ξ_{2}^{(k)} = x^{2} \frac{∂}{∂x} − ∑_{r=0}^{k} ∑_{j=3}^{n} [ 2x(j + r)a_{n−j}^{(r)} \frac{∂}{∂a_{n−j}^{(r)}}

+ (j − 1)(n − (j − 1))a_{n−(j−1)}^{(r)} \frac{∂}{∂a_{n−j}^{(r)}}

+ (2j + r − 1)ra_{n−j}^{(r−1)} \frac{∂}{∂a_{n−j}^{(r)}} ] ,  (4.11)

where a_{n−2}^{(r)} = 0.

4.4.2. Invariant subbundles. Let E^{i} , i = n − 3, n − 4, …, 0, −1 , be the subspaces of the total space E of τ defined by

E^{i} = { ( x, a_{n−3}, a_{n−4}, …, a_{0} ) ∈ E | a_{j} = 0 if j > i } .

Consider the subbundle τ|_{E^{i}} : E^{i} → ℝ of the bundle τ.

Proposition 4.12. Every subbundle E^{i} is invariant w.r.t. G^{(0)}.

Proof. From (4.9) – (4.11), we have that the restrictions of the vector fields ξ_{0}^{(0)}, ξ_{1}^{(0)}, ξ_{2}^{(0)} on E^{i} are defined by

ξ_{0}^{(0)}|_{E^{i}} = \frac{∂}{∂x} ,  (4.12)
ξ_{1}^{(0)}|_{E^{i}} = x\frac{∂}{∂x} − ( (n − i)a_{i} \frac{∂}{∂a_{i}} + ⋯ + na_{0} \frac{∂}{∂a_{0}} ) ,  (4.13)
ξ_{2}^{(0)}|_{E^{i}} = x^{2} \frac{∂}{∂x} − 2x( (n − i)a_{i} \frac{∂}{∂a_{i}} + ⋯ + na_{0} \frac{∂}{∂a_{0}} )
− ( i(n − i)a_{i} \frac{∂}{∂a_{i−1}} + ⋯ + (n − 1)a_{1} \frac{∂}{∂a_{0}} ) .  (4.14)

It now is clear that ξ_{0}^{(0)}|_{E^{i}}, ξ_{1}^{(0)}|_{E^{i}}, ξ_{2}^{(0)}|_{E^{i}} are vector fields tangent to E^{i}. Therefore every subbundle E^{i} is invariant w.r.t. G_{+}^{(0)}. From (4.5), we have μ^{(0)}( E^{i}) = E^{i}. This concludes the proof. □

Thus, we have the following sequence of the subbundles invariant w.r.t. G^{(0)}.

E = E^{n−3} ⊃ E^{n−4} ⊃ … ⊃ E^{0} ⊃ E^{−1} .

Let E_{i} , i = n − 3, n − 4, …, 0, −1 , be the subsets of the total space E of τ defined by

E_{i} = E^{i} \ E^{i−1} if i ≥ 0 and E_{−1} = E^{−1} .

Consider the subbundle

τ^{i} = τ|_{E_{i}} : E_{i} → ℝ

of the bundle τ.

Corollary 4.13. Every subbundle E_{i} is invariant w.r.t. G^{(0)}.

Thus, E is the union

E = E_{n−3} ∪ E_{n−4} ∪ … ∪ E_{0} ∪ E_{−1}  (4.15)

of nonintersecting subbundles invariant w.r.t. G^{(0)}.

The following proposition is needed for the sequel.

Proposition 4.14. The symmetric differential (n−i)-form ω_{i} = a_{i}dx^{n−i}|_{E_{i}} on E_{i} is invariant w.r.t. G^{(0)}.

Proof. Let us calculate the Lie derivatives of ω_{i} w.r.t. vector fields ξ_{0}^{(0)}|_{E_{i}}, ξ_{1}^{(0)}|_{E_{i}}, ξ_{2}^{(0)}|_{E_{i}}. From (4.12) – (4.14), we have

ξ_{0}^{(0)}|_{E_{i}}( ω_{i}) = 0 ,
ξ_{1}^{(0)}|_{E_{i}}( ω_{i}) = a_{i} (n − i) dx^{n−i} − (n − i) a_{i}dx^{n−i} = 0 ,
ξ_{2}^{(0)}|_{E_{i}}( ω_{i}) = a_{i} (n − i) 2x dx^{n−i} − 2x (n − i) a_{i}dx^{n−i} = 0 .

Hence ω_{i} is invariant w.r.t. G_{+}^{(0)}.

It follows from (4.5) that (μ^{(0)})^{∗}(ω_{i}) = ω_{i}. Thus ω_{i} is invariant w.r.t. G^{(0)}. □

Corollary 4.15. (The transformation law of the first nonzero component.)

Let θ_{0} = ( x, 0, …, 0, a_{i}, …, a_{0} ) ∈ E_{i}, let f ∈ G, and let f^{(0)}( θ_{0} ) = ( f(x), 0, …, 0, A_{i}, …, A_{0} ) ∈ E_{i}. Then

a_{i} = ( f′(x) )^{n−i}A_{i} .

4.4.3. Scalar differential invariants of linear ODEs. In this subsection, we calculate scalar differential invariants of linear ODEs. For a general theory of scalar differential invariants refer to [1], [11].

It was proved in subsection 4.4.2 that the bundle τ^{i} = τ|_{E_{i}} : E_{i} → ℝ is invariant w.r.t. G^{(0)}. It follows that the jet bundle J^{k}τ^{i} are invariant w.r.t. G^{(k)} , k = 1, 2 …, ∞. Hence J^{k}τ^{i} are invariant w.r.t. the subgroup G_{+}^{(k)} ⊂ G^{(k)} , k = 0, 1, 2 …, ∞.

A function I ∈ C^{∞}(J^{k}τ^{i}) is called a scalar differential invariant of G ( G_{+} ) if

(f^{(k)})^{∗}I = I ∀f ∈ G ( G_{+} ) .

Let I be a scalar differential invariant of G ( G_{+} ) and let S be a section of τ^{i}. By definition, put

I(S) = ( j_{k}S)^{∗}I  (4.16)

For any f ∈ G, we have

I( f(S) ) ∘ f = I(S) .  (4.17)

Indeed,

I( f(S) ) = ( j_{k}f(S) )^{∗}I = ( j_{k}(f^{(0)} ∘ S ∘ f^{−1}) )^{∗}I = ( f^{(k)} ∘ j_{k}S ∘ f^{−1} )^{∗}I =

( f^{−1} )^{∗} ∘ ( j_{k}S)^{∗} ∘ ( f^{(k)} )^{∗}I = ( f^{−1} )^{∗} ∘ ( j_{k}S)^{∗}I = ( f^{−1} )^{∗}I(S) =

I(S) ∘ f^{−1} .

Let S be a section of τ admitting a 1-dimensional algebra of projective symmetries. Then I(S) is a constant for any scalar differential invariant I. Indeed, let ξ be a projective symmetry of S and let f_{t} be its flow. Then

I(S) = I( f_{t}(S) ) = I( f_{t}(S) ) ∘ f_{t} = I(S) ∘ f_{t} .

It is not hard to prove that I ∈ C^{∞}(J^{k}τ^{i}) is a scalar differential invariant of G_{+} iff I is a solution of the system of linear PDEs

\{ \begin{matrix}ξ̄_{0}^{(k)}(I) & = 0 \\ ξ̄_{1}^{(k)}(I) & = 0 \\ ξ̄_{2}^{(k)}(I) & = 0 , \end{matrix}  (4.18)

where ξ̄_{0}^{(k)}, ξ̄_{1}^{(k)}, ξ̄_{2}^{(k)} are the restrictions of ξ_{0}^{(k)}, ξ_{1}^{(k)}, ξ_{2}^{(k)} to J^{k}τ^{i}. From (4.9) – (4.11), we have

ξ̄_{0}^{(k)} = ξ_{0}^{(k)}|_{J^{k}τ^{i}} = \frac{∂}{∂x} ,  (4.19)
ξ̄_{1}^{(k)} = ξ_{1}^{(k)}|_{J^{k}τ^{i}} = x\frac{∂}{∂x} − ∑_{r=0}^{k} ∑_{j=i}^{0}(n − j + r)a_{j}^{(r)} \frac{∂}{∂a_{j}^{(r)}} ,  (4.20)

ξ̄_{2}^{(k)} = ξ_{2}^{(k)}|_{J^{k}π^{i}} = x^{2} \frac{∂}{∂x} − ∑_{r=0}^{k} ∑_{j=0}^{i} [ 2x(n − j + r)a_{j}^{(r)} \frac{∂}{∂a_{j}^{(r)}}

+ (n − j − 1)(j + 1)a_{j+1}^{(r)} \frac{∂}{∂a_{j}^{(r)}}

+ ( 2(n − j) + r − 1 )ra_{j}^{(r−1)} \frac{∂}{∂a_{j}^{(r)}} ] .  (4.21)

By 𝒜_{i}^{k} we denote the algebra of scalar differential invariants of G_{+} on J^{k}τ^{i}. We identify A_{i}^{k} with its image (τ_{l,k}^{i})^{∗}(𝒜_{i}^{k}) , l > k. As a result we have the following filtration

A_{i} = A_{i}^{∞} ⊃ … ⊃ A_{i}^{k} ⊃ … ⊃ A_{i}^{1} ⊃ A_{i}^{0}.

By 𝒟_{i}^{k} we denote the distribution on J^{k}τ^{i} generated by vector fields ξ̄_{0}^{(k)} , ξ̄_{1}^{(k)} , ξ̄_{2}^{(k)}. From (4.19) – (4.21), we have that dim 𝒟_{i}^{k} = 2 if i = 0 and k = 0, otherwise dim 𝒟_{i}^{k} = 3.

By N_{i}^{k} we denote the number of functionally independent scalar differential invariant in A_{i}^{k}. Clearly,

N_{i}^{k} = dim J^{k}π^{i} − dim 𝒟_{i}^{k}

It is easy to prove that

N_{0}^{0} = 0 , N_{0}^{1} = 0 , N_{0}^{k} = k − 1 if k ≥ 2 ,  (4.22)
N_{1}^{0} = 0 , N_{1}^{k} = 2k if k ≥ 1 ,  (4.23)
N_{i}^{0} = i − 1 , N_{i}^{k} = (k + 1)(i − 1) + 2k if k ≥ 1 .  (4.24)

Consider the vector field on J^{∞}τ^{i}

ζ_{i} = |a_{i}|^{−1/(n−i)}D̄_{x} ,  (4.25)

where D̄_{x} = D_{x}|_{J^{∞}τ^{i}} = ∂/∂x + ∑_{r=0}^{∞} ∑_{j=i}^{0} a_{j}^{(r+1)}∂/∂a_{j}^{(r)} is the operator of

total derivation w.r.t. x restricted on J^{∞}τ^{i}.

Proposition 4.16. The vector field ζ_{i} is invariant w.r.t. G_{+}^{(∞)}.

Proof. By ξ̄_{r}^{(∞)} , r = 0, 1, 2 , we denote the restriction of ξ_{r}^{(∞)} to J^{∞}τ^{i}. Let us check that [ ζ_{i}, ξ̄_{r}^{(∞)} ] = 0 for all r.

Taking into account (4.9), we have

[ ζ_{i}, ξ̄_{0}^{(∞)} ] = [ |a_{i}|^{−1/(n−i)}D̄_{x}, \frac{∂}{∂x} ] = 0 .

Taking into account (4.8), we consider the vector fields ξ̄_{1}^{(∞)} and ξ̄_{2}^{(∞)} in the form (4.6):

ξ̄_{1}^{(∞)} = xD̄_{x} + ¯�_{( (n−i)a_{i}+xa_{i}^{(1)} )} ,
ξ̄_{2}^{(∞)} = x^{2}D̄_{x} + ¯�_{( (i+1)(n−i+1)a_{i+1}+2(n−i)xa_{i}+x^{2}a_{i}^{(1)} )} ,

where ¯�_{ψ} is the restriction of �_{ψ} on J^{∞}τ^{i}. Now taking into account that [ D̄_{x} , ¯�_{ψ} ] = 0 for any ψ, we easily obtain that [ ζ_{i}, ξ̄_{1}^{(∞)} ] = 0 and [ ζ_{i}, ξ̄_{2}^{(∞)} ] = 0 . □

Obviously, for any I ∈ 𝒜_{i}, its Lie derivative ζ_{i}(I) ∈ 𝒜_{i}. Thus, ζ_{i} and I generate the sequence I , ζ_{i}(I) , …, ζ_{i}^{k}(I) , … of scalar differential invariants from 𝒜_{i}.

Theorem 4.17. The algebra 𝒜_{i} is generated by the following free generators

ζ_{i}^{k}(I_{i−m}), m = 0, 1, …, i , k = 0, 1, 2, …,

where

I_{i} = [ 2a_{i} a_{i}^{(2)} − \frac{2(n − i) + 1}{n − i} (a_{i}^{(1)})^{2} ] · (a_{i})^{−2(n−i+1)/(n−i)} ; (4.26)
I_{i−1} = [ a_{i−1} − \frac{i}{2} a_{i}^{(1)} ] · |a_{i}|^{−(n−i+1)/(n−i)} ;  (4.27)
for 2 ≤ m ≤ i ,

I_{i−m} = [ a_{i−m} + \frac{(−1)^{m}}{m!} ∏_{r=n−i+1}^{n−i+m−1} \frac{(n − r)r}{(n − i)i} (a_{i})^{1−m} (a_{i−1})^{m} +

+ ∑_{l=n−i+1}^{n−i+m−1} \frac{(−1)^{n−i+m−l}}{(n − i + m − l)!} ∏_{r=l}^{n−i+m−1} \frac{(n − r)r}{(n − i)i} (a_{i})^{i−n+l−m} ·

(a_{i−1})^{n−i+m−l} a_{n−l} ] · |a_{i}|^{−(n−i+m)/(n−i)} .  (4.28)

Proof. It is not hard to check that I_{i}, …, I_{0} are solutions of system (4.18).

Let i = 0. We have that I_{0} ∈ 𝒜_{0}^{2}. For any k = 0, 1, 2, …, the invariants I_{0}, ζ_{0}(I_{0}), ζ_{0}^{2}(I_{0}), …, ζ_{0}^{k}(I_{0}) belong to 𝒜_{0}^{k+2} and they are functionally independent. The number of them is equal to (k + 2) − 1. Now from (4.22), we obtain that (k + 2) − 1 = N_{0}^{k+2}. This concludes the proof for i = 0.

Suppose i ≥ 1. We have

ζ_{i}(I_{i−1}) = [ −\frac{i}{2} |a_{i}| a_{i}^{(2)} + …] · (a_{i})^{−2(n−i+1)/(n−i)}

The manifold J^{∞}τ^{i} has two connected components defined by the inequalities a_{i} > 0 and a_{i} < 0 respectively. Comparing ζ_{i}(I_{i−1}) with I_{i}, we can define the scalar differential invariant J ∈ 𝒜_{i}^{1} by the formula

J = \{ \begin{matrix}I_{i} + \frac{4}{i} ζ_{i}(I_{i−1}) if a_{i} > 0 \\ I_{i} − \frac{4}{i} ζ_{i}(I_{i−1}) if a_{i} < 0 . \end{matrix}

It is easy to calculate that

J = [ \frac{4}{i} a_{i} a_{i−1}^{(1)} −\frac{4(n − i + 1)}{i(n − i)} a_{i}^{(1)} a_{i−1}+ \frac{1}{n − i} (a_{i}^{(1)})^{2}]·(a_{i})^{−2(n−i+1)/(n−i)} .

Let i = 1. Then I_{i−1}, J ∈ 𝒜_{i}^{1} and they are functionally independent. The invariants

I_{i−1}, J, ζ_{i}(I_{i−1}), ζ_{i}(J), … ζ_{i}^{k}(I_{i−1}), ζ_{i}^{k}(J)

belong to 𝒜_{i}^{k+1} , k = 0, 1, 2, …, they are functionally independent, and the number of them is equal to 2(k + 1). Now from (4.23), we obtain 2(k + 1) = N_{1}^{k+1}. This concludes the proof for i = 1.

Let i > 1. Then the invariants I_{i−2}, …, I_{0} are functionally independent and they belong to 𝒜_{i}^{0}. The invariants I_{i−2}, …, I_{0}, I_{i−1}, J are functionally independent and they belong to 𝒜_{i}^{1}. At last, the invariants

I_{i−2}, …, I_{0}, I_{i−1}, J, …, ζ_{i}^{k}(I_{i−2}), …, ζ_{i}^{k}(I_{0}), ζ_{i}^{k}(I_{i−1}), ζ_{i}^{k}(J)

are functionally independent, they belong to 𝒜_{i}^{k} , k = 1, 2, …, and the number of them is equal to (k + 1)(i − 2) + 2k. Now from (4.24), we obtain that (k + 1)(i − 1) + 2k = N_{i}^{k}. This concludes the proof for i > 1. □

From (4.5), we obtain

Remark 4.18. The invariant

I_{i} = [ 2a_{i} a_{i}^{(2)} − \frac{2(n − i) + 1}{n − i} (a_{i}^{(1)})^{2} ] · (a_{i})^{−2(n−i+1)/(n−i)}

is an invariant of the group G.

4.4.4. The equivalence problem of linear ODEs. Let

S_{1}(x) = ( 0, …, 0, a_{i}(x), …, a_{0}(x) )

and

S_{2}(X) = ( 0, …, 0, A_{i}(X), …, A_{0}(X) )

be sections of τ^{i} in neighborhoods of points p ∈ ℝ and P ∈ ℝ respectively.

Sections S_{1} and S_{2} are locally equivalent at ( p, P ) w.r.t. G_{+} ( G) if there exist f ∈ G_{+} ( G) and neighborhoods V of p and U of P that

f(p) = P and f(S_{1}|_{V}) =^{def} f^{(0)} ∘ S_{1}|_{V} ∘ f^{−1} = S_{2}|_{U}.

Theorem 4.19. Sections S_{1} and S_{2} of τ^{i} are locally equivalent at ( p, P ) w.r.t. G_{+} iff the following conditions hold:

(1) a_{i}(p) · A_{i}(P) > 0 ,

(2) the solution f of the Cauchy problem

f′ = |a_{i}(x)|^{1/(n−i)} · |A_{i}(f(x))|^{−1/(n−i)} , f(p) = P  (4.29)

satisfies to the equations

I_{m}(S_{2}) ∘ f = I_{m}(S_{1}) , m = i, i − 1, …, 0  (4.30)

in some neighborhood of p.

Proof. Suppose S_{1} and S_{2} are locally equivalent at ( p, P ) w.r.t. G_{+}. Then there exist f ∈ G_{+} and neighborhoods V of p and U of P that f(p) = P and f(S_{1}|_{V}) = S_{2}|_{U}. Consider the symmetric differential (n − i)-form ω_{i} on E_{i} (see proposition 4.14). We have

f^{∗}( S_{2}^{∗}(ω_{i}) ) = S_{1}^{∗}(ω_{i}) .  (4.31)

Indeed,

f^{∗}( f(S_{1})^{∗}(ω_{i}) ) = f^{∗}( (f^{(0)}∘S_{1}∘f^{−1})^{∗}(ω_{i}) ) = S_{1}^{∗}( (f^{(0)})^{∗}(ω_{i}) ) = S_{1}^{∗}(ω_{i}) .

Equality (4.31) means that

a_{i}(x) = (f′)^{n−i}A_{i}(f(x)) .

It follows both that a_{i}(p)·A_{i}(P) > 0 and that f is a solution of Cauchy problem (4.29). Further, from (4.17), we have that equations (4.30) hold.

Conversely, let a_{i}(p) · A_{i}(P) > 0, let f be a solution of Cauchy problem (4.29), and let f be a solution of equations (4.30).

Let us show that f ∈ G_{+}. From (4.29), we can obtain f′′ and f′′′ in terms of a_{i} , A_{i} and their 1-st and 2-nd derivatives:

f′′ = \frac{1}{r}[ |a_{i}|^{\frac{1−r}{r}} |A_{i}|^{\frac{−1}{r}} sgn(a_{i})a′_{i} − |a_{i}|^{\frac{2}{r}} |A_{i}|^{\frac{−r−2}{r}} sgn(A_{i})A′_{i} ] , (4.32)
f′′′ = \frac{1}{r}[ \frac{1 − r}{r}|a_{i}|^{\frac{1−2r}{r}} |A_{i}|^{\frac{−1}{r}} (a′_{i})^{2} + |a_{i}|^{\frac{1−r}{r}} |A_{i}|^{\frac{−1}{r}} sgn(a_{i})a′′_{i}

− \frac{3}{r}|a_{i}|^{\frac{2−r}{r}} |A_{i}|^{\frac{−r−2}{r}} a′_{i}A′_{i} + \frac{2 + r}{r}|a_{i}|^{\frac{3}{r}} |A_{i}|^{\frac{−2r−3}{r}} (A′_{i})^{2}

−|a_{i}|^{\frac{3}{r}} |A_{i}|^{\frac{−r−3}{r}} sgn(A_{i})A′′_{i} ] ,  (4.33)

where r = n − i. Substituting expressions (4.29),(4.32), and (4.33) for f′ , f′′, and f′′′ in the left side of equation (3.4), we obtain

2 f′′′ f′ − 3 (f′′)^{2} = \frac{1}{n − i} |a_{i}|^{4/(n−i)}|A_{i}|^{−2/(n−i)}( I_{i}(S_{1})−I_{i}(S_{2})∘f ) = 0 .

Thus, f ∈ G_{+}.

Let S_{3} = f(S_{1}). Then I_{m}(S_{3}) ∘ f = I_{m}( f(S_{1}) ) ∘ f = I_{m}(S_{1}) = I_{m}(S_{2}) ∘ f, m = i, i − 1, …, 0. Hence

I_{m}(S_{3}) = I_{m}(S_{2}) , m = i, i − 1, …, 0 .

Let S_{3} = ( 0, …, 0, B_{i}, …, B_{0} ). Then obviously, B_{i} = (f′)^{−(n−i)}a_{i} = A_{i} in some neighborhood of P. It now follows from I_{i−1}(S_{3}) = I_{i−1}(S_{2}) that B_{i−1} = A_{i−1} in this neighborhood. From I_{i−2}(S_{3}) = I_{i−2}(S_{2}), we have B_{i−2} = A_{i−2} in this neighborhood and so on. Thus S_{3} = S_{2} in some neighborhood of P. □

Corollary 4.20. Sections S_{1} , S_{2} of τ^{i} are locally equivalent at the ( p, P ) w.r.t. G iff S_{1} local equivalent w.r.t. G_{+} either to S_{2} at ( p, P ) or to μ(S_{2}) at ( p, −P ).

Corollary 4.21. Let the invariants I_{m}(S_{1}), I_{m}(S_{2}) , m = i, i−1, …, 0 be constants. Then S_{1} , S_{2} are locally equivalent at ( p, P ) w.r.t. G_{+} iff the following conditions hold:

(1) a_{i}(p) · A_{i}(P) > 0 ,

(2) I_{m}(S_{1}) = I_{m}(S_{2}) , m = i, i − 1, …, 0 .

Proposition 4.22. Let S be a section of τ^{i}; then dim Prj S = 1 iff I_{i}(S), I_{i−1}(S), …, I_{0}(S) are constants.

Proof. The necessary is proved in the beginning of this subsection. Prove the sufficiency. Let S(x) = ( x, 0, …, 0, a_{i}(x), …, a_{0}(x) ) and invariants I_{i}(S), I_{i−1}(S), …, I_{0}(S) are constants.

From proposition 3.2, we have that a vector field φ(x)∂/∂x ∈ 𝔤 is a symmetry of section S iff φ(x) is a solution of system (2.11). In our case, this system has the following form:

\{ \begin{matrix}φ′′′ = 0 \\ (n − i)a_{i}φ′ + a′_{i}φ = 0 \\ \frac{(n − j − 1)(j + 1)}{2} a_{j+1}φ′′ + (n − j)a_{j}φ′ + a′_{j}φ = 0 , \\ j = i − 1, i − 2, …, 0 . \end{matrix}  (4.34)

It follows from the second equation of this system that φ = C|a_{i}|^{−1/(n−i)}, C ∈ ℝ.

From the identities d I_{m}(S)/dx ≡ 0 , m = i, i − 1, …, 0, we can obtain by direct calculations that |a_{i}|^{−1/(n−i)} is a solution of system (4.34). Thus, the vector field |a_{i}|^{−1/(n−i)}∂/∂x is a symmetry of the section S. □

4.5. Classification of linear ODEs with n + 1 – dimensional algebra of point symmetries. In this subsection, we use the scalar differential invariants obtained in the previous subsection to classify linear ODEs (1.1) with n+1 – dimensional algebra of point symmetries in a neighborhood of a regular point up to a transformation (1.3).

4.5.1. Regular germs. Let S be a section of π, let p be an arbitrary point of the domain of S, and let f be a Lagguere-Forsyth transformation of S defined in a neighborhood of p. We say that p is a point of class i for S (for the ODE ℰ_{S}) if there exist a neighborhood U′ of f(p) and subbundle E_{i} of τ (see section 4.4.2) with Im f(S)|_{U′} ⊂ E_{i}. It is easy to prove that a point of class i is well defined.

We say that S is a section of class i if every point of the domain of S is a point of class i.

Clearly, if p is a point of class i for S then, for some neighborhood U of p , S|_{U} is a section of class i.

Let S be a section of class i with dim Sym S = 0, let f be a Laguerre-Forsyth transformation of S, let S′ = f(S), and let I_{i}, I_{i−1}, …, I_{0} be the generators of the algebra of scalar differential invariants 𝒜_{i} defined in theorem 4.17. Consider the smooth functions I_{i}(S′), I_{i−1}(S′), …, I_{0}(S′) in ℝ^{1} (see formula (4.16)). From dim Sym S = 0 we have dim Sym S′ = 0. It follows from proposition 4.22 that there exist integer j , i ≤ j ≤ 0, such that I_{j}(S′) is not constant. We say that a point p of domain of S is regular point of S (of the ODE ℰ_{S}) if dI_{j}(S′)|_{f(p)} ≠ 0.

Obviously, the set of all regular points of a section of class i is an everywhere dense subset of the domain of this section.

We say that the germ {S}_{p} with dim Sym S = 0 is a regular germ of class i if p is both a point of class i for S and a regular point for S.

By ℱ̃_{i} , i = n − 3, n − 4, …, 0 we denote the set of all regular germs {S}_{0} of class i.

It is easy to prove that ℱ̃_{i} is invariant w.r.t. diffeomorphisms from Γ_{0}.

4.5.2. Classification of regular germs. Let {S}_{0} ∈ ℱ̃_{i}, let f be a Laguerre-Forsyth transformation of S, let S′ = f(S), and let p_{0} = f(0). By definition put

m = max{ j ∈ {i, i − 1, …, 0} | dI_{j}(S′)|_{p_{0}} ≠ 0 }

From (4.17), we have that the number m is an invariant of the action of Γ.

It follows from dI_{m}(S′)|_{p_{0}} ≠ 0 that there exists a neighborhood U of p_{0} such that we can consider I_{m}(S′)|_{U} as an element of Γ. Let

f̃ = I_{m}(S′)|_{U} − I_{m}(S′)(p_{0}) .

Obviously, f̃ ∈ Γ. Below, it will be useful to represent the function f̃ in the form

f̃ = R_{I_{m}(S′)(p_{0})} ∘ I_{m}(S′)|_{U} ,

where the function R_{a} is defined by R_{a} : x ↦ x − a.

We say that the germ { f̃(S′) }_{0} ∈ ℱ̃_{i} is the canonical form of the germ {S}_{0} . It follows from the next theorem that the canonical form of {S}_{0} is well defined.

Theorem 4.23. Let {S_{1}}_{0}, {S_{2}}_{0} ∈ ℱ̃_{i}. Then {S_{1}}_{0} and {S_{2}}_{0} are equivalent iff their canonical forms are the same.

Proof. Let {S_{1}}_{0} and {S_{2}}_{0} are equivalent. It follows that there exist g ∈ Γ_{0} with {g(S_{1})}_{0} = {S_{2}}_{0}. Let f_{1} and f_{2} be Laguerre-Forsyth transformations in neigborhood of 0 ∈ ℝ^{1} for S_{1} and S_{2} respectively. Let S′_{1} = f_{1}(S_{1}) , S′_{2} = f_{2}(S_{2}) and let p_{1} = f_{1}(0) , p_{2} = f_{2}(0). Then in some neighborhood of p_{2}, we have

f̃_{2} = I_{m}(S′_{2}) − I_{m}(S′_{2})(p_{2})

= I_{m}( (f_{2} ∘ g ∘ f_{1}^{−1})(S′_{1}) ) − I_{m}( (f_{2} ∘ g ∘ f_{1}^{−1})(S′_{1}) )(p_{2})
= I_{m}(S′_{1}) ∘ (f_{1} ∘ g^{−1} ∘ f_{2}^{−1}) − I_{m}(S′_{1})(p_{1})

= R_{I_{m}(S′_{1})(p_{1})} ∘ I_{m}(S′_{1}) ∘ (f_{1} ∘ g^{−1} ∘ f_{2}^{−1}) .

Hence in the correspondence neighborhood of 0 ∈ ℝ^{1}, we have

f̃_{2}(S′_{2}) = R_{I_{m}(S′_{1})(p_{1})} ∘ I_{m}(S′_{1}) ∘ (f_{1} ∘ g^{−1} ∘ f_{2}^{−1})( (f_{2} ∘ g ∘ f_{1}^{−1})(S′_{1}) )

= ( R_{I_{m}(S′_{1})(p_{1})} ∘ ( I_{m}(S′_{1}) )(S′_{1}) = f̃_{1}(S′_{1})

The sufficiency is obvious. □

Let ℱ_{i} be the set of canonical forms of all germs containing in ℱ̃_{i} and

let ℱ = ⋃_{i=0}^{n−3} ℱ_{i}. Obviously, we get the following theorem.

Theorem 4.24. (Classification of linear ODEs with n+1–dimensional algebras of point symmetries in neighborhoods of regular points up to equivalence.)

(1) Any two germs from ℱ are not equivalent.

(2) Any regular germ of section of π is equivalent to some germ from ℱ.

References

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[12] E. J. Wilczynski, Projective differential geometry of curves and ruled surfaces, B. G. Teubner, Leipzig, 1906.

[13] V.A.Yumaguzhin, Point transformations and classification of 3-order linear ODEs, Russian Journal of Mathematical Physics, Vol. 4, No. 3, pp. 403-410, 1996.

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