# Journal of Nonlinear Mathematical Physics

Volume 26, Issue 2, March 2019, Pages 188 - 201

# Generalized Solvable Structures and First Integrals for ODEs Admitting an π°π©(2, β) Symmetry Algebra

Authors
Paola Morando
DISAA, UniversitΓ  degli Studi di Milano, Via Celoria, 2 Milano, 20133, Italy,paola.morando@unimi.it
ConcepciΓ³n Muriel
Departamento de MatemΓ‘ticas, Universidad de CΓ‘diz, Puerto Real (CΓ‘diz), 11510, Spain,concepcion.muriel@uca.es
Received 11 May 2018, Accepted 24 October 2018, Available Online 6 January 2021.
DOI
10.1080/14029251.2019.1591712How to use a DOI?
Keywords
Generalized solvable structure; first integral; nonsolvable symmetry algebra
Abstract

The notion of solvable structure is generalized in order to exploit the presence of an π°π©(2, β) algebra of symmetries for a kth-order ordinary differential equation β° with k > 3. In this setting, the knowledge of a generalized solvable structure for β° allows us to reduce β° to a family of second-order linear ordinary differential equations depending on k β3 parameters. Examples of explicit integration of fourth and fifth order equations are provided in order to illustrate the procedure.

Open Access

## 1. Introduction

It is well known that the existence of a k-dimensional solvable Lie symmetry algebra for a kth-order ordinary differential equation (ODE) ensures the (local) integrability of the equation by quadratures. However, it is not difficult to provide examples of equations lacking of Lie point symmetries but such that their solutions can be computed by quadratures [4, 15, 22, 23, 27]. This fact triggered, in recent decades, a number of generalizations of the classical Lie reduction method such as hidden symmetries [1, 2], nonlocal symmetries [3, 18], Ξ»-symmetries [20, 21], ΞΌ-symmetries [12, 13], Ο-symmetries [14, 19], and solvable structures [6β8, 11, 16, 26].

The paper is organized as follows. We start Section 2 briefly recalling the main definitions and facts about distributions of vector fields and their symmetries and the concept of solvable structure, as well as its role in the integration by quadratures of ODEs. Afterwards we define generalized solvable structures for kth-order ODEs admitting an π°π©(2, β) symmetry algebra and we prove a result which allows us to exploit the vector fields belonging to the generalized solvable structure for computing by quadratures k β 3 functionally independent first integrals shared by the equation and the symmetry generators of π°π©(2, β). In particular, it is proved that the restriction of the ODE to the generic leaf defined by these first integrals provides a third-order equation which inherits π°π©(2, β) as Lie symmetry algebra. Since the classical Lie reduction of order for this kind of equations ends with a Riccati-type equation that does not permit to recover a closed form expression for the general solution to the third-order ODE [5, 17]. We exploit the recent results on the first integrals and general solutions to this class of equations [24, 25], in order to complete the set of first integrals. In particular we obtain the remaining three functionally independent first integrals in terms of the solutions to a related second-order ODE. The complete set of k independent first integrals can be used to provide the general solution to the original kth-order equation, expressed, in explicit or parametric form, in terms of a fundamental set of solutions to a (k β3)-parameter family of second-order linear equations.

In Section 3 our results are applied to fourth and fifth order ODEs, whose symmetry algebra is three-dimensional and isomorphic to π°π©(2, β). Since the symmetry algebra is nonsolvable and its dimension is strictly lower than the order of the ODE, the integrability by quadratures cannot be guaranteed by the classical Lie theory. For each one of the two examples of fourth-order equations, the construction of a generalized solvable structure requires the determination of a single vector field, and allows us to provide the general solutions to the ODEs (in explicit and parametric form, respectively) in terms of the solutions to a one-parameter family of second-order linear equations. For the fifth-order ODE presented in Section 3.3, the generalized solvable structure is formed by two vector fields and the general solution to the equation is expressed in parametric form in terms of the solutions to a two-parameter family of SchrΓΆdinger-type equations.

## 2.1. Distributions of vector fields and their symmetries

Given a set of vector fields {A1,...,Anβk} defined on an n-dimensional manifold N, we denote by π := β©A1,...,Anβkβͺ the distribution generated by {A1,...,Anβk}. Moreover, given a set of one-forms {Ξ²1,...,Ξ²k}, we denote by β©Ξ²1,...,Ξ²kβͺ the corresponding Pfaffian system (i.e. the sub-module over Cβ(N)) generated by {Ξ²1,...,Ξ²k}.

The distribution π is integrable (in Frobenius sense) if [A, B] β π, for any A, B β π. If U is an open domain of N where the vector fields {A1,...,Anβk} are pointwise linearly independent, we say that π is a distribution of maximal rank n β k (or of codimension k) on U. It is well known that any integrable distribution π of maximal rank on U β N determines a (n β k)-dimensional foliation of U. If this foliation is described through the vanishing of k functions of the form Ih β ch, where Ih β πβ(U) and ch β β, we can chose β©dI1,...,dIkβͺ as generators for the Pfaffian system annihilating the distribution π. A submanifold S β N is an integral manifold for π if π|S β TS. If in particular π|S = TS we say that S is a maximal integral manifold for π.

Given a distribution π, a vector field X is a symmetry of π if [X, A] β π, for any A β π. Let π and β¬ be two distributions on N. We say that π and β¬ are transversal at p β N if they do not vanish at p and π (p) β© β¬(p) = {0}. Analogously, π and β¬ are transversal in U if they are transversal at any point of U. An algebra π’ of symmetries for a distribution π is nontrivial if π’ generates a distribution which is transversal to π.

## 2.2. Solvable structures for ODEs

It is well known that, given a (n β k)-dimensional integrable distribution π on an n-dimensional manifold N, the knowledge of a solvable k-dimensional algebra π’ of nontrivial symmetries for π guarantees that a complete set of first integrals for π can be found by quadratures. Solvable structures provide an extension of this classical result, significantly enlarging the class of vector fields which can be used to integrate by quadratures a distribution of vector fields.

In this section we recall some basic definitions and facts on solvable structures. The interested reader is referred to [6, 8, 16, 26] for further details.

### Definition 2.1.

Let π be a (n β k)-dimensional distribution on an n-dimensional manifold N. A set of vector fields {Y1,...,Yk} is a solvable structure for π in an open domain U β N if, denoting by π0 = π and πh = π β β©Y1,...,Yhβͺ (h β€ k), the following conditions hold:

1. (1)

The distribution β©Y1, Y2,...,Yhβͺ has maximal rank h and is transversal to π in U, for any h β€ k;

2. (2)

πh has maximal rank (n β k + h) in U;

3. (3)

βYhπhβ1 β πhβ1, for 1 β€ h β€ k.

### Theorem 2.1.

Let π = β©A1,...,Anβkβͺ be an integrable (n β k)-dimensional distribution defined on an orientable n-dimensional manifold N and let {Y1,...,Yk} be a solvable structure for π. Denoting by Ξ© a volume form on N and by Ξ± the k-form A1β ...βAnβkβΞ©, the distribution π can be described as the annihilator of the Pfaffian system generated by

Οi=1Ξ(Y1ββ¦βY^iββ¦βYkβΞ±),βββ(i=1,β¦,k)(2.1)
where the hat denotes omission of the corresponding vector field and Ξ is the function on N defined by
Ξ=Y1βY2ββ¦βYkβΞ±.

Moreover, the forms Οi satisfy

dΟk=0,dΟi=0βββmod{Οi+1,β¦,Οk}
for i β {1,...,k β 1}. Therefore the integral manifolds of the distribution π can be described in implicit form as the level manifolds I1 = c1, I2 = c2, ..., Ik = ck, ci β β, where
Οk=dIk,βββΟkβ1|{Ik=ck}=dIkβ1,β¦,Ο1|{Ik=ck,Ikβ1=ckβ1,β¦,I2=c2}=dI1.

### Proof.

The interested reader is referred to the original papers [6β8, 16, 26] for a proof of this theorem.

We remark that, if the distribution π admits an Abelian Lie algebra of symmetries generated by the vector fields Y1,...,Yk, then all the 1-forms Οi are closed, i.e. the function 1/Ξ provides an integrating factor for all the 1-forms (Y1β... βΕΆiβ...βYkβΞ±), for 1 β€ i β€ k.

The main difference between a solvable structure and a solvable symmetry algebra for a completely integrable distribution π is that the fields belonging to a solvable structure do not need to be symmetries of π. This, of course, gives more freedom in the choice of the vector fields which can be exploited to find integral manifolds of π by quadratures.

Let (x, u(kβ1)) = (x, u, u1,...,ukβ1) denote the coordinates of the jet space Jkβ1(β, β), for k β₯ 2. In order to apply Theorem 2.1 to the integration of ODEs, we recall that, with any kth-order ODE in normal form

uk=F(x,u(kβ1)),(2.2)
we can associate the vector field A defined on a suitable domain U β Jkβ1(β, β) and given by
A=βx+u1+βu+β―+F(x,u(kβ1))βukβ1.(2.3)

### Definition 2.2.

If β° is a kth-order ODE of the form (2.2) and A is the corresponding vector field given by (2.3), the vector fields {Y1, Y2,...,Yk} defined on a domain U β Jkβ1(β, β) are a solvable structure for β° if they are a solvable structure for the one-dimensional distribution π = β©Aβͺ.

### Corollary 2.1.

The knowledge of a k-dimensional solvable structure for a kth-order ODE, allows us to obtain the solution to the ODE by quadratures.

### Proof.

The distribution π is obviously integrable, being one-dimensional. Moreover the functions Ih of Theorem 2.1 provide a complete set of first integrals for A.

## 2.3. Generalized solvable structures for ODEs

In this section we generalize the notion of solvable structure for ODEs, in order to exploit the knowledge of a symmetry algebra isomorphic to π°π©(2, β) for kth-order ODEs with k > 3. In particular we give the following

### Definition 2.3.

Let β° be a kth-order ODE of the form (2.2) with k > 3 and let A be the corresponding vector field defined by (2.3). If β° admits a symmetry algebra isomorphic to π°π©(2, β) with generators {X1, X2, X3}, we call generalized solvable structure for β° a set of vector fields {Y1,...,Ykβ3} which form a standard solvable structure for the integrable distribution π = β©A, X1, X2, X3βͺ.

We remark that the k vector fields {X1, X2, X3, Y1,...,Ykβ3} do not form a solvable structure for the distribution generated by A, due to the commutation relations between Xi. Despite this fact we have the following result.

### Theorem 2.2.

Let β° be a kth-order ODE with k > 3 admitting a symmetry algebra isomorphic to π°π©(2, β) and let A be the corresponding vector field defined by (2.3). The knowledge of a generalized solvable structure {Y1,...,Ykβ3} for β° allows us to construct k first integrals for the ODE (2.2) in terms of two independent solutions to a linear second order ODE.

### Proof.

Since {Y1,...,Ykβ3} provide a solvable structure for the integrable distribution π = β©A, X1, X2, X3βͺ, we can use Theorem 2.1 in order to find suitable functions {I1,...,Ikβ3} such that Ξ£ := {Ii = ci}i=1,...,kβ3 is an integrable manifold for π. Since in this setting the manifold N = Jkβ1(β, β) has dimension (k + 1), we have that the dimension of Ξ£ is k + 1 β (k β 3) = 4. Moreover, since the vector fields A, X1, X2, X3 are tangent to Ξ£, equation (2.2) restricted to Ξ£ turns out to be a third-order ODE admitting π°π©(2, β) as symmetry algebra. Therefore, we can exploit the results of [24,25] in order to obtain the remaining three first integrals. In particular, starting from the restriction of equation (2.2) to the 4-dimensional submanifold Ξ£, we get Ikβ2, Ikβ1 and Ik in terms of two dependent solutions to a second-order linear ODE and of the real constants ci, i = 1,...,k β 3.

## 3.1. Example 1

Let us consider the fourth-order equation

u4u12β2(3u2u3u1β3u22βuu16)=0(3.1)
which corresponds, when u1 β  0, to the vector field
A=βx+u1βu+u2βu1+u3βu2+2(3u2u3u1β3u22βuu16)u12βu3.

The symmetry algebra of equation (3.1) is spanned by

v1=βx,βββv2=x2βx,βββv3=xβx,(3.2)
and hence it is three-dimensional and isomorphic to π°π©(2, β). Since equation (3.1) lacks of further symmetries, we can look for a generalized solvable structure. In particular we have to find a vector field Y on a suitable domain U β J3(β,β) such that Y is a symmetry of the distribution π = β©A, X1, X1, X3βͺ, where Xi stands for the third-order prolongation of v i, for i = 1, 2, 3, i.e.
X1=v1(3)=βx,X2=v2(3)=x2βxβ2xu1βu1β(2u1+4u2x)βu2β(6u2+6u3x)βu3,X3=v3(3)=xβxβu1βu1β2u2βu2β3u3βu3.(3.3)

It is worthwhile to remark that finding a symmetry for the distribution π = β©A, X1, X1, X3βͺ is definitely simpler than finding a symmetry for the vector field A. For instance, it is easy to check that the vector field

Y=u13βu3,(3.4)
provides a generalized solvable structure for ODE (3.1), due to the following commutation relations
[X1,Y]=[X2,Y]=[X3,Y]=0,[A,Y]=(12u12x2)X1+(12u12)X2βu12xX3.(3.5)

Once we have got Y we can apply Theorem 2.2. by considering the volume form Ξ© = dx β§ du β§ du1 β§ du2 β§ du3 and the differential 1-form Ξ± = AβX1βX2βX3βΞ© given by

Ξ±=4uu16duβ(6u3u12β12u22u1)du1β6u12u2du2+2u13du3.

Since Y is a symmetry of Ξ±, the function 1/Ξ, with

Ξ=YβΞ±=2u16,
is an integrating factor of Ξ±. Therefore, the one-form
Ο=1ΞΞ±=2uduβ3(u3u1β2u22)u15du1β3u2u14du2+1u13du3
is closed and so (locally) exact, i.e. Ο = dI, where the function I = I(x, u(3)) is given (up to an additive constant) by:
I=u2+u3u13β3u222u14.(3.6)

Function (3.6) provides a first integral for the integrable distribution β©A, X1, X2, X3βͺ and the restriction of (3.1) to the submanifold I = c1, where c1 β β, leads to the reduced equation

2u2u14+2u1u3β3u22=2u14c1.(3.7)

Equation (3.7) inherits a symmetry algebra isomorphic to π°π©(2, β) and spanned by the restrictions of the vector fields (3.2) to I = c1. Moreover (3.7) corresponds to the equation presented in [25], Case 1 in Table 2, for the function C(u) = u2 β c1. By Proposition 4.1 in [25], a complete set of first integrals for equation (3.7) can be expressed in terms of two independent solutions to the second-order linear ODE (see Case 1, Table 6 in [25]):

Ξ¨c1β³(u)+(u2βc1)Ξ¨c1(u)=0.(3.8)

Let Ξ¨c1;1(u) and Ξ¨c1;2(u) denote two linearly independent solutions to (3.8), such that the corresponding Wronskian is equal to 1 (note that the Wronskian is constant by the Liouvilleβs formula).

According to Table 7 (Case 1) in [25], three independent first integrals for equation (3.7) are given by

I1(x,u,u1,u2;c1)=u2Ξ¨c1;1(u)β2u12Ξ¨c1;1β²(u)u2Ξ¨c1;2(u)β2u12Ξ¨c1;2β²(u),I2(x,u,u1,u2;c1)=(2u1+xu2)Ξ¨c1;1(u)β2xu12Ξ¨c1;1β²(u)(2u1+xu2)Ξ¨c1;2(u)β2xu12Ξ¨c1;2β²(u),I3(x,u,u1,u2;c1)=((2u1+xu2)Ξ¨c1;2(u)β2xu12Ξ¨c1;2β²(u))24u13.(3.9)

If we put c1=u2+u3u13β3u222u14 in (3.9), the resulting functions I1, I2, I3 provide, together with (3.6), a complete set of first integrals for equation (3.1). Therefore, the general solution to (3.1) is given by (see Section 5 in [25]):

x=c4(c2βc3)c2Ξ¨c1;1(u)βΞ¨c1;2(u)c3Ξ¨c1;1(u)βΞ¨c1;2(u),(3.10)
where ci β β for i = 1, 2, 3, 4, c4 β  0, c2 β  c3.

We finally remark that the solutions to (3.8) can be expressed in terms of Whittaker functions: if MΞΌ,Ξ½(z) and WΞΌ,Ξ½(z) denote the Whittaker functions of parameters ΞΌ=ΞΉ4c1 and v=14 , then two linearly independent solutions to (3.8) are given by

Ξ¨c1;1(u)=1uMΞΌ,Ξ½(iu2)βββandβββΞ¨c1;2(u)=1ΞΌWΞΌ,Ξ½(iu2).

Consequently, the general solution (3.10) can be expressed as follows:

x=c4(c2βc3)c2WΞΌ,Ξ½(iu2)βMΞΌ,Ξ½(iu2)c3WΞΌ,Ξ½(iu2)βMΞΌ,Ξ½(iu2).

## 3.2. Example 2

Let us consider the fourth-order ODE

u1(xβu)u4β2xu2u3β6u22u1+4u12u3+2u2u3uβ12u22+8u3u1=0(3.11)
and let A denote the corresponding vector field of the form (2.3) on a suitable domain U β J4(β, β) such that u1(x β u) β  0.

The Lie symmetry algebra of equation (3.11) is three-dimensional and isomorphic to π°π©(2, β). The third-order prolongations of the corresponding Lie symmetry generators are given by

X1=βx+βu,X2=x2βu+u2βuβ2u1(xβu)βu1β(2u1+4xu2β2u12β2uu2)βu2ββββ(6u2+6xu3β6u2u1β2u3u)βu3,X3=xβx+uβuβu2βu2β2u3βu3.

A symmetry of the integrable distribution β©A, X1, X2, X3βͺ can be easily found searching, for instance, a symmetry of the form Ξ·(x, u(3))βu. In particular, if we consider the vector field

Y=(u12u3(xβu)+3u2(u1+1))βu,(3.12)
we have the following commutation relations
[X1,Y]=[X3,Y]=0,[X2,Y]=ΞΌ0A+ΞΌ1X1+ΞΌ2X2+ΞΌ3X3,[A,Y]=Ο0A+Ο1X1+Ο2X2+Ο3X3,(3.13)
for suitable functions ΞΌi, Οi (i = 0, 1, 2, 3) whose expressions are omitted, not being involved in the following discussion. Since the vector field (3.12) provides a generalized solvable structure for equation (3.11), an integrating factor for the 1-form Ξ± = Aβ X1βX2βX3βΞ© is given by 1/Ξ, where
and we assume that Ξ β  0.

Therefore, the differential one-form Ο=1ΞΞ± is closed and, locally, exact. A corresponding first integral is given by

I1=12u12(6u2(uβx)βu3(xβu)2+6u2u1(uβx)β6u1β6u13).(3.14)

The restriction of equation (3.11) to a generic leaf I1 = c1, c1 β β, provides the following third-order ODE:

u3=β2(c1u12+3u13+3u1β3uu1u2+3u1u2xβ3u2x+3xu2)(xβu)2.(3.15)

Equation (3.15) inherits the symmetry algebra π°π©(2, β) and corresponds to the third-order ODE appearing in [25, Case 3 in Table 1], for the particular case of

C(s)=13s2+4c1β24,βββwhereβββs=2u1+2u12+u2(xβu)u13/2.

According to [25], a complete set of first integrals to (3.15) can be expressed in terms of a system of solutions to the second-order linear equation

Ξ¨c1β³(s)+7s3s2+4c1β24Ξ¨c1β²(s)+4(3s2+4c1β24)Ξ¨c1(s)=0.(3.16)

In particular, if Ξ¨c1;1 = Ξ¨c1;1(s), Ξ¨c1;2 = Ξ¨c1;2(s) denote two linearly independent solutions to (3.16) and W = W(Ξ¨c1;1, Ξ¨c1;2)(s) denotes the corresponding Wronskian, a complete system of first integrals for equation (3.15) is given by (see [25, Case 3 in Table 8]):

I2(x,u,u1,u2;c1)=2u1Ξ¨c1;1(s)+(3s2+4c1β24)Ξ¨c1;1β²(s)2u1Ξ¨c1;2(s)+(3s2+4c1β24)Ξ¨c1;2β²(s),I3(x,u,u1,u2;c1)=2xu1Ξ¨c1;1(s)+u(3s2+4c1β24)Ξ¨c1;1β²(s)2xu1Ξ¨c1;2(s)+u(3s2+4c1β24)Ξ¨c1;2β²(s),I4(x,u,u1,u2;c1)=(2xu1Ξ¨c1;2(s)+3u(3s2+4c1β24)Ξ¨c1;2β²(s))26u1(uβx)(3s2+4c1β24)W.(3.17)

By setting Ji = Ji(x, u(3)) = Ii(x, u, u1, u2; I1), for i = 2, 3, 4, we finally obtain that {I1, J2, J3, J4} is a complete set of first integrals for the fourth-order equation (3.11). Hence, the general solution to (3.15), and therefore to the original equation (3.11), is implicitly defined by

I2(x,u,u1,u2;c1)=c2,I3(x,u,u1,u2;c1)=c3,I4(x,u,u1,u2;c1)=c4,(3.18)
where ci β β for i = 1, 2, 3, 4.

In order to obtain a parametric expression for the implicit solution (3.18), we consider a new parameter t such that s = s(t) is determined as follows (see [25], Section 5 for further details):

sβ²(t)=1C(s(t))=3s(t)2+4c1β24.(3.19)

Three possible cases arise:

• If c1 > 6 then (3.19) yields

s(t)=2cβ12tan(2t2c1β12),
and the second-order equation (3.16) is transformed by means of the change Οc1(t) = Ξ¨c1(s(t)) into
Οc1β³(t)+2cβ12tan(2t2c1β12)Οc1β²(t)+4Οc1(t)=0.(3.20)

• If c1 < 6 then (3.19) yields

s(t)=β12β2c1tanh(2t12β2c1),
and through the transformation Οc1(t) = Ξ¨c1(s(t)) the second-order equation (3.16) becomes
Οc1β³(t)β12β2ctanh(2t12β2c)Οc1β²(t)+4Οc1(t)=0.(3.21)

• If c1 = 6 then a solution to (3.19) is locally given by

s(t)=β12t
and by means of the transformation Οc1(t) = Ξ¨c1(s(t)), equation (3.16) is mapped into
Οc1β³(t)β12tΟc1β²(t)+4Οc1(t)=0.(3.22)

For each one of the three cases considered above, let Ο1;c1(t) = Ξ¨c1;1(s(t)) and Ο2;c1(t) = Ξ¨c1;2 (s(t)) denote two linearly independent solutions to the second-order linear equations (3.20), (3.21), and (3.22), respectively. In terms of these functions, the general solution to equation (3.11) can be expressed in parametric as follows [25]:

{x(t)=c4(c2βc3)c3Ο2;c1β²(t)βΟ1;c1β²(t)c2Ο2;c1β²(t)βΟ1;c1β²(t),u(t)=c4(c2βc3)c3Ο2;c1(t)βΟ1;c1(t)c2Ο2;c1(t)βΟ1;c1(t).(3.23)

## 3.3. Example 3

In this section we consider the fifth-order ODE

uu3u5+u1u3u4+2u2u32βuu42+4u3u34=0(3.24)
on a suitable domain U β J5(β, β) such that uu3 β  0 and its associated vector field
A=βx+u1βu+u2βu1+u3βu2+u4βu3β(u4u1u3+2u2u32βuu42+4u3u34uu3)βu4.

The Lie symmetry algebra of equation (3.24) is three dimensional and isomorphic to π°π©(2, β). The corresponding fourth-order prolongations of the symmetry generators are

X1=βx,X2=x2βx+2xuβu+2uβu1β(2xu2β2u1)βu2β4xu3βu3β(4u3+6xu4)βu4,X3=xβx+uβuβu2βu2β2u3βu3β3u4βu4.

A symmetry Y1 of the integrable distribution β©A, X1, X2, X3βͺ can be easily determined by searching, for instance, a vector field of the form Ξ·(x, u(4))βu2. For example, the vector field

Y1=1uβu2,
satisfies the following commutation relations:

[X1,Y1]=[X2,Y1]=[X3,Y1]=0,[A,Y1]=βx22u2X1β12u2X2+xu2X3.(3.25)

The next step in order to obtain a generalized solvable structure for equation (3.24) is looking for a symmetry Y2 of the integrable distribution β©A, X1, X2, X3, Y1βͺ. It is easy to check that the vector field

Y2=β2uβuβ3u1βu1+u3βu3
satisfies
[X1,Y2]=[X3,Y2]=0,[X2,Y2]=βx2X1βX2+2xX3+8uu1Y1,[Y1,Y2]=β2Y1,[A,Y2]=Aβ(u+2x2u2)uX1β2u2uX2+4xu2uX3+(4u2u1β2uu3)Y1.

Therefore β©Y1, Y2βͺ provides a generalized solvable structure for equation (3.24).

In order to apply Theorem 2.2. we consider the volume form Ξ© = dxβ§duβ§du1 β§du2 β§du3 β§du4 and the two one-forms

An integrating factor for Ξ±2 is given by 1/Ξ, where

Ξ=Y2βΞ±2=6(4u4u34+u2u42+4u4u1u3u+4u32u12)u
and we assume that Ξ β  0. Therefore we can chose as first integral for the (locally) exact form Ο2=1ΞΞ±2 the function ln(I2)/6, where
I2=u324u4u34+u2u42+4u4u1u3u+4u32u12(3.26)
is a first integral for the distribution β©A, X1, X2, X3, Y1βͺ.

Since Ο1=1ΞΞ±1 is closed on each leaf I2 = c2, with c2 β β, c2 > 0, we have that Ο1 is locally exact module Ο2 and a corresponding first integral is given by

I1=12u12βu2u+12arctan(2u32u22u3u1+uu4).(3.27)

Hence, by restricting to a generic leaf {I1 = c1, I2 = c2}, we get the reduced equation

u3=βsin(2c1+2uu2βu12)2u2c2,(3.28)
which is a third-order ODE admitting π°π©(2, β) as Lie symmetry algebra. Equation (3.28) corresponds to the third-order ODE given in [25, Case 2 in Table 2] for
C(s)=c24sin(2c1βs),βββwhereβββs=u12β2uu2.

According to the results obtained in [25], if Ξ¨c1,c2;1 = Ξ¨c1,c2;1(s) and Ξ¨c1,c2;2 = Ξ¨c1,c2;2(s) are two linearly independent solutions to the linear second-order equation [25, Case 2 in Table 6],

β4sin2(2c1βs)Ξ¨β³(s)+2sin(4c1β2s)Ξ¨β²(s)+c2sΞ¨(s)=0,(3.29)
three functionally independent first integrals for equation (3.28) are given by
I3(x,u(2);c1,c2)=c2u1Ξ¨c1,c2;1(s)+2sin(2c1βs)Ξ¨c1,c2;1β²(s)c2u1Ξ¨c1,c2;2(s)+2sin(2c1βs)Ξ¨c1,c2;2β²(s),I4(x,u(2);c1,c2)=c2(β2u+u1x)Ξ¨c1,c2;1(s)+2xsin(2c1βs)Ξ¨c1,c2;1β²(s)c2(β2u+u1x)Ξ¨c1,c2;2(s)+2xsin(2c1βs)Ξ¨c1,c2;2β²(s),I5(x,u(2);c1,c2)=(c2(β2u+u1x)Ξ¨c1,c2;2(s)+2xsin(2c1βs)Ξ¨c1,c2;2β²(s))24c2usin(2c1βs)W,
where W = W(Ξ¨c1,c2;1, Ξ¨c1,c2;2)(s) denotes the Wronskian corresponding to the solutions Ξ¨c1,c2;1 and Ξ¨c1,c2;2. Therefore, as in the previous examples, the general solution to the original fifth-order equation (3.24) is implicitly given by
I3(x,u(2);c1,c2)=c3,I4(x,u(2);c1,c2)=c4,I5(x,u(2);c1,c2)=c5.(3.30)

In order to obtain a parametric general solution from (3.30), we consider a new parameter t such that s = s(t) is determined as follows:

sβ²(t)=1C(s(t))=4sin(2c1βs(t))c2,
which yields
s(t)=βarctan(2eβ4t/c2eβ8t/c2β1)+2c1.(3.31)

By means of the transformation Ο(t) = Ξ¨(s(t)) equation (3.29) is transformed into the SchrΓΆdinger-type equation

Οβ³(t)+4(arctan(2eβ4t/c2eβ8t/c2β1)β2c1)Ο(t)=0.(3.32)

If Ο1 = Ο1(t; c1, c2) and Ο2 = Ο2(t; c1, c2) are two linearly independent solutions to (3.32) such that W(Ο1, Ο2)(t; c1, c2) = 1, then the parametric general solution to equation (3.24) can be expressed as follows

{x(t)=c5(c3βc4)Ο1(t;c1,c2)βc4Ο2(t;c1,c2)Ο1(t;c1,c2)βc3Ο2(t;c1,c2),u(t)=c5(c3βc4)24(Ο1(t;c1,c2)βc3Ο2(t;c1,c2))2,(3.33)
where ci β β for i = 1, 2, 3, 4, 5, c2 > 0, c3 β  c4, c5 β  0.

## 4. Conclusions

Generalized solvable structures for ODEs of order k > 3 admitting a symmetry algebra isomorphic to π°π©(2, β) have been introduced, providing a powerful tool for the explicit determination of the general solutions to these equations.

A generalized solvable structure is formed by k β 3 vector fields, which, in general, are not symmetries of the equation. Moreover, the symmetry generators of π°π©(2, β) and the vector fields of the generalized solvable structure do not form a solvable structure. Nevertheless, the knowledge of a generalized solvable structure allows the determination by quadratures of a complete set of first integrals, and hence of the general solution to the equation, in terms of two independent solutions to an associated family of second-order linear ODEs. This link between equations admitting π°π©(2, β) and second-order linear ODEs is, to the best of our knowledge, new in the literature and generalizes the recent results obtained for third-order equations.

The concept of generalized solvable structure gives more freedom in the choice of the vector fields which can be used to integrate the equation under study. The effectiveness of the proposed method is illustrated by several examples of equations admitting only an π°π©(2, β) algebra of Lie symmetries. In these cases, after finding a generalized solvable structure, we provide a complete set of first integrals as well as the general solutions in parametric or explicit forms.

## Acknowledgments

P. Morando acknowledges the financial support from Gruppo Nazionale di Fisica Matematica (GNFM-INdAM).

C. Muriel and A. Ruiz acknowledge the financial support from the University of CΓ‘diz by means of the project PR2017-090 and from Grupo de InvestigaciΓ³n de la Junta de AndalucΓ­a FQM-377.

A. Ruiz acknowledges the financial support from the Ministry of Education, Culture and Sport of Spain (FPU grant FPU15/02872).

Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
26 - 2
Pages
188 - 201
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2019.1591712How to use a DOI?
Open Access

TY  - JOUR
AU  - Paola Morando
AU  - ConcepciΓ³n Muriel
PY  - 2021
DA  - 2021/01/06
TI  - Generalized Solvable Structures and First Integrals for ODEs Admitting an π°π©(2, β) Symmetry Algebra
JO  - Journal of Nonlinear Mathematical Physics
SP  - 188
EP  - 201
VL  - 26
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1591712
DO  - 10.1080/14029251.2019.1591712
ID  - Morando2021
ER  -