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1ВЕСТНИК ОШСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА МАТЕМАТИКА, ФИЗИКА, ТЕХНИКА. 2023, №1
УДК 517.912
https://doi.org/10.52754/16948645 2023 1 262
INTEGRATION OF A THIRD-ORDER ODE VIA ANALYTICAL AND
GEOMETRICAL METHODS
Beltrán de la Flor, Master's Degree, Ph. D. Student,
beltran.delaflor@uca. es Concepción Muriel, Ph. D., Full Professor, concepcion.muriel@uca. es Adrián Ruiz, Ph. D., Assistant Professor, adrian.ruiz@uca.es
Universidad de Cádiz, Facultad de Ciencias, 11510, Puerto Real, Spain.
Abstracts. Analytical and geometrical methods are applied to integrate an ordinary differential equation of third order. The main objective is to compare both approaches and show the possibilities that each one of them offers in the integration process of the considered equation, specially when not only Lie point symmetries but also generalized symmetries are involved. The analytical method of order reduction by using a generalized symmetry provides the general solution of the equation but in terms of a primitive that cannot be explicitly evaluated. On the other hand, the application of geometrical tools previously reported in the recent literature leads to two functionally independent first integrals of the equation without any kind of integration. In order to complete the integration of the given third-order equation, a third independent first integral arises by quadrature as the primitive of a closed differential one-form. From these first integrals, the expression of the general solution of the equation can be expressed in parametric form and in terms of elementary functions.
Key words: Lie symmetries, first-order symmetries, first integrals, involutive distributions.
1. Introduction.
The Lie symmetry approach for reducing the order and integrating ordinary differential equations (ODEs) is one of the most powerful and used tools available for handling ODEs and their exact solutions. It is well known that an nth-order equation admitting an r-dimensional algebra of Lie point symmetries can be reduced to an (n — r)th-order equation in terms of the common differential invariants of the symmetries. Furthermore, if this algebra is solvable then one can recover the solution of the original equation by solving the reduced ODE and carrying out r consecutive quadratures [1, 2, 3, 4, 5]. In this paper, we will refer to these procedures of order reduction or integration of ODEs as analytical (or classical) methods. This method can be extended to use higher-order symmetries to reduce the order of the equation [3, 6, 7], although in this case the calculation of the differential invariants gets more complicated since, in general, it is not possible to obtain a complete set of differential invariants by derivation of lower order invariants. Furthermore, the reduction process in this situation for an nth-order ODE leads to a system of n — 1 first-order ODEs that is not equivalent to an (n — 1)th-order ODE.
In the last decade of the past century, P. Basarab-Horwath [8], J. Sherring and G. Prince [9], T. Hartl and C. Athorne [10] and M. A. Barco and G. Prince [11] obtained powerful geometrical results regarding the integration by quadratures of involutive distributions of vector fields. Such results are based on the concept of solvable structure, an object that generalizes the notion of solvable Lie algebra of symmetries of an involutive distribution. These geometrical results can be applied to integrate ODEs by quadratures by considering the involutive distribution generated by the vector associated to the equation and determining a solvable structure for it. Remarkably, the vector fields involved in a solvable structure are not necessarily Lie point nor generalized symmetries of the equation. Once a solvable structure is known, the equation can be integrated by quadratures by following a procedure that has been studied, applied and generalized by many authors over the years since its introduction [10, 11, 12, 13].
In particular, in the general setting of an n-dimensional manifold, it was developed a procedure to integrate by quadratures involutive distributions of vector fields of dimension r admitting an (n — r)-dimensional solvable structure ([8, prop. 3], [9, prop. 4.6 and 4.7]). This method works by constructing n — r closed differential 1-forms which belong to the annihilator of the distribution and which can be integrated successively. This procedure generalizes a classical result known as Lie-Bianchi's Theorem [14, th. 1.7.2], which uses solvable Lie algebras of symmetries instead of solvable structures. We will refer to the application of these last procedures to the involutive distribution associated to an ODE as the geometrical method of integration of the equation.
In the present paper we aim to perform a comparison of the results obtained when both the analytical and the geometrical methods are applied to integrate a third-order ODE. The considered equation can be found in [3, eq. (3.245)], where the authors proved that the equation admits three Lie point symmetries and seven first-order symmetries. The analytical method of reduction of order is applied using one of the first-order symmetries. We will review this procedure and its flaws, and we will see how the geometrical method can give us some advantages in this situation.
The work is organized as follows. In Section 2 the basics definitions and results regarding the method of solvable structures to integrate involutive distributions of vector fields are briefly introduced. In Section 3, with the aim of being self-contained, we describe the reduction of order of ODEs via Lie point symmetries and generalized symmetries, as well as the application of the geometrical methods in the particular case of ODEs. In Section 4, we introduce the ODE under study and the Lie point and generalized symmetries admitted by the equation. We also review the analytical method of reduction of order described in [3], showing some of the problems that may appear when using generalized symmetries. As a consequence, an implicit expression for the general solution of the ODE in terms of a primitive that cannot be evaluated is obtained. In Section 5 we apply geometrical tools to the study of the third-order ODE. It is proved that at least two functionally independent first integrals can be obtained without any kind of integration. A third functionally independent first integral can be obtained by quadrature as a primitive of certain 1-form. The geometric approach allows us to give the general solution of the ODE in parametric form and expressed in terms of elementary functions, greatly improving the results obtained by the analytical method.
2. Preliminaries.
Consider an n-dimensional differentiable manifold M. Given a connected, open set U £ M, the real vector space of smooth functions defined on U will be denoted as C™(U). The C™(U)-module of smooth vector fields defined on U will be denoted as %(U). The COT(^)-module of differential p-forms in U will be denoted as (U) [15, def. 2.15], and the exterior algebra will be denoted as Q.*(U) [15, def. 2.14]. The exterior product of differential forms will be denoted by A. The contraction of a p-form o> by a vector field X [16, pg. 72 (d)] will be written as ixu>, while the exterior derivative of a p-form o> [16, pg. 70 (b)] will be represented by dti>.
A collection of r vector fields A1, ...,Ar e %(U) will be pointwise linearly independent on U (or simply independent) if the vectors A-^(p), ...,Ar(p) are linearly independent for each p e U. The same applies for a collection of r differential p-forms.
The COT(y)-module generated by A1 ,...,Ar eX(U) will be called the r-dimensional distribution V generated by the vector fields (see [15, def. 1.56], [13, sec. 2.2]) and will be denoted by
© = (A1.....Ar). (1)
We will say that XCBor that X is transversal to V if the vector fields X, A1, ...,Ar are pointwise linearly independent in U.
An r-dimensional distribution V is said to be involutive if it is closed under the Lie bracket [15, def. 1.56]. This condition guarantees, via the well-known Frobenius theorem, the local existence of n — r functionally independent first integrals lt, ...,In_r e C°°(U), in which case
N = {peU: Ij (p) = Cj, j = 1.....n — r] (2)
are integral manifolds of V for Ct, ...,Cn_r e R [15, th. 1.60]. Nevertheless, Frobenius theorem does not provide a procedure to compute these first integrals. In order to find them, the concept of symmetry of a distribution [9, p. 441] is useful, as we will shortly see:
Definition 1. Let U £ M be an open set, and T) an r-dimensional distribution. A vector field Y is a symmetry of T) if for every A e V it is [Y, A] e T). A symmetry of T) is called non-trivial if it is transversal to T>.
The set of symmetries of V will be denoted by Sym {V]. Using Jacobi's identity, it can be proved that Sym {T>} is a Lie algebra, that is, it is a real vector space and [X, Y] e Sym {T>} whenever X, Ye Sym {D}. Nevertheless, in general, Sym {D} is not a COT(^)-module, since the product of a symmetry by a smooth function which does not vanish on U generally is not a symmetry.
The knowledge of an (n — r)-dimensional solvable Lie algebra of symmetries of an r-dimensional involutive distribution allows the computation of n — r functionally independent first integrals of the distribution by quadratures alone, a result known in the literature as Lie-Bianchi's theorem [14, th. 1.7.2].
The next concept generalizes the notion of solvable Lie algebra of symmetries of an r-dimensional distribution [8, def. 4]:
Definition 2. Let V = (At, ...,Ar) be an r-dimensional distribution and let {Yr, ...,Yn_r] be an ordered set of pointwise linearly independent vector fields on U. We will say that the previous ordered set is a solvable structure for V if:
• {Y-l, ..., Yn_r, A1,..., Ar] are pointwise linearly independent in U.
• Yn_r is a symmetry of V = {At,..., Ar).
• Yj is a symmetry of Dy = (YJ+1,..., Yn_r, A1,..., Ar) for every j = 1,...,n — r — 1.
Observe that, as we announced, an (n — r)-dimensional solvable Lie algebra of symmetries of an r-dimensional distribution is a particular case of a solvable structure, as according to [17, prop 1.23] there exists a basis of the Lie algebra, Yt,..., Yn_r e %(U), such that
[Yi, Yj] = lil\cfjYk, cfje R, 1 <i< j < n — r. (3)
The next result, whose proof can be found, for instance, in [9, prop. 4.6 and 4.7] or in [8, prop. 3], allows us to integrate by n - r successive quadratures any r-dimensional distribution
admitting a solvable structure. It generalizes Lie-Bianchi's theorem by requiring the knowledge of a solvable structure instead of an (n — r)-dimensional solvable Lie algebra of symmetries. Before proceeding let us introduce the following notation: given m1, ...,u>r e D.P(U) pointwise linearly independent,
J(Mlt...,Mr) (4)
will be the ideal generated by the previous p-forms undertaking exterior products [16, lemma 2.19 (ii)].
Theorem 1. Let V = {At, ...,Ar) be an r-dimensional, involutive distribution. Let ft e D.n(U) be a non-zero n-form. Suppose that [Y-^,..., Yn_r} is a solvable structure of V and define the following 1-forms, where the hat denotes omission of the element:
O);
■(iYl ... iYj . iYn.riA1 . , j = 1,...,n — r. (5)
J iY1 -Wn-r-iAi
Then the previous 1-forms are pointwise linearly independent in U and satisfy
d.M1 = 0, du>j e 0{u)1,...,Mj_1), for j = 2,...,n — r. (6)
Expressions (6) imply the local existence of a function lt such that
Wj = dl^ (7)
By construction of (see equation (5) for j = 1), we have that
iAjM1 = iYkM1 = 0, for j = 1,...,r, k = 2,...,n — r. (8)
From (7) and (8) it follows that is a first integral of the involutive distribution tD1, and in particular, of T>. The restriction of w2 to the submanifold defined by keeping lt constant is closed, because according to (6), e J(dl1). Therefore, there exists a function /2 such that, locally,
= dI2—Y1(I2)dI1. (9)
We can continue in this way until we have finally found a complete set of functionally independent first integrals [I1,.,In_r} of the distribution T>. More details and examples on the theory of solvable structures and its generalizations can be consulted in [8, 9, 10, 11, 12, 13] and the references therein.
3. Geometrical and analytical methods of reduction of order for ODEs.
In this section we first review the main aspects of the analytical methods of reduction of order of ODEs via Lie (point and generalized) symmetries [1, 2, 3, 4].
3.1 Symmetry methods for ODEs.
Let us consider an nth-order ordinary differential equation of the form
un = F(x,u,u1,...,un_1), (10)
where F is a smooth function defined on an open set M £ Mn+1 and
UJ = 0, J = 1.....n. (11)
In what follows, A e %(M) will be the restriction of the total derivative operator
= Tx + + - + — + + - (12)
to the submanifold defined by equation (10):
A = -T + "i^T + ^ + un-iT— + F 7T~—■ (13)
OX OU OUn-2 OUn-1
It can be checked that the graph of the (n — 1)th-order prolongation of any solution of (10) is an integral curve of the distribution Conversely, any integral curve of this distribution can be locally written as the graph of the (n — 1)th-order prolongation of a solution of (10) [14, ex. 1.1.2].
Following [2, sec. 3.4], the Lie point symmetries of (10) can be characterized as the vector
fields
d d
V = t(x, u) — + V(x, u) — (14)
such that
[v(n~1), A] = —AÇQA, (15)
where v(n_1) denotes the (n — 1)th-order prolongation of v (see [2, sec. 4.1] and [1, th. 2.36]).
It is well-known that the knowledge of a Lie point symmetry (14) of an nth-order ODE (10) leads to reducing the equation to an (n — 1)th-order ODE plus a quadrature. This can be done through canonical coordinates or differential invariants (see, for instance, [3, sec. 3.3.1 and sec 3.3.2]):
1. Canonical coordinates: let r = r(x, u) and s = s(x, u) be corresponding canonical coordinates for v satisfying v(r) = 0 and v(s) = 1. Then equation (10) reduces to an (n — 1)th-order ODE
=^ ^ 2.....(16)
drn~
. ds
where z = —.
dr
In the particular case when n = 1, then (16) can be written as
^ = G(r), (17)
which can be integrated by a single quadrature.
2. Differential invariants: The first step is to find two functionally independent invariants y = y(x, u) and m = m(x, u, u1) of the first-order prolongation of v, through the characteristic equations of v(1) (see [3, eq. (3.102)]). Then by successive derivations
„ m_dm = Mm) _ drnj_1 = A(rnj_1) . = 2 ^ ml-dy = A(y") ' mj_ dy = A(y) , J =2.....n 1 (18)
we obtain invariants for the nth-order prolongation of v. Moreover, it can be checked that the
invariants [y,m,m1,...,mn_1} are functionally independent, i.e., [y,m,m1,...,mn_1} is a
complete set of invariants for p(n). Then equation (10) can be written in terms of these invariants
as an (n — V)th-order ODE
A(y, m, m1,...,mn_1) = 0, (19)
where y is the independent variable and m is the dependent variable. If m = G(y; Ct, C2,..., Cn_1), where Ct, C2,..., Cn_1 e R, denotes the general solution of equation (19), then the general solution of (10) arises from the first-order ODE:
m(x, u, %) = G(y(x, u); C1, C2,...,Cn_1) (20)
which reduces to a quadrature because it admits v as a Lie point symmetry.
When, for n> 2, the infinitesimals and ^ of a vector field (14) are allowed to depend on derivatives of u with respect to x up to some order I < n — 1, we get an extension of the notion of symmetry, known in the literature with the name of generalized symmetries [1] (also higherorder symmetries [3] or dynamical symmetries [2]). A generalized vector field [1, def. 5.1]
d d v = %(x,u,ut, ...,Ui) — + r\(x,u,ut, ...,Uj)— (21)
can be prolonged in accordance with the prolongation formula [1, th. 2.36] and generalized symmetries can be characterized through the condition (15).
In the calculation and use of generalized symmetries it is very convenient to consider the evolutionary (or characteristic) form of the generalized vector field (21), which takes the form
= qb (22)
where Q = r\{x,u,u1,...,uf) — ^(x, i)u1 denotes the characteristic of (21). The
generalized vector field (21) is a generalized symmetry of equation (10) if and only if its evolutionary representative (22) is [1, prop. 5.5]. This permits to consider generalized symmetries of the form
Q
v = rj(x, u, u1,.,ui) —, l<n — 1, (23) whose prolongations take a particularly simple form:
V-t + Ti-.A" (rj) (24)
The determination of a generalized symmetry in the form (23) is done through the condition (15) or equivalently, through the invariance criterion [3, th. 3.5.1-1], which provides a symmetry determining equation for the infinitesimal rj (see [3, eq. (3.239)]). In general, it is quite complicated to find solutions for such determining equation. It is usual to try to find some particular solutions by some ad hoc ansatz, assuming that rj has a special dependency on one or more of its arguments.
The independent variable x is always a zeroth-order invariant of a generalized symmetry of the form (23). A system of higher-order invariants {w1, ..,wn~1} for v(n) can be determined by solving the characteristic system associated to (24), where x is considered as a constant (see [3, eq. (3.298)]).
The symmetry condition (15) implies that, for j = 1,...,n — 1, A(w->) is also an invariant
of v(n),
that can therefore be expressed in terms of the complete set of invariants {x, w1,..., wn in the form
A(wj) = GJ(x, w1, ...,wn~1), j = 1,...,n — 1. (25)
In this way we get a reduction of ODE (10) to a system of (n — 1) first-order ODEs:
^ = G*(x, w1.....w-'1),
X : (26) ^^ = Gn~1(x, w1.....wn~1).
The details about this procedure can be consulted, for instance, in [3, sec. 3.5.4] and [2]. As far as we are concerned, there are few references that actually present examples of equations admitting higher-order symmetries and use them to reduce the order of the equation. One of these examples, taken from [3], will be analyzed in the Section 4.
3.2 Geometrical methods of reduction for ODEs.
The problem of reducing or integrating the nth-order ordinary differential equation (10) can be formulated in terms of the geometric notions of symmetry and Frobenius integrability [15]. The vector field (13) associated with equation (10) generates a trivially involutive distribution V = {A), that by Frobenius Theorem [15, prop. 1.59 and th. 1.60] is completely integrable.
The (n — 1)th-order prolongation of a Lie (point or generalized) symmetry v defines a symmetry of the distribution V = {A) in the sense of Definition 1, because the vector field Y ■= v(n~1') satisfies relation (15).
If equation (10) admits an n-dimensional solvable symmetry algebra of Lie point or generalized symmetries, the procedure described in Theorem 1 can be used to find by quadratures a complete set {/1, ...,ln} of first integrals of equation (10), because such symmetry algebra is a particular case of a solvable structure (see also [9, prop. 5.5]).
In order to do that, since the symmetry algebra is solvable, we can choose a basis such that the (n — 1)th-order prolongations Y1,.,Yn satisfy (3). Let Ü = dx A du A ■■■ A dun_1 be the volume form and denote
A = iYi-iYniAn. (27)
Observe that A is the determinant formed by the coordinates of the vector fields A, Yt,..., Yn, which are pointwise linearly independent. The corresponding l-form in (5) for j = 1 becomes
«i = ^iY2-iYniAn. (28)
By Theorem 1, is closed and hence locally exact. A corresponding primitive arises by quadrature, and it is a first integral of A. Next, we construct the corresponding l-form in (5) for j = 2. The restriction of such l-form w2 to the submanifold defined by = Ct, where Ct e R, is closed, and hence, locally exact. This permits to determine a primitive
T2 = f2(x, u,ut ...,un_1; C-¡) (29)
by quadrature. Replacing Ct by l1(x, u, ut,... ,un_1) in (29), we get a function /2 satisfying (9), which is a first integral of A. Clearly the process is inductive, and it can be continued until we have calculated a complete system {lt,..., In} of first integrals for A.
Moreover, when more symmetries than the order of the equation are known, the following result can be really powerful, because it allows to obtain first integrals algebraically, without any kind of integration [9, prop. 5.6]:
Proposition 1. Let A e %(M) be the vector field associated with equation (10). Let £ be an involutive distribution containing A. Suppose that X, Ye Sym{£} are transversal to £ and that we can write Y = aX + Z for some Z e£. Then the function a is a (possibly trivial) first integral of
A.
As a natural consequence, the knowledge of more extra symmetries may provide several first integrals without integration [9, cor. 5.7]:
Proposition 2. Let A E %(M) be the vector field associated with equation (10) and assume that Xt, ...,Xj are independent, non-trivial symmetries of V — (A). If Y is an additional non-trivial symmetry of V such that Y — atXt + —+ ajXj + fiA, then ar, ...,aj are (possibly trivial) first integrals of A.
In Section 5, we will apply these geometrical tools and results to derive new strategies of integration of a third-order ODE that has been studied in [3] by analytical methods based on generalized symmetries.
4. The ODE and its general solution via the analytical method.
We consider the third-order equation
"3 +1)- (30)
This equation was introduced by G. W. Bluman and S. C. Anco in [3, eq. (3.245)] as an example of how to determine generalized symmetries and use them to reduce the equation.
The corresponding vector field A E %(M) associated to equation (30) becomes
A = Tx + + + + 6t\xt + (31)
OX UU UU-ft—2 U^ \ U^ / UU-ft— i
which is defined on the open set:
M = {(x, u, ut, u2) EM: 0}. (32)
In the cited reference, the authors prove that (30) admits seven generalized symmetries (of first order) given, in evolutionary form, by
_ 1 9 -1 9 _ 29 _245
— , — 7 „ , — XU, , — X U-i
U1 OU Ul ou ou ou
v5 — (9x2u2 — 12xuu1 + 4u2) ^, v6 — (%x — (33)
v7 — (3x2uf — 2xuul)—~.
ou
In addition, in [3] was also proved that equation (30) admits the following three independent Lie point symmetries:
d d d VB — ^, — U -, V10 — X-, (34)
which span a three-dimensional, solvable Lie algebra because the respective commutations relationships become
[v8, v9] — v8, [v8, vw] — [v9, v10] — 0. (35)
In [3, p. 179-181] the generalized symmetry v^ defined in (33) was used to reduce equation (30) to a system of two first-order ODEs. The first step is to calculate a complete set of second-order differential invariants of v1. As we said in Section 3, x is already a zeroth-order invariant. For the remaining ones, G. W. Bluman and S. C. Anco solved the characteristic equations for the
corresponding second-order prolongation (24) of v1. By choosing u1 as the independent variable, such characteristic system becomes [3, eq. (3.304)]:
' du -, Uo . w?
= 6x-§ + 4 —,
~ + 4-,
(36)
UJ U1
U1
u2 ■
After using symmetry methods for system (36), the authors found the following second-order invariants of v^:
w1 =
2xul + —, w2 =u —^ w1 — 2xu1 ■ (37)
u2 2u1
It can be checked that, in this case, A(w1) = A(w2) = 0. Therefore, according to (26), the corresponding reduced system of two first-order equations becomes
=o,
ax
2 (38)
=0.
dx
System (38) is trivial and its general solution is w1 = Ct, w2 = C2, where Ct, C2 e R. Substituting the expressions (37) and eliminating u2, they obtain the first-order ODE
2xul + (C2 — u)ul + ^ = 0. (39)
Solving this ODE yields the general solution of (30). Equation (39) can be written in explicit form as
ut = G(x, u; Ct, C2), (40)
and it inherits a Lie point symmetry from the first-order symmetry vt of (30),
~ Id , ., N
Vi = -r—. (41)
1 G(X'U;C1'C2) du v y
However, working with this symmetry is not really convenient, since the expression G(x, u; Ct, C2) requires to solve (39) as a cubic equation in u1. In order to avoid this difficulty, Bluman and Anco determined a new Lie point symmetry for equation (39):
* = Xfx + 2S (U — C2) Tu (42)
Applying the method of canonical coordinates, equation (39) is reduced to a quadrature. In particular, we can choose
^ U) = -ISf, (43)
U(x, u) = Inx,
so that
V = (44)
Writing now (39) in terms of (r, s), carrying out the quadrature and writing the resulting expression back to the original coordinates, the general solution of (39), and thus of (30), is obtained in implicit form:
u — c2 + exp (1h + C3), (45)
where H is a function such that
i
H'(Z) — ,2_3ZHSlv<&, where p(z) — (z3 + 3y/3C1(27C1 — 2z*) — 27C1)'. (46)
In the integration procedure that has been applied in this section, several analytical methods based on symmetries have been successively used. First, the generalized symmetry v^ given in (33) has been determined. Second, in order to find second-order differential invariants for v1, symmetry methods have been used to find a particular solution of the characteristic system (36). Luckily, in this example, the reduced system (38) can be trivially integrated; however, in general, additional symmetries might be necessary to solve the reduced system. Third, a new symmetry (42) has been determined in order to solve the first-order ODE (39). Finally, the method of canonical coordinates has been used to integrate equation (39). As a result, the general solution of the ODE has been obtained in (45), although it is expressed in implicit form and in terms of a primitive that cannot be explicitly evaluated (see equation (46)).
In the following section, we investigate if the application of geometrical methods to equation (30) can improve the results that have been obtained so far by using only analytical methods.
5. Integration by geometrical methods.
In this section we apply the geometric tools described in Section 3.2 with the aim of providing a more convenient expression for the general solution of equation (30):
"3 — +1)- <47>
As in Section 4, we consider the associated vector field (31),
A — T* + ^ + ■ + + 6t{xlT + 1)^Г-, (48)
OX OU OU-ft—2 \ U^ / OU-ft— i
defined on the open set M introduced in (32),
M — {(x, u, ut, u2) EM: 0}. (49)
In order to achieve our objective of obtaining the general solution of (47) by geometrical methods, we can use some of the Lie point and first-order symmetries of the ODE given in (33) and (34). The integration procedure that will be presented in this section does not need to calculate any differential invariants nor canonical coordinates at all.
First, we observe that among the symmetries (33) and (34) it is possible to select three pointwise independent symmetries satisfying the hypothesis of Theorem 1. This can be done, for instance, by considering the second-order prolongations of the symmetries (34) because, according to (35), they span a three-dimensional, solvable Lie algebra, which is a particular case of a solvable structure of the distribution V — (A). However, since for equation (47) we know an oversupply of symmetries, in the next section we will apply the theoretical results presented in Section 3.2, in order to obtain first integrals algebraically, without any kind of integration (see Proposition 1 and Corollary 1). This will be done by conveniently choosing the symmetries that will be used in the
integration process. With this aim, we choose the following symmetries (in the sense of Definition 1) of the distribution T>:
Y = v(2) = ± ^ = = du'
v (2) 5 , d , d
Y2 = vy9 ) = u— + u-L — + U2—,
^ du du1 du2
„_(2)_ d d d (50) y3 = v10 = xgx Ulgui 2u2gu2,
y = (2) = 13 2u2 3 6M|(2XM2+M1) 3 4 2 u\du u\ du1 M^ du2
It can be checked that the commutator relationships become [Y1, A] = [Y2, A] = [Y3, A] = [Y4, A] = 0, [Y!, Y2] = Y1, [Y1, Y3] = [Y1, Y4] = [Y2, Y3] = 0, [Y2, Y4] = —3Y4, [Y3, Y4] = 2Y4. (51)
5.1 Two first integrals without integration.
In this section we aim to apply the theoretical results presented in Section 3.2 in order to obtain two first integrals of A by an algebraic procedure, without any kind of integration.
It can be checked that the set [A, Yt, Y3, Y4} is linearly independent on the open set
V = [(x, u, ut, u2) e M: u2(3xu2 + %) ^ 0}. (52)
This implies that Y1, Y3, Y4 are pointwise linearly independent and transversal symmetries of the distribution V = {A) on V. Consequently, they can be used as the non-trivial symmetries Xt (i = 1, 2, 3) required in Corollary 1.
Since [A, Yt, Y3, Y4} is a basis of X(V), any additional non-trivial symmetry of V can be expressed in terms of Y1, Y3, Y4 and A. For instance, the symmetry Y2 given in (50) is a transversal symmetry of V that can be written as follows:
y _ _6xu1u2—2uu2+uj y _ 2y + u:[(2xu2+u1) y + 2xA (53)
2u2 2u2
As a direct consequence of Corollary 1 we conclude that the following functions, corresponding to the coefficients of Yt and Y4 in (53), are non-trivial first integrals of A:
u2 u4
I1 = 6xu1 — 2u + —, I2 = 2xul + —. (54)
u2 u2
These functions are defined on the open set
U = [(x, u, ut, u2) e R4: u1u2 ^ 0} Q M, (55)
and it can be checked that dlt A dl2 does not vanish on U, so and l2 are functionally independent first integrals of A on U.
Since the coefficient of Y3 in (53) is constant, the application of Corollary 1 by using the symmetries Yt, Y2, Y3, Y4 only provides two functionally independent first integrals of A. In order to complete the integration of the distribution V = {A), one more functionally independent first integral is required. Although Corollary 1 could be applied by using other sets of symmetries of the equation, there is not a criterion to know a priori which ones will produce non-trivial and functionally independent first integrals. In the worst case, it might happen that none of the admitted
symmetries gave rise to the remaining first integral. In this situation, an alternative strategy must be followed. In the next subsection we illustrate how Theorem 1 can be applied to overcome this possible obstacle.
5.2 A remaining first integral and the general solution of equation (47).
Besides the first integrals and l2 given in (54), one more functionally independent first integral of A is required in order to complete the integration of equation (47). In order to determine such first integral, we first observe that the distribution £ — {Y3, Y4, A) is involutive by commutator relationships (51). Moreover, since
A(I1)=Y3(I1)=Y4(I1) = 0, (56)
we conclude that is a first integral of £.
Consider the following local change of variables on the open set U defined in (55),
9: U - 9(U) (57)
(x,u,u1,u2) ^ (u,U!,U2,Ii).
By means of the push-forward by 9 (see [18, pg. 46]), the vector fields A, Y4,Y3 are expressed in terms of local coordinates (u, u2, Ii) as follows:
A — 9 A-u ± + u _L + uj(2uu2 + I1u2 + 5u21^^ A — - U1 + u2 +
3u z Su-L uf 3u2
Y3 - 9*Y3 - -uiyu- 2u^, (58)
^ _ _ 1 d 2u2 d 2U|(2UUz + I1U2 + 2U5) d
Y4 - Y4 - -7 — - "X— - ■
uij Su-l u^j du2
We can restrict each one of the previous vector fields to the submanifold defined by the level set — C1, where e E, by substituting by in (58). We keep denoting the restricted vector fields by A, Y3 and Y4 respectively.
By using (51), it can be checked that Y3 and Y4 span a 2-dimensional, solvable Lie algebra of symmetries of the distribution V — (A). In particular, they generate a solvable structure for V. In order to apply Theorem 1, we consider the non-zero 3-form fi — du A dux A du2 and construct the corresponding 1-form given by (5) for j — 1:
Y4 A ,2 x (59)
1 /„ , 2uu2+C1u2+3uf , uf , \
-r[2u2du +--2——-- du! —1 du2).
C1u2+uf V. u2 /
2uu2+C1u2+u
Theorem 1 ensures that (59) is closed and, therefore, locally exact. A primitive of and hence a first integral of A, can be obtained by quadratures:
Js — ln ((2uu2 + C1u2 + u2)u1), (60)
Thus, the following function
J3 — exp(-J3) — g +2u + c) ut (61)
is also a first integral of A. Substituting now Cx by in (61) and writing the obtained expression back in terms of (x, u, u2), we obtain a function I3 e (U) given by
I3 = 3xu^ +
1 u2
which is a first integral of A. It can be checked that ^, I2 (defined in (54)) and I3 are functionally independent on U, since dIx A dI2 A dI3 does not vanish on U.
Therefore, the general solution of ODE (47) can be implicitly defined by equations ^ = C1( I2 = C2, I3 = C3, where C1( C2, C3 G R:
,2
6xu — 2u + — = C1(
u2
2xu? + ^ =C2,
1 u2 ^
3xu^ + ^ =C3.
u2
(63)
In order to obtain a parametric expression for the general solution of ODE (47), we eliminate u2 from the last equation in (63) and choose ux =t as a parameter:
¡x(t) = ' u(t) =
c3t-c2 t3 ' -C1t2+4C3t-3C2 2t2
(64)
We depict the graphs of two of the solutions in Figures 1 and 2, obtained by setting different values to the integration constants.
Figure 1:
Cj = 10, C2 = 0,C3 = -1,0.5 < t < 10.
Figure 2:
Ci = -21, C2 = -5, C3 = 7,0.5 < t < 10.
6. Concluding remarks.
Different analytical and geometrical methods have been applied in the study of a third-order ODE for which abundant Lie point and generalized symmetries had been previously reported.
Regarding analytical methods, a generalized symmetry of the equation had been used in the previous literature to reduce the given ODE to a system of two first-order ODEs. In this
reduction process, additional symmetry methods had been necessary to determine differential invariants for the generalized symmetry. After solving the reduced system, it remains the problem of reconstruction of the solution for the original equation. Although theoretically this can be done by a quadrature, the difficulty of obtaining an explicit expression for the underlying symmetry (41) forced the search of a new symmetry (42). After application of the canonical coordinates method, the implicit general solution (45) was finally obtained. However, this expression involves a primitive that cannot be explicitly evaluated.
In this work we have shown that the application of geometrical methods greatly simplifies the integration of the given third-order ODE. Remarkably, two functionally independent first integrals of the equation have been calculated by simple algebraic manipulations, avoiding the use of differential invariants, canonical coordinates or any kind of integration. Moreover, a remaining first integral has been calculated by quadrature, as a primitive of a 1-form defined in an open set of a three-dimensional space. From the complete system of first integrals of the equation derived by using these geometrical tools we have obtained the general solution of the equation in parametric form (see equation (64)). The obtained solution is given in terms of simple rational expressions, greatly improving the solution (45) derived via the analytical procedures.
It can be concluded that the geometrical approach to integrating ODEs is a powerful alternative to the classical approach of differential invariants or canonical coordinates, specially when there are higher-order symmetries involved. An additional advantage that must be taken into account is that the geometrical methods allow us to use not only symmetries of the associated distribution, i.e., not only prolongations of Lie (point or generalized) symmetries of the equation. In a solvable structure, only the first element must be a symmetry of the associated distribution while, in general, the remaining vector fields are not symmetries of the equation. This fact greatly expands the strategies that can be followed to find exact solutions of differential equations.
7. Acknowledgements.
B. de la Flor acknowledges the financial support from the Ministry of Universities of Spain (FPU grant FPU21/01046). A. Ruiz and C. Muriel thank the financial support from Junta de Andalucía (Spain) by means of the project ProyExcel_00780. The authors also appreciate the funding support from Junta de Andalucía (Spain) to the research group FQM-377.
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