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A semi-algorithm to find elementary first order invariants of rational second order ordinary differential equations
Applied Mathematics and Computation 184 (2007) 2–11 www.elsevier.com/locate/amc
A semi-algorithm to find elementary first order invariants of rational second order ordinary differential equations J. Avellar a
a,b,*
, L.G.S. Duarte a, S.E.S. Duarte c, L.A.C.P. da Mota
a
Universidade do Estado do Rio de Janeiro, Instituto de Fı´sica, Depto. de Fı´sica Teo´rica, 20559-900 Rio de Janeiro, RJ, Brazil b Fundac¸a˜o de Apoio a` Escola Te´cnica, E.T.E. Juscelino Kubitschek, 21311-280 Rio de Janeiro, RJ, Brazil c Fundac¸a˜o de Apoio a` Escola Te´cnica, E.T.E. Visconde de Maua´, 20537-200 Rio de Janeiro, RJ, Brazil
Abstract Here we present a method to find elementary first integrals of rational second order ordinary differential equations (SOODEs) based on a Darboux type procedure [L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, A method to tackle first order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A: Math. Gen. 35 (2002) 3899–3910, L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, Analyzing the structure of the integrating factors for first order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A: Math. Gen. 35 (2002) 1001– 1006]. Apart from practical computational considerations, the method will be capable of telling us (up to a certain polynomial degree) if the SOODE has an elementary first integral and, in the positive case, finds it via quadratures. 2006 Elsevier Inc. All rights reserved. Keywords: Elementary first integrals; Semi-algorithm; Darboux; Lie symmetry
1. Introduction The differential equations (DEs) are the most widespread way to formulate the evolution of any given system in many scientific areas. Therefore, for the last three centuries, much effort has been made in trying to solve them. Broadly speaking, we may divide the approaches to solving ODEs in the ones that classify the ODE and the ones that do not (classificatory and non-classificatory methods). Up to the end of the nineteenth century, we only had many (unconnected) classificatory methods to try to deal with the solving of ODEs. Sophus Lie then introduced his method [1–3] that was meant to be general and try to solve any ODE, i.e., non-classificatory. Despite this appeal, the Lie approach had a shortcoming: namely, in order to deal with the ODE, one has to
*
Corresponding author. Address: Universidade do Estado do Rio de Janeiro, Instituto de Fı´sica, Depto. de Fı´sica Teo´rica, 20559-900 Rio de Janeiro, RJ, Brazil. E-mail addresses: [email protected] (J. Avellar), [email protected] (L.G.S. Duarte), [email protected] (S.E.S. Duarte), [email protected] (L.A.C.P. da Mota). 0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.06.017
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
3
know the symmetries of the given ODE. Unfortunately, this part of the procedure was not algorithmic (the classificatory approach is algorithmic by nature). So, for many decades, the Lie method was not put to much ‘‘practical’’ use since to ‘‘guess’’ the symmetries was considered to be as hard as guessing the solution to the ODE itself. In [4,5], an attempt was made to make this searching for the symmetries to the ODE practical and, consequently, make the Lie method more used. Even thought the attempt mentioned above was very successful, the procedures applied to find the symmetries were heuristic. So, a non-classificatory algorithmic approach was still missing. The first algorithmic approach applicable to solving first order ordinary differential equations (FOODEs) was made by Prelle and Singer [6]. The attractiveness of this approach (the PS method) lies not only in the fact that it is based on a totally different theoretical point of view but, also in the fact that, if the given FOODE has a solution in terms of elementary functions, the method guarantees that this solution will be found (though, in practice it can take a long time to do so). The original PS method was built around a system of two autonomous FOODEs of the form x_ ¼ P ðx; yÞ; y_ ¼ Pðx; yÞ with P and P in C[x, y] or, equivalently, the form y 0 = R(x, y), with R(x, y) a rational function of its arguments. The PS method has its limitations, for instance, it deals only with rational FOODEs. But, since it is so powerful in many respects, it has generated many extensions [7–14]. Nevertheless, all these extensions deal only with FOODEs. In particular, the second order ordinary differential equations (SOODEs) play a very important role in the physical sciences. So, with this in mind, some of us have produced [16] a PS-type approach to deal with SOODEs. This approach dealt with SOODEs that presented elementary1 solutions (with two elementary first order invariants). Here, we present a different approach that, besides dealing with a much broader class of SOODEs (those with at least one elementary first order invariant), does not depend on a conjecture about the general structure of the first order invariants. In Section 2, we present the state of the art up to the present paper. In Section 3, we introduce some important theoretical results for the building of the algorithm to find the integrating factor. In Section 4, we present the algorithm for finding the integrating factor with examples of its application. Finally, we conclude and point out some directions to further our work. 2. Earlier results In the paper [6], one can find an important result that, translated to the case of SOODEs of the form y 00 ¼
Mðx; y; y 0 Þ ¼ /ðx; y; y 0 Þ; N ðx; y; y 0 Þ
ð1Þ
where M and N are polynomials in (x, y, y 0 ), can be stated as Theorem 1. If the SOODE (1) has a first order invariant that can be written in terms of elementary functions, then it has one of the form: m X ci lnðwi Þ; ð2Þ I ¼ w0 þ i
where m is an integer and the w 0 s are algebraic functions2 of (x, y, y 0 ). The integrating factor for a SOODE of the form (1) is defined by Rð/ y 00 Þ ¼ where
1 2
d dx
dIðx; y; y 0 Þ ; dx
represents the total derivative with respect to x.
For a formal definition of elementary function, see [15]. For a formal definition of algebraic function, see [15].
ð3Þ
4
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
Below we will present some results and definitions (previously presented on [16]) that we will need. First let us remember that, on the solutions, dI ¼ I x dx þ I y dy þ I y 0 dy 0 ¼ 0. So, from Eq. (3), we have Rð/ dx dy 0 Þ ¼ I x dx þ I y dy þ I y 0 dy 0 ¼ dI ¼ 0:
ð4Þ
Since y 0 dx = dy, we have R½ð/ þ Sy 0 Þ dx S dy dy 0 ¼ dI ¼ 0;
ð5Þ
0
0
adding the null term Sy dx S dy, where S is a function of (x, y, y ). From Eq. (5), we have I x ¼ Rð/ þ Sy 0 Þ; I y ¼ RS;
ð6Þ
I y 0 ¼ R; that must satisfy the compatibility conditions. Thus, defining the differential operator D: D ox þ y 0 oy þ /oy 0 ;
ð7Þ
after a little algebra, that can be shown to be equivalent to D½R ¼ RðS þ /y 0 Þ;
ð8Þ
D½RS ¼ R/y :
ð9Þ
3. New theoretical results concerning the function S Let us start this section by stating a corollary to Theorem 1 concerning S and R. Corollary 1. If a SOODE of the form (1) has a first order elementary invariant then the integrating factor R for such an SOODE and the function S defined in the previous section can be written as algebraic functions of (x, y, y 0 ). Proof. Using Pm the above mentioned result by Prelle and Singer, there is always a first order invariant I ¼ w0 þ i ci lnðwi Þ for the SOODE. So we have, using Eq. (3), M y 00 ¼ I x þ y 0 I y þ y 00 I y 0 ) R ¼ I y 0 ; R ð10Þ N where Iu ouI. From Eq. (2), we have m X wiy 0 ci : I y 0 ¼ w0y 0 þ wi i
ð11Þ
Then I y 0 is an algebraic function of (x, y, y 0 ) and, by Eq. (10), so is R. From Eq. (6), one can see that Pm w w0y þ i ci wiiy Iy S¼ ¼ P w : I y 0 w0y 0 þ mi ci wi y0 i Therefore, S is also an algebraic function of (x, y, y 0 ).
ð12Þ
h
Besides that, working on Eqs. (8) and (9), we get [16]: D½S ¼ S 2 þ /y 0 S /y ¼
M ; N
ð13Þ
where M and N are given by 2
M ðNSÞ þ ðNM y 0 MN y 0 ÞS ðNM y MN y Þ; 2
NN :
ð14Þ ð15Þ
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
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Concerning Eq. (13) we can demonstrate the following theorem: Theorem 2. Consider the operator defined by DS MoS þ ND. If P is an eigenpolynomial of DS (i.e., DS[P] = kP, where k is a polynomial) that contains S, then P = 0 defines a particular solution of Eq. (13). Conversely, if P is a polynomial that contains S, such that P = 0 defines a particular solution of Eq. (13), then P is either an eigenpolynomial of DS or P is an absolute invariant of the Lie transformation group defined by DS. Proof. In order to demonstrate Theorem 2 we will, first, prove the following lemma: P i Lemma 1. If Eq. (13) ðD½S ¼ M=NÞ has an algebraic solution defined by i ai S ¼ 0, where the ai are 0 polynomials in (x, y, y ), then X ðND½ai S i þ Mai iS i1 Þ ¼ 0: ð16Þ i
P P Conversely, if i ðND½ai S i þ Mai iS i1 Þ ¼ 0 then i ai S i ¼ 0 defines an algebraic function (S) as a particular solution of Eq. (13). P Proof of Lemma 1. We begin by proving the first part of P the lemma. Let i ai S i ¼ 0 define an algebraic particular solution of Eq. (13). Applying the operator D on i ai S i ¼ 0, one gets: X X M ðD½ai S i þ ai iS i1 D½SÞ ¼ 0 ) D½ai S i þ ai iS i1 ¼ 0: N i i Multiplying this by N we get Eq. (16). Let us now prove the converse. Consider that Eq. (16) applies. Multiplying it by N and remembering that S obeys D½S ¼ M=N, one gets: , X X X M i i1 M i D½ai S þ ai iS ðD½ai S Þ ðai iS i1 Þ ¼ : ¼0 ) ð17Þ N N i i i P However, if we have an algebraic function (B) defined by i bi Bi ¼ 0, then D[B] can be found as follows: , " # X X X X i i i1 i D bi B ¼ 0 ) ðD½bi B þ bi iB D½BÞ ¼ 0 ) D½B ¼ ðD½bi B Þ ðbi iBi1 Þ: i
i
i
i
Therefore, one can see that Eq. (17) can be put in the form: D½S ¼
M ; N
where S is an algebraic function defined by
ð18Þ P
i ai S
i
¼ 0.
h
Now, using Lemma 1, we are going to demonstrate Theorem 2. Consider that P is an eigenpolynomial of DS that contains S. By definition, DS[P] = kP, where k is a polyP nomial. So, writing P ¼ i ai S i , where the ai’s are polynomials in (x, y, y 0 ), we have X ðND½ai S i þ Mai iS i1 Þ ¼ kP : ð19Þ i
P Over the algebraic function defined by P = 0, we get i ðND½ai S i þ Mai iS i1 Þ ¼ 0. This, using Lemma 1, imP plies that i ai S i ¼ 0 defines a particular solution to Eq. (13). P Conversely, consider that P is a polynomial that contains S, such that P ¼ i ai S i ¼ 0 defines a particular solution of Eq. (13). Again, via Lemma 1, we have that X ðND½ai S i þ Mai iS i1 Þ ¼ 0: ð20Þ i
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Consider now the following operator: Mðx1 ; x2 ; x3 Þ 2 O ¼ N ðx1 ; x2 ; x3 Þ ox1 þ x3 ox2 þ ox þ Mðx1 ; x2 ; x3 ; x4 Þox4 : ð21Þ N ðx1 ; x2 ; x3 Þ 3 P Applying O to a generic polynomial P ¼ i ci xi4 , where the ci’s are polynomials in (x1, x2, x3), one obtains: X N 2 ðx1 ; x2 ; x3 ÞD½ci xi4 þ Mðx1 ; x2 ; x3 ; x4 Þci ixi1 O½P ¼ ; ð22Þ 4 i 1 ;x2 ;x3 Þ o Þ. where D ðox1 þ x3 ox2 þ Mðx N ðx1 ;x2 ;x3 Þ x3
Since the terms multiplying the partial derivatives are polynomials, applying O to a polynomial will generate a polynomial. So, from Eq. (22), we have X ðN 2 ðx1 ; x2 ; x3 ÞD½ci xi4 þ Mðx1 ; x2 ; x3 ; x4 Þci ix4i1 Þ ¼ Q; ð23Þ i
where Q is a polynomial in (x1, x2, x3, x4). Note that the left-hand side P of Eqs. (23) and (20) are formally equivalent. Consider that the hypothesis of Theorem 2 apply, i.e., P ¼ i ci xi4 ¼ 0 defines an algebraic function x4(x1, x2, x3) that is a particular solution of D½x4 ¼ Mðx P1 ; x2 ; x3 ; x4 Þ=Nðx1 ; x2 ; x3 Þ. Lemma 1 implies then that, if we substitute the function x4 ¼ Root ofð i ci xi4 Þ into Eq. (23), we get 0 = Q. Therefore, P ¼ 0 ) Q ¼ 0. Since both P and Q are polynomials we have two possibilities: • Q is identically null – in that case, P is an absolute invariant of the Lie transformation group defined by the operator O. • P is a factor of Q – in that case, Q ¼ kP, where k is a polynomial in (x1, x2, x3, x4) and Q is a relative invariant of the Lie transformation group defined by the operator O. This completes the proof of Theorem 2.
h
From the results above we may finally conclude: Corollary 2. If the SOODE (1) has an elementary first integral, then there is a polynomial P (containing S) that is either an eigenpolynomial of DS or is an absolute invariant of the Lie transformation group defined by DS. Proof. From Corollary 1, since Eq. (1) has a first order elementary invariant, there is an algebraic function S that satisfies Eq. (13). So, by definition, there is a polynomial P (that contains S) such that P = 0 defines S. From Theorem 2, this implies that the polynomial P is either an eigenpolynomial of DS or is an absolute invariant of the Lie transformation group defined by DS. h These theoretical results provide us an algorithm to find S. Briefly, in words, what we have to do is the following: • Find the eigenpolynomials of the DS operator containing S. Let us call them Pi. • In order to find S, we have to choose one of those Pi (let us call it P) that contains S and solve the equation P = 0 for S. • If you succeed in doing the above steps, you would have found S for the SOODE you are dealing with. The importance of these results and the above sketched method is that, as we shall briefly see, the finding of the algebraic function S will allow us to produce a semi-algorithmic method to find the integrating factor (and, consequently, the first order invariant) for SOODEs of the type described in Eq. (1).
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
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4. Finding the integrating factor and the first integral In what follows, we are going to demonstrate a result concerning the general structure of R. Let us state that result as a theorem. Theorem 3. Consider a SOODE of the form (1), where M and N are polynomials in (x, y, y 0 ), that presents an elementary first order invariant I. Then the integrating factor R for this SOODE can be written as Y R¼ f ni ; ð24Þ i i where fi are irreducible polynomials in (x, y, y 0 , S), which are eigenpolynomials of the operator D ND or are factors of N and ni are non-zero rational numbers. Proof. Suppose that the hypothesis of the theorem is satisfied. Re-writing Eq. (4): R ½ðM þ SNy 0 Þ dx SN dy N dy 0 ¼ dI ¼ 0: ð25Þ N For the sake of simplicity, let us establish some notation: M ðM þ y 0 NSÞ, N ðNSÞ and R NR . We can write I x ¼ RM and I y ¼ RN and imposing the compatibility condition Ixy = Iyx, we have oy ðRMÞ ¼ ox ðRN Þ. Expanding this D½R ¼ ðM y þ N x Þ; Ry M þ Rx N ¼ RðM y þ N x Þ ) ð26Þ R where D N ox þ Moy . Since M and N are polynomials in (x, y, S),3 Eq. (26) is formally equivalent to the equation which establish the conditions for the theorem by Prelle and Singer [6] with the extension by Shtokhamer [7].4 More formally, consider the FOODE defined by e M ð27Þ y0 ¼ ; e N e and N e are the polynomials obtained by replacing y 0 by a constant k in M and N respectively. Then where M we can write the solution for the FOODE (27) as eI ¼ C, where eI is defined by replacing y 0 (by the constant k) on I. By hypothesis, I is elementary and, consequently, eI is also elementary. Therefore, by the theorem due to 0 e Prelle and Singer [6] and the Shtokhamer’s extension Q [7], the integrating factor R (defined by replacing y (by the constant k) on R) for Eq. (27) can be written as i f~ ni i , where f~ i are irreducible polynomials in (x, y, S). We could proceed analogously for the other two possible pairings of variables (i.e., (x, y 0 ) and (y, y 0 )). More explicitly, in the reasoning leading to Eq. (26), we could have imposed the compatibility condition I xy 0 ¼ I y 0 x and, in the same fashion we did above for the pairing (x, y), conclude that the integrating factor can be written as a product of irreducible polynomials in (x, y 0 , S). The same can be done for the pairing (y, y 0 ). So, R we can con0 , and N is a clude that R can be written as a product of irreducible polynomials in (x, y, y , S). Since R N Q polynomial in (x, y, y 0 ), finally we have that R ¼ i fini , where fi are irreducible polynomials in (x, y, y 0 , S) and ni are non-zero rational numbers. Let us now prove that the fi are eigenpolynomials of the operator D or factors of N. From Eq. (8) and remembering that R NR , we have D½RN NM y 0 MN y 0 ¼ S RN N2 multiplying both sides of (28) by N2, we get D½R N2 þ ND½N ¼ N 2 S ðNM y 0 MN y 0 Þ R 3
ð28Þ
ð29Þ
Note that M and N are polynomials in y 0 as well. P Notice that, since S is an algebraic function of (x, y, y 0 ), it is a root of a polynomial equation of the form i pi S i ¼ 0, where pi are 0 polynomials in (x, y, y ). Therefore, its derivatives can be expressed in terms of rational powers of itself. So, one may consider that would be dealing with a Shtokhamer extension with U = S. 4
8
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
and finally, D½R ¼ ND½N N 2 S ðNM y 0 MN y 0 Þ: ð30Þ R Since the right-hand side of (30) is a polynomial in (x, y, y 0 , S), so is N D½R . Using (24) this implies that R D½R X D½fi ¼ ni N ð31Þ N fi R i N
is a polynomial. Since fi are irreducible (independent) polynomials, we may conclude that, either fi jD½fi or fi is a factor of N. h In order to obtain the integrating factor R, we are going to use Eq. (8). Dividing it by R, we get NM y 0 MN y 0 D½R=R ¼ ðS þ /y 0 Þ ¼ S þ : N2
ð32Þ
Multiplying both sides of the equation above by N2, one has N 2 D½R=R ¼ ðSN 2 þ NM y 0 MN y 0 Þ ) N D½R=R ¼ ðSN 2 þ NM y 0 MN y 0 Þ:
ð33Þ
Due to the results of Theorem 3, Eq. (33) above can be solved in the same manner as in the methods inspired on the Prelle–Singer procedure. Once R is found as a function of (x, y, y 0 , S), we can substitute the known S (see Section 3) and, finally, from Eq. (6), find the first order invariant via quadratures: Z Z Z o 0 0 0 Rð/ þ Sy Þ dx dy Iðx; y; y Þ ¼ Rð/ þ Sy Þ dx RS þ oy Z Z Z Z o o 0 0 Rð/ þ Sy Þ dx dy dy 0 : Rð/ þ Sy Þ dx Rþ 0 RS þ ð34Þ oy oy That concludes the presentation of our proposed approach which is a semi-algorithm procedure to reduce SOODEs of the form of Eq. (1). 4.1. Example Here we are going to briefly introduce an example of a non-trivial5 SOODE that illustrates the results displayed above. Consider the SOODE: 2
y 00 ¼
3
ðy 0 Þ þ 2ðy 0 Þ 1 2y 0 x ðy 0 Þx ðy 0 Þy 1 þ ðy 0 Þ2
ð35Þ
:
For this SOODE, the operator D becomes: D ¼ ðy 02 þ 2y 03 1 2y 0 Þoy 0 þ ðx þ y 0 x þ y 0 y þ 1 y 02 Þy 0 oy þ ðx þ y 0 x þ y 0 y þ 1 y 02 Þox ;
ð36Þ
and the operator DS is DS ¼ ND þ Mos ; 2
0
ð37Þ 2 02
02
03
2
2 2
2
2 02
2 04
02
04
03
where M ¼ 2s xy y þ 2s y xy þ 7sy x þ 4sy x þ s þ s x þ 2s x 2s y þ s y sx þ 4sy 2sy þ sy þ 4sy y 2s2 y 0 2 x þ s2 y 0 2 y 2 2s2 y 0 3 y þ 2s2 y 0 x2 þ 2sy 0 x 2s y 0 þ s2 y 0 2 x2 2s2 y 0 3 x þ 2s2 y 0 y 2y 0 2 þ y 0 3 þ 2y 0 4 þ sy 0 2 y þ 2s2 y 0 x. The polynomial P ¼ ðx þ y 0 x þ y 0 y þ 1 y 0 2 ÞS þ 1 y 0 2 is an eigenpolynomial of the DS operator. So, P = 0 defines S as S¼
5
1 þ y 0 2 : x þ y0x þ y0y þ 1 y02
For instance, the MAPLE package, release 7, could not reduce it.
ð38Þ
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
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In order to calculate the integrating factor R, we need (see Eqs. (31) and (32)) to find the eigenpolynomials of the D operator. These are found to be (up to the first degree): f1 ¼ 1 þ y 0 ; f2 ¼ 1 y 0 ; f3 ¼ 1 þ 2y 0 and, for this particular case, these are sufficient (together with N) to build the integrating factor R. Combining Eqs. (31) and (32) and solving for the ni’s we have that {n1 = 3/2, n2 = 3/2, n3 = 0}. So, R is found to be6: R¼
x þ y0x þ y0y þ 1 y02 ððy 0 1Þð1 þ y 0 ÞÞ
3=2
ð39Þ
:
This leads to the first order invariant given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x þ y þ y0x I ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ lnðy 0 þ 1 þ y 0 2 Þ: 1 þ y 0 2
ð40Þ
5. Computational considerations Here, we are going to introduce an alternative way of calculating the integrating factor for a given SOODE. Why do we call this section computational considerations? The reason is that the material contained in this section is not applicable to all SOODEs7 of the form of Eq. (1) and, indeed, we do not have a criteria of applicability. But, on the other hand, the method we are about to present is ‘‘less costly’’ computationally, therefore could be useful in practical applications. In the following, we will be using some results from the Lie theory. Let us show a relation between the function S and a symmetry of the SOODE (1). Making the following transformation: S¼
D½g : g
ð41Þ
Eq. (13) becomes D2 ½g ¼ /y 0 D½g þ /y g:
ð42Þ
From Lie theory we can see that Eq. (42) represents the condition for a SOODE (1) to have a symmetry [0,g]. So, from (41) we can find a symmetry given by R g ¼ e S dx : ð43Þ Looking at Eq. (5) we can infer that R is also an integrating factor for the auxiliary FOODE defined by dy 0 ¼ S; dy
ð44Þ
where x is regarded as a parameter. Besides, for this FOODE, [g, D[g]] is a point symmetry. So, R is given by R¼
1 : gS þ D½g
ð45Þ
If in (45) we use g defined by (43) we would get a singular R. However, Eq. (42) has another solution independent from g given by
3=2 3=2
Note that R is formed by the product f1n1 f2n2 f3n3 N 1 ¼ f1 f2 N 1 . Note that the method presented on the previous section is a general semi-algorithmic approach to deal with SOODEs of the form (1) that present an elementary first order invariant. 6 7
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J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
g¼g
Z
e
R
/y 0 dx
g2
ð46Þ
dx:
Using this g in (45) we get R and, once this is done, we can calculate the first integral I by using simple quadratures. R Actually, trying to solve Eq. (43) could be complex. The operator S dx that appears on Eq. (43) is meant 8 to be the inverse of operator D (defined on (13)). As one might expect, actually finding g from (43) is trick. It could be impossible to integrate (43) and that is the reason why this shortcut is not applicable in general. 5.1. Example Consider the SOODE y 00 ¼
2y 0 2 x2 2y 0 2 2xy 0 y y 2 x4 þ 2y 2 x2 y 2 : 2yðx2 1Þ
ð47Þ
Constructing the DS operator we get that 2
y 2 ðx2 1Þ S 2 y 2 y 0 þ 2Sy 0 y
pffiffiffiffiffiffiffiffiffiffiffiffiffi is an eigenpolynomial of it. Then S is given by S ¼ y 0 =y þ x2 1 and g, g are respectively pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffipffiffiffiffiffiffi xpffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi R pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi x1 xþ1 e x 1 2 1 pffiffiffiffiffiffiffi 2 dx x þ x xþ x 1 xþ x2 1 pffiffiffiffiffiffi pffiffiffiffiffiffi g¼ ; g¼ : 2 2 x x 1 x x 1 ye 2 ye 2 R can be written as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi y x þ x2 1 R ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi x x2 1 x 2 1e 2
ð48Þ
ð49Þ
ð50Þ
leading to the following first integral: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi x þ x2 1ð2y 0 y þ x2 1y 2 Þ I¼ : pffiffiffiffiffiffiffiffiffiffiffiffiffi xpffiffiffiffiffiffi x2 1 x 2 1e 2
ð51Þ
6. Conclusion In [16], some of us have developed a method, based on a conjecture, to deal with SOODEs that presented an elementary solution (possessing two elementary first order invariants). In that same paper, a function S was introduced to transform the Pfaffian equation related to the particular SOODE under consideration into a 1-form proportional to differential of the first order differential invariant. That function S was instrumental in finding the integrating factor for the SOODE. Here, in the present paper, we have introduced many theoretical results concerning that function S and we have presented a method to calculate it via a Darboux-type procedure. Here also the function S was used to produce the integrating factor. We have demonstrated general results about the structure of the integrating factor R and that we can calculate it using a procedure very similar to the one inspired by the original work by Prelle and Singer [6] (applicable for FOODEs). These results assure that we have a semi-algorithmic procedure to deal with rational SOODEs presenting at least one elementary first order invariant. Consequently, we can cover a much broader class of SOODEs than before [16].
8
Actually, this meaning for the operator
R
is true for the whole of the present section.
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In the above section, we have introduced an alternative way to calculate R from the knowledge of S for restricted cases. The motivation for that is that the general method can be computationally very demanding. We are searching for better algorithms for the many steps of the method presented here in order to make it computationally more efficient. We are also working on a full computational implementation of the method as it stands today. References [1] H. Stephani, Differential equations: their solution using symmetries, in: M.A.H. MacCallum (Ed.), Cambridge University Press, New York, London, 1989. [2] G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Applied Mathematical Sciences, vol. 81, Springer Verlag, 1989. [3] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, 1986. [4] E.S. Cheb-Terrab, L.G.S. Duarte, L.A.C.P. da Mota, Computer algebra solving of first order ODEs using symmetry methods, Comput. Phys. Commun. 101 (1997) 254. [5] E.S. Cheb-Terrab, L.G.S. Duarte, L.A.C.P. da Mota, Computer algebra solving of second order ODEs using symmetry methods, Comput. Phys. Commun. 108 (1998) 90. [6] M. Prelle, M. Singer, Elementary first integral of differential equations, Trans. Am. Math. Soc. 279 (1983) 215. [7] R. Shtokhamer, Solving first order differential equations using the Prelle–Singer algorithm, Technical report 88-09, Center for Mathematical Computation, University of Delaware, 1988. [8] C.B. Collins, Algebraic invariants curves of polynomial vector fields in the plane, Preprint. Canada: University of Waterloo, 1993.; C.B. Collins, Quadratic vector fields possessing a centre, Preprint. Canada: University of Waterloo, 1993. [9] C. Christopher, Liouvillian first integrals of second order polynomial differential equations, Electron. J. Differen. Equat. (49) (1999) 7, electronic. [10] C. Christopher, J. Llibre, Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. Differen. Equat. 16 (1) (2000) 5–19. [11] J. Llibre, Integrability of polynomial differential systems, in: A. Can˜ada, P. Dra´bek, A. Fonda (Eds.), Handbook of Differential equations, Ordinary Differential Equations, vol. 1, Elsevier B.V., 2004, pp. 437–531 (Chapter 5). [12] L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, A method to tackle first order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A: Math. Gen. 35 (2002) 3899–3910. [13] L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, Analyzing the structure of the integrating factors for first order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A: Math. Gen. 35 (2002) 1001–1006. [14] L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, J.F.E. Skea, Extension of the Prelle–Singer method and a MAPLE implementation, Comput. Phys. Commun., Holanda 144 (1) (2002) 46–62. [15] J.H. Davenport, Y. Siret, E. Tournier, Computer Algebra: Systems and Algorithms for Algebraic Computation, Academic Press, Great Britain, 1993. [16] L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, J.E.F. Skea, Solving second order ordinary differential equations by extending the Prelle–Singer method, J. Phys. A: Math. Gen. 34 (2001) 3015–3024.
A semi-algorithm to find elementary first order invariants of rational second order ordinary differential equations J. Avellar a
a,b,*
, L.G.S. Duarte a, S.E.S. Duarte c, L.A.C.P. da Mota
a
Universidade do Estado do Rio de Janeiro, Instituto de Fı´sica, Depto. de Fı´sica Teo´rica, 20559-900 Rio de Janeiro, RJ, Brazil b Fundac¸a˜o de Apoio a` Escola Te´cnica, E.T.E. Juscelino Kubitschek, 21311-280 Rio de Janeiro, RJ, Brazil c Fundac¸a˜o de Apoio a` Escola Te´cnica, E.T.E. Visconde de Maua´, 20537-200 Rio de Janeiro, RJ, Brazil
Abstract Here we present a method to find elementary first integrals of rational second order ordinary differential equations (SOODEs) based on a Darboux type procedure [L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, A method to tackle first order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A: Math. Gen. 35 (2002) 3899–3910, L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, Analyzing the structure of the integrating factors for first order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A: Math. Gen. 35 (2002) 1001– 1006]. Apart from practical computational considerations, the method will be capable of telling us (up to a certain polynomial degree) if the SOODE has an elementary first integral and, in the positive case, finds it via quadratures. 2006 Elsevier Inc. All rights reserved. Keywords: Elementary first integrals; Semi-algorithm; Darboux; Lie symmetry
1. Introduction The differential equations (DEs) are the most widespread way to formulate the evolution of any given system in many scientific areas. Therefore, for the last three centuries, much effort has been made in trying to solve them. Broadly speaking, we may divide the approaches to solving ODEs in the ones that classify the ODE and the ones that do not (classificatory and non-classificatory methods). Up to the end of the nineteenth century, we only had many (unconnected) classificatory methods to try to deal with the solving of ODEs. Sophus Lie then introduced his method [1–3] that was meant to be general and try to solve any ODE, i.e., non-classificatory. Despite this appeal, the Lie approach had a shortcoming: namely, in order to deal with the ODE, one has to
*
Corresponding author. Address: Universidade do Estado do Rio de Janeiro, Instituto de Fı´sica, Depto. de Fı´sica Teo´rica, 20559-900 Rio de Janeiro, RJ, Brazil. E-mail addresses: [email protected] (J. Avellar), [email protected] (L.G.S. Duarte), [email protected] (S.E.S. Duarte), [email protected] (L.A.C.P. da Mota). 0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.06.017
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
3
know the symmetries of the given ODE. Unfortunately, this part of the procedure was not algorithmic (the classificatory approach is algorithmic by nature). So, for many decades, the Lie method was not put to much ‘‘practical’’ use since to ‘‘guess’’ the symmetries was considered to be as hard as guessing the solution to the ODE itself. In [4,5], an attempt was made to make this searching for the symmetries to the ODE practical and, consequently, make the Lie method more used. Even thought the attempt mentioned above was very successful, the procedures applied to find the symmetries were heuristic. So, a non-classificatory algorithmic approach was still missing. The first algorithmic approach applicable to solving first order ordinary differential equations (FOODEs) was made by Prelle and Singer [6]. The attractiveness of this approach (the PS method) lies not only in the fact that it is based on a totally different theoretical point of view but, also in the fact that, if the given FOODE has a solution in terms of elementary functions, the method guarantees that this solution will be found (though, in practice it can take a long time to do so). The original PS method was built around a system of two autonomous FOODEs of the form x_ ¼ P ðx; yÞ; y_ ¼ Pðx; yÞ with P and P in C[x, y] or, equivalently, the form y 0 = R(x, y), with R(x, y) a rational function of its arguments. The PS method has its limitations, for instance, it deals only with rational FOODEs. But, since it is so powerful in many respects, it has generated many extensions [7–14]. Nevertheless, all these extensions deal only with FOODEs. In particular, the second order ordinary differential equations (SOODEs) play a very important role in the physical sciences. So, with this in mind, some of us have produced [16] a PS-type approach to deal with SOODEs. This approach dealt with SOODEs that presented elementary1 solutions (with two elementary first order invariants). Here, we present a different approach that, besides dealing with a much broader class of SOODEs (those with at least one elementary first order invariant), does not depend on a conjecture about the general structure of the first order invariants. In Section 2, we present the state of the art up to the present paper. In Section 3, we introduce some important theoretical results for the building of the algorithm to find the integrating factor. In Section 4, we present the algorithm for finding the integrating factor with examples of its application. Finally, we conclude and point out some directions to further our work. 2. Earlier results In the paper [6], one can find an important result that, translated to the case of SOODEs of the form y 00 ¼
Mðx; y; y 0 Þ ¼ /ðx; y; y 0 Þ; N ðx; y; y 0 Þ
ð1Þ
where M and N are polynomials in (x, y, y 0 ), can be stated as Theorem 1. If the SOODE (1) has a first order invariant that can be written in terms of elementary functions, then it has one of the form: m X ci lnðwi Þ; ð2Þ I ¼ w0 þ i
where m is an integer and the w 0 s are algebraic functions2 of (x, y, y 0 ). The integrating factor for a SOODE of the form (1) is defined by Rð/ y 00 Þ ¼ where
1 2
d dx
dIðx; y; y 0 Þ ; dx
represents the total derivative with respect to x.
For a formal definition of elementary function, see [15]. For a formal definition of algebraic function, see [15].
ð3Þ
4
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
Below we will present some results and definitions (previously presented on [16]) that we will need. First let us remember that, on the solutions, dI ¼ I x dx þ I y dy þ I y 0 dy 0 ¼ 0. So, from Eq. (3), we have Rð/ dx dy 0 Þ ¼ I x dx þ I y dy þ I y 0 dy 0 ¼ dI ¼ 0:
ð4Þ
Since y 0 dx = dy, we have R½ð/ þ Sy 0 Þ dx S dy dy 0 ¼ dI ¼ 0;
ð5Þ
0
0
adding the null term Sy dx S dy, where S is a function of (x, y, y ). From Eq. (5), we have I x ¼ Rð/ þ Sy 0 Þ; I y ¼ RS;
ð6Þ
I y 0 ¼ R; that must satisfy the compatibility conditions. Thus, defining the differential operator D: D ox þ y 0 oy þ /oy 0 ;
ð7Þ
after a little algebra, that can be shown to be equivalent to D½R ¼ RðS þ /y 0 Þ;
ð8Þ
D½RS ¼ R/y :
ð9Þ
3. New theoretical results concerning the function S Let us start this section by stating a corollary to Theorem 1 concerning S and R. Corollary 1. If a SOODE of the form (1) has a first order elementary invariant then the integrating factor R for such an SOODE and the function S defined in the previous section can be written as algebraic functions of (x, y, y 0 ). Proof. Using Pm the above mentioned result by Prelle and Singer, there is always a first order invariant I ¼ w0 þ i ci lnðwi Þ for the SOODE. So we have, using Eq. (3), M y 00 ¼ I x þ y 0 I y þ y 00 I y 0 ) R ¼ I y 0 ; R ð10Þ N where Iu ouI. From Eq. (2), we have m X wiy 0 ci : I y 0 ¼ w0y 0 þ wi i
ð11Þ
Then I y 0 is an algebraic function of (x, y, y 0 ) and, by Eq. (10), so is R. From Eq. (6), one can see that Pm w w0y þ i ci wiiy Iy S¼ ¼ P w : I y 0 w0y 0 þ mi ci wi y0 i Therefore, S is also an algebraic function of (x, y, y 0 ).
ð12Þ
h
Besides that, working on Eqs. (8) and (9), we get [16]: D½S ¼ S 2 þ /y 0 S /y ¼
M ; N
ð13Þ
where M and N are given by 2
M ðNSÞ þ ðNM y 0 MN y 0 ÞS ðNM y MN y Þ; 2
NN :
ð14Þ ð15Þ
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
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Concerning Eq. (13) we can demonstrate the following theorem: Theorem 2. Consider the operator defined by DS MoS þ ND. If P is an eigenpolynomial of DS (i.e., DS[P] = kP, where k is a polynomial) that contains S, then P = 0 defines a particular solution of Eq. (13). Conversely, if P is a polynomial that contains S, such that P = 0 defines a particular solution of Eq. (13), then P is either an eigenpolynomial of DS or P is an absolute invariant of the Lie transformation group defined by DS. Proof. In order to demonstrate Theorem 2 we will, first, prove the following lemma: P i Lemma 1. If Eq. (13) ðD½S ¼ M=NÞ has an algebraic solution defined by i ai S ¼ 0, where the ai are 0 polynomials in (x, y, y ), then X ðND½ai S i þ Mai iS i1 Þ ¼ 0: ð16Þ i
P P Conversely, if i ðND½ai S i þ Mai iS i1 Þ ¼ 0 then i ai S i ¼ 0 defines an algebraic function (S) as a particular solution of Eq. (13). P Proof of Lemma 1. We begin by proving the first part of P the lemma. Let i ai S i ¼ 0 define an algebraic particular solution of Eq. (13). Applying the operator D on i ai S i ¼ 0, one gets: X X M ðD½ai S i þ ai iS i1 D½SÞ ¼ 0 ) D½ai S i þ ai iS i1 ¼ 0: N i i Multiplying this by N we get Eq. (16). Let us now prove the converse. Consider that Eq. (16) applies. Multiplying it by N and remembering that S obeys D½S ¼ M=N, one gets: , X X X M i i1 M i D½ai S þ ai iS ðD½ai S Þ ðai iS i1 Þ ¼ : ¼0 ) ð17Þ N N i i i P However, if we have an algebraic function (B) defined by i bi Bi ¼ 0, then D[B] can be found as follows: , " # X X X X i i i1 i D bi B ¼ 0 ) ðD½bi B þ bi iB D½BÞ ¼ 0 ) D½B ¼ ðD½bi B Þ ðbi iBi1 Þ: i
i
i
i
Therefore, one can see that Eq. (17) can be put in the form: D½S ¼
M ; N
where S is an algebraic function defined by
ð18Þ P
i ai S
i
¼ 0.
h
Now, using Lemma 1, we are going to demonstrate Theorem 2. Consider that P is an eigenpolynomial of DS that contains S. By definition, DS[P] = kP, where k is a polyP nomial. So, writing P ¼ i ai S i , where the ai’s are polynomials in (x, y, y 0 ), we have X ðND½ai S i þ Mai iS i1 Þ ¼ kP : ð19Þ i
P Over the algebraic function defined by P = 0, we get i ðND½ai S i þ Mai iS i1 Þ ¼ 0. This, using Lemma 1, imP plies that i ai S i ¼ 0 defines a particular solution to Eq. (13). P Conversely, consider that P is a polynomial that contains S, such that P ¼ i ai S i ¼ 0 defines a particular solution of Eq. (13). Again, via Lemma 1, we have that X ðND½ai S i þ Mai iS i1 Þ ¼ 0: ð20Þ i
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Consider now the following operator: Mðx1 ; x2 ; x3 Þ 2 O ¼ N ðx1 ; x2 ; x3 Þ ox1 þ x3 ox2 þ ox þ Mðx1 ; x2 ; x3 ; x4 Þox4 : ð21Þ N ðx1 ; x2 ; x3 Þ 3 P Applying O to a generic polynomial P ¼ i ci xi4 , where the ci’s are polynomials in (x1, x2, x3), one obtains: X N 2 ðx1 ; x2 ; x3 ÞD½ci xi4 þ Mðx1 ; x2 ; x3 ; x4 Þci ixi1 O½P ¼ ; ð22Þ 4 i 1 ;x2 ;x3 Þ o Þ. where D ðox1 þ x3 ox2 þ Mðx N ðx1 ;x2 ;x3 Þ x3
Since the terms multiplying the partial derivatives are polynomials, applying O to a polynomial will generate a polynomial. So, from Eq. (22), we have X ðN 2 ðx1 ; x2 ; x3 ÞD½ci xi4 þ Mðx1 ; x2 ; x3 ; x4 Þci ix4i1 Þ ¼ Q; ð23Þ i
where Q is a polynomial in (x1, x2, x3, x4). Note that the left-hand side P of Eqs. (23) and (20) are formally equivalent. Consider that the hypothesis of Theorem 2 apply, i.e., P ¼ i ci xi4 ¼ 0 defines an algebraic function x4(x1, x2, x3) that is a particular solution of D½x4 ¼ Mðx P1 ; x2 ; x3 ; x4 Þ=Nðx1 ; x2 ; x3 Þ. Lemma 1 implies then that, if we substitute the function x4 ¼ Root ofð i ci xi4 Þ into Eq. (23), we get 0 = Q. Therefore, P ¼ 0 ) Q ¼ 0. Since both P and Q are polynomials we have two possibilities: • Q is identically null – in that case, P is an absolute invariant of the Lie transformation group defined by the operator O. • P is a factor of Q – in that case, Q ¼ kP, where k is a polynomial in (x1, x2, x3, x4) and Q is a relative invariant of the Lie transformation group defined by the operator O. This completes the proof of Theorem 2.
h
From the results above we may finally conclude: Corollary 2. If the SOODE (1) has an elementary first integral, then there is a polynomial P (containing S) that is either an eigenpolynomial of DS or is an absolute invariant of the Lie transformation group defined by DS. Proof. From Corollary 1, since Eq. (1) has a first order elementary invariant, there is an algebraic function S that satisfies Eq. (13). So, by definition, there is a polynomial P (that contains S) such that P = 0 defines S. From Theorem 2, this implies that the polynomial P is either an eigenpolynomial of DS or is an absolute invariant of the Lie transformation group defined by DS. h These theoretical results provide us an algorithm to find S. Briefly, in words, what we have to do is the following: • Find the eigenpolynomials of the DS operator containing S. Let us call them Pi. • In order to find S, we have to choose one of those Pi (let us call it P) that contains S and solve the equation P = 0 for S. • If you succeed in doing the above steps, you would have found S for the SOODE you are dealing with. The importance of these results and the above sketched method is that, as we shall briefly see, the finding of the algebraic function S will allow us to produce a semi-algorithmic method to find the integrating factor (and, consequently, the first order invariant) for SOODEs of the type described in Eq. (1).
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
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4. Finding the integrating factor and the first integral In what follows, we are going to demonstrate a result concerning the general structure of R. Let us state that result as a theorem. Theorem 3. Consider a SOODE of the form (1), where M and N are polynomials in (x, y, y 0 ), that presents an elementary first order invariant I. Then the integrating factor R for this SOODE can be written as Y R¼ f ni ; ð24Þ i i where fi are irreducible polynomials in (x, y, y 0 , S), which are eigenpolynomials of the operator D ND or are factors of N and ni are non-zero rational numbers. Proof. Suppose that the hypothesis of the theorem is satisfied. Re-writing Eq. (4): R ½ðM þ SNy 0 Þ dx SN dy N dy 0 ¼ dI ¼ 0: ð25Þ N For the sake of simplicity, let us establish some notation: M ðM þ y 0 NSÞ, N ðNSÞ and R NR . We can write I x ¼ RM and I y ¼ RN and imposing the compatibility condition Ixy = Iyx, we have oy ðRMÞ ¼ ox ðRN Þ. Expanding this D½R ¼ ðM y þ N x Þ; Ry M þ Rx N ¼ RðM y þ N x Þ ) ð26Þ R where D N ox þ Moy . Since M and N are polynomials in (x, y, S),3 Eq. (26) is formally equivalent to the equation which establish the conditions for the theorem by Prelle and Singer [6] with the extension by Shtokhamer [7].4 More formally, consider the FOODE defined by e M ð27Þ y0 ¼ ; e N e and N e are the polynomials obtained by replacing y 0 by a constant k in M and N respectively. Then where M we can write the solution for the FOODE (27) as eI ¼ C, where eI is defined by replacing y 0 (by the constant k) on I. By hypothesis, I is elementary and, consequently, eI is also elementary. Therefore, by the theorem due to 0 e Prelle and Singer [6] and the Shtokhamer’s extension Q [7], the integrating factor R (defined by replacing y (by the constant k) on R) for Eq. (27) can be written as i f~ ni i , where f~ i are irreducible polynomials in (x, y, S). We could proceed analogously for the other two possible pairings of variables (i.e., (x, y 0 ) and (y, y 0 )). More explicitly, in the reasoning leading to Eq. (26), we could have imposed the compatibility condition I xy 0 ¼ I y 0 x and, in the same fashion we did above for the pairing (x, y), conclude that the integrating factor can be written as a product of irreducible polynomials in (x, y 0 , S). The same can be done for the pairing (y, y 0 ). So, R we can con0 , and N is a clude that R can be written as a product of irreducible polynomials in (x, y, y , S). Since R N Q polynomial in (x, y, y 0 ), finally we have that R ¼ i fini , where fi are irreducible polynomials in (x, y, y 0 , S) and ni are non-zero rational numbers. Let us now prove that the fi are eigenpolynomials of the operator D or factors of N. From Eq. (8) and remembering that R NR , we have D½RN NM y 0 MN y 0 ¼ S RN N2 multiplying both sides of (28) by N2, we get D½R N2 þ ND½N ¼ N 2 S ðNM y 0 MN y 0 Þ R 3
ð28Þ
ð29Þ
Note that M and N are polynomials in y 0 as well. P Notice that, since S is an algebraic function of (x, y, y 0 ), it is a root of a polynomial equation of the form i pi S i ¼ 0, where pi are 0 polynomials in (x, y, y ). Therefore, its derivatives can be expressed in terms of rational powers of itself. So, one may consider that would be dealing with a Shtokhamer extension with U = S. 4
8
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
and finally, D½R ¼ ND½N N 2 S ðNM y 0 MN y 0 Þ: ð30Þ R Since the right-hand side of (30) is a polynomial in (x, y, y 0 , S), so is N D½R . Using (24) this implies that R D½R X D½fi ¼ ni N ð31Þ N fi R i N
is a polynomial. Since fi are irreducible (independent) polynomials, we may conclude that, either fi jD½fi or fi is a factor of N. h In order to obtain the integrating factor R, we are going to use Eq. (8). Dividing it by R, we get NM y 0 MN y 0 D½R=R ¼ ðS þ /y 0 Þ ¼ S þ : N2
ð32Þ
Multiplying both sides of the equation above by N2, one has N 2 D½R=R ¼ ðSN 2 þ NM y 0 MN y 0 Þ ) N D½R=R ¼ ðSN 2 þ NM y 0 MN y 0 Þ:
ð33Þ
Due to the results of Theorem 3, Eq. (33) above can be solved in the same manner as in the methods inspired on the Prelle–Singer procedure. Once R is found as a function of (x, y, y 0 , S), we can substitute the known S (see Section 3) and, finally, from Eq. (6), find the first order invariant via quadratures: Z Z Z o 0 0 0 Rð/ þ Sy Þ dx dy Iðx; y; y Þ ¼ Rð/ þ Sy Þ dx RS þ oy Z Z Z Z o o 0 0 Rð/ þ Sy Þ dx dy dy 0 : Rð/ þ Sy Þ dx Rþ 0 RS þ ð34Þ oy oy That concludes the presentation of our proposed approach which is a semi-algorithm procedure to reduce SOODEs of the form of Eq. (1). 4.1. Example Here we are going to briefly introduce an example of a non-trivial5 SOODE that illustrates the results displayed above. Consider the SOODE: 2
y 00 ¼
3
ðy 0 Þ þ 2ðy 0 Þ 1 2y 0 x ðy 0 Þx ðy 0 Þy 1 þ ðy 0 Þ2
ð35Þ
:
For this SOODE, the operator D becomes: D ¼ ðy 02 þ 2y 03 1 2y 0 Þoy 0 þ ðx þ y 0 x þ y 0 y þ 1 y 02 Þy 0 oy þ ðx þ y 0 x þ y 0 y þ 1 y 02 Þox ;
ð36Þ
and the operator DS is DS ¼ ND þ Mos ; 2
0
ð37Þ 2 02
02
03
2
2 2
2
2 02
2 04
02
04
03
where M ¼ 2s xy y þ 2s y xy þ 7sy x þ 4sy x þ s þ s x þ 2s x 2s y þ s y sx þ 4sy 2sy þ sy þ 4sy y 2s2 y 0 2 x þ s2 y 0 2 y 2 2s2 y 0 3 y þ 2s2 y 0 x2 þ 2sy 0 x 2s y 0 þ s2 y 0 2 x2 2s2 y 0 3 x þ 2s2 y 0 y 2y 0 2 þ y 0 3 þ 2y 0 4 þ sy 0 2 y þ 2s2 y 0 x. The polynomial P ¼ ðx þ y 0 x þ y 0 y þ 1 y 0 2 ÞS þ 1 y 0 2 is an eigenpolynomial of the DS operator. So, P = 0 defines S as S¼
5
1 þ y 0 2 : x þ y0x þ y0y þ 1 y02
For instance, the MAPLE package, release 7, could not reduce it.
ð38Þ
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
9
In order to calculate the integrating factor R, we need (see Eqs. (31) and (32)) to find the eigenpolynomials of the D operator. These are found to be (up to the first degree): f1 ¼ 1 þ y 0 ; f2 ¼ 1 y 0 ; f3 ¼ 1 þ 2y 0 and, for this particular case, these are sufficient (together with N) to build the integrating factor R. Combining Eqs. (31) and (32) and solving for the ni’s we have that {n1 = 3/2, n2 = 3/2, n3 = 0}. So, R is found to be6: R¼
x þ y0x þ y0y þ 1 y02 ððy 0 1Þð1 þ y 0 ÞÞ
3=2
ð39Þ
:
This leads to the first order invariant given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x þ y þ y0x I ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ lnðy 0 þ 1 þ y 0 2 Þ: 1 þ y 0 2
ð40Þ
5. Computational considerations Here, we are going to introduce an alternative way of calculating the integrating factor for a given SOODE. Why do we call this section computational considerations? The reason is that the material contained in this section is not applicable to all SOODEs7 of the form of Eq. (1) and, indeed, we do not have a criteria of applicability. But, on the other hand, the method we are about to present is ‘‘less costly’’ computationally, therefore could be useful in practical applications. In the following, we will be using some results from the Lie theory. Let us show a relation between the function S and a symmetry of the SOODE (1). Making the following transformation: S¼
D½g : g
ð41Þ
Eq. (13) becomes D2 ½g ¼ /y 0 D½g þ /y g:
ð42Þ
From Lie theory we can see that Eq. (42) represents the condition for a SOODE (1) to have a symmetry [0,g]. So, from (41) we can find a symmetry given by R g ¼ e S dx : ð43Þ Looking at Eq. (5) we can infer that R is also an integrating factor for the auxiliary FOODE defined by dy 0 ¼ S; dy
ð44Þ
where x is regarded as a parameter. Besides, for this FOODE, [g, D[g]] is a point symmetry. So, R is given by R¼
1 : gS þ D½g
ð45Þ
If in (45) we use g defined by (43) we would get a singular R. However, Eq. (42) has another solution independent from g given by
3=2 3=2
Note that R is formed by the product f1n1 f2n2 f3n3 N 1 ¼ f1 f2 N 1 . Note that the method presented on the previous section is a general semi-algorithmic approach to deal with SOODEs of the form (1) that present an elementary first order invariant. 6 7
10
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
g¼g
Z
e
R
/y 0 dx
g2
ð46Þ
dx:
Using this g in (45) we get R and, once this is done, we can calculate the first integral I by using simple quadratures. R Actually, trying to solve Eq. (43) could be complex. The operator S dx that appears on Eq. (43) is meant 8 to be the inverse of operator D (defined on (13)). As one might expect, actually finding g from (43) is trick. It could be impossible to integrate (43) and that is the reason why this shortcut is not applicable in general. 5.1. Example Consider the SOODE y 00 ¼
2y 0 2 x2 2y 0 2 2xy 0 y y 2 x4 þ 2y 2 x2 y 2 : 2yðx2 1Þ
ð47Þ
Constructing the DS operator we get that 2
y 2 ðx2 1Þ S 2 y 2 y 0 þ 2Sy 0 y
pffiffiffiffiffiffiffiffiffiffiffiffiffi is an eigenpolynomial of it. Then S is given by S ¼ y 0 =y þ x2 1 and g, g are respectively pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffipffiffiffiffiffiffi xpffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi R pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi x1 xþ1 e x 1 2 1 pffiffiffiffiffiffiffi 2 dx x þ x xþ x 1 xþ x2 1 pffiffiffiffiffiffi pffiffiffiffiffiffi g¼ ; g¼ : 2 2 x x 1 x x 1 ye 2 ye 2 R can be written as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi y x þ x2 1 R ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi x x2 1 x 2 1e 2
ð48Þ
ð49Þ
ð50Þ
leading to the following first integral: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi x þ x2 1ð2y 0 y þ x2 1y 2 Þ I¼ : pffiffiffiffiffiffiffiffiffiffiffiffiffi xpffiffiffiffiffiffi x2 1 x 2 1e 2
ð51Þ
6. Conclusion In [16], some of us have developed a method, based on a conjecture, to deal with SOODEs that presented an elementary solution (possessing two elementary first order invariants). In that same paper, a function S was introduced to transform the Pfaffian equation related to the particular SOODE under consideration into a 1-form proportional to differential of the first order differential invariant. That function S was instrumental in finding the integrating factor for the SOODE. Here, in the present paper, we have introduced many theoretical results concerning that function S and we have presented a method to calculate it via a Darboux-type procedure. Here also the function S was used to produce the integrating factor. We have demonstrated general results about the structure of the integrating factor R and that we can calculate it using a procedure very similar to the one inspired by the original work by Prelle and Singer [6] (applicable for FOODEs). These results assure that we have a semi-algorithmic procedure to deal with rational SOODEs presenting at least one elementary first order invariant. Consequently, we can cover a much broader class of SOODEs than before [16].
8
Actually, this meaning for the operator
R
is true for the whole of the present section.
J. Avellar et al. / Applied Mathematics and Computation 184 (2007) 2–11
11
In the above section, we have introduced an alternative way to calculate R from the knowledge of S for restricted cases. The motivation for that is that the general method can be computationally very demanding. We are searching for better algorithms for the many steps of the method presented here in order to make it computationally more efficient. We are also working on a full computational implementation of the method as it stands today. References [1] H. Stephani, Differential equations: their solution using symmetries, in: M.A.H. MacCallum (Ed.), Cambridge University Press, New York, London, 1989. [2] G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Applied Mathematical Sciences, vol. 81, Springer Verlag, 1989. [3] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, 1986. [4] E.S. Cheb-Terrab, L.G.S. Duarte, L.A.C.P. da Mota, Computer algebra solving of first order ODEs using symmetry methods, Comput. Phys. Commun. 101 (1997) 254. [5] E.S. Cheb-Terrab, L.G.S. Duarte, L.A.C.P. da Mota, Computer algebra solving of second order ODEs using symmetry methods, Comput. Phys. Commun. 108 (1998) 90. [6] M. Prelle, M. Singer, Elementary first integral of differential equations, Trans. Am. Math. Soc. 279 (1983) 215. [7] R. Shtokhamer, Solving first order differential equations using the Prelle–Singer algorithm, Technical report 88-09, Center for Mathematical Computation, University of Delaware, 1988. [8] C.B. Collins, Algebraic invariants curves of polynomial vector fields in the plane, Preprint. Canada: University of Waterloo, 1993.; C.B. Collins, Quadratic vector fields possessing a centre, Preprint. Canada: University of Waterloo, 1993. [9] C. Christopher, Liouvillian first integrals of second order polynomial differential equations, Electron. J. Differen. Equat. (49) (1999) 7, electronic. [10] C. Christopher, J. Llibre, Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. Differen. Equat. 16 (1) (2000) 5–19. [11] J. Llibre, Integrability of polynomial differential systems, in: A. Can˜ada, P. Dra´bek, A. Fonda (Eds.), Handbook of Differential equations, Ordinary Differential Equations, vol. 1, Elsevier B.V., 2004, pp. 437–531 (Chapter 5). [12] L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, A method to tackle first order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A: Math. Gen. 35 (2002) 3899–3910. [13] L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, Analyzing the structure of the integrating factors for first order ordinary differential equations with Liouvillian functions in the solution, J. Phys. A: Math. Gen. 35 (2002) 1001–1006. [14] L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, J.F.E. Skea, Extension of the Prelle–Singer method and a MAPLE implementation, Comput. Phys. Commun., Holanda 144 (1) (2002) 46–62. [15] J.H. Davenport, Y. Siret, E. Tournier, Computer Algebra: Systems and Algorithms for Algebraic Computation, Academic Press, Great Britain, 1993. [16] L.G.S. Duarte, S.E.S. Duarte, L.A.C.P. da Mota, J.E.F. Skea, Solving second order ordinary differential equations by extending the Prelle–Singer method, J. Phys. A: Math. Gen. 34 (2001) 3015–3024.