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A MAPLE routine for second-order ordinary differential equations based on λ-symmetries

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A MAPLE routine for second-order ordinary differential equations based on λ-symmetries
A MAPLE routine for second-order ordinary
differential equations based on λ-symmetries
We present a MAPLE procedure to check if a given equation is in the class A of
the 2nd-order ODEs that admit first integrals of the form
A(t, x)ẋ + B(t, x).
(1)
Computational methods based on the algorithms developed in [1] and [2] permit
to obtain the functions A and B, λ-symmetries associated to the first integrals and
local and Sundman transformations that linearize the equations in class A.
first integrals of the form (1). Such functions are used to give the general solution in An example
implicit form:
A2 (c1 − B1 ) = A1 (c2 − B2 ), c1 , c2 ∈ R.
(3) We apply the routine to nonlinear ODE
The equations in subclass A2 have an unique (up to multipliers) first integral of
x − 1 2 te−x
ẍ +
ẋ −
ẋ + e−x = 0
(8)
the form (1). In this case the the corresponding functions A1 , B1 can readily be obx
x
tained by quadratures. The pair A1 , B1 can be used to obtained the general solution
by solving the auxiliar first-order ODE
that does not have Lie point symmetries and can not be integrated by Lie’s method.
A1 ẋ + B1 = c1 ∈ R.
(4)
It should be noted that in both cases the corresponding function A is an integrating
factor of the equation in A.
The Maple procedure
The procedure Muriel, described and illustrated in this work, together with some necessary technical routines, have been implemented for the computer algebra system
Maple15.
The second-order ODEs with first integrals of the form (1) must be of the form
ẍ + a2 (t, x)ẋ2 + a1 (t, x)ẋ + a0 (t, x) = 0.
J.Vidal
Centro Andaluz de Ciencia y Tecnologı́a Marinas
C. Muriel,J.L. Romero,
Dept. de Matemáticas
Universidad de Cádiz, Spain
(2)
Input:
var- array with the dependent and independent variables.
coef- array with the coefficients a2 , a1 and a0 .
Syntax:
Muriel(coef,var)
Intrinsic characterization
Output:
firstIntegral-array with the first integrals of the equation. The output is a twodimensional array for equations in A1 . The coefficient functions that define the first
integrals can be obtained by writing directly Asub1,Bsub1,Asub2,Bsub2.The
equations in A2 only admits one first integral (1) defined by a pair of functions
Asub1,Bsub1.
solution-if S1 = S2 = 0 the output is expression (3). For the equations in A2 the
output is the general solution of (4) provided by the MAPLE system if succeed.
λ-symmetries
The equations in A can be characterized in terms of λ-symmetries ([3]) because they
admit the vector field ∂x as λ−symmetry for specific functions of the form
λ = α(t, x)ẋ + β(t, x).
(5)
An intrinsic characterization of the equations in the class A can be given in terms of the
Equation (8) is in class A2 because its coefficients satisfy S1 6= 0, and S3 = S4 = 0.
coefficients a0 , a1 , a2 . Two subclasses appear depending on the following possibilities: In the routine the functions α and β are calculated directly from the functions A and
B. An alternative method that uses the coefficients of the equation appears in [1].
The procedure gives the first integral
Subclass A1 :
It should be remarked that the subclass A2 contains equations that lack Lie point
I = ex /xẋ + t/x.
S1 = a1x − 2a2t = 0,
symmetries and the classical Lie method ([5]) can not be used to integrate them (see,
S2 = (a0 a2 + a0x )x + (a2t − a1x )t + (a2t − a1x )a1 = 0.
The λ-symmetry is given by ∂x and
for example, eq. (8)).
x−1
te−x
Subclass A2 :
Output:
λ(t, x, ẋ) = −
ẋ +
.
x
x
S2
lambdasym-array with the functions α and β that defines (5). The output
S3 =
− (a2t − a1x ) = 0,
Equation (8) can be linearized by a generalized Sundman transformation (6) where
S
is a two-dimensional array for equations in A1 corresponding to two different λ 1 x 2
2
2
symmetries. The equations in A2 only admits one λ-symmetry of this form.
S2
S2
S2
t
t
S4 =
+
+ a1
+ a0 a2 + a0x = 0.
+ ex , G(t, x) = xϕ0
+ ex ,
F (t, x) = ϕ
S1 t
S1
S1
2
2
Output:
s-array with the functions S1 , S2 , S3 , S4 . These functions can also be obtained by
writing directly s1, s2, s3 and s4. When S1 = 0, the functions S3 and S4 are not
defined and the output returns literally s3 and s4.
test-if S1 = S2 = 0 the output is Subclass A1 . If S1 6= 0 and S3 = S4 = 0 the
output is Subclass A2 . Otherwise test returns Failed, meaning that the equation
does not pass the test and does not admit first integrals of the form (1).
First integrals and general solution
Linearization problem
for any arbitrary differentiable function ϕ. It should be observed that equation (8)
By using the results of Duarte et al. in [4], the equations in A can be characterized as does not pass the Lie test of linearization.
the equations that are linearizable by a generalized Sundman transformation ([6],[7],
[8]) of the form
X = F (t, x),
dT = G(t, x)dt.
(6) References
The routine includes an algorithm to calculate generalized Sundman transforma- [1] Muriel C and Romero J L, J. Nonlinear Math. Phys. 16 (2009), 209–222
tions that linearize the equations in A. Such linearizing transformations can be chosen
[2] Muriel C and Romero J L, J. Phys. A: Math. Theor. 43 (2010), 434025.
to be local and of the form
X = F (t, x),
T = G(t)
(7)
The procedure to compute the functions A and B of the first integrals (1) involves if and only if the equation is in A1 . The algorithm computes both types of linearizing
compatible systems of first-order PDEs that can be solved directly by integration (see transformations: local and nonlocal.
Theorem 2 in [1]).
Output:
For the equations in subclass A1 such systems are constructed in terms of the colinearization-array with the functions F and G that define linearizing transforefficients a2 , a1 and a0 and two independent solutions of a second-order linear ODE, mations. The output for equations in A1 is a two dimensional array with two pairs
g 00 (t) + f (t)g(t) = 0, where f (t) = a0 a2 + a0x − 21 a1t − 41 a21 . In this stage we have used of functions of the form F (t, x), G(t) such that, by means of (7), the equation in
the Maple solver pdsolve. If the MAPLE system calculates such solutions, the proce- A1 becomes XT T = 0.
dure provides two pair of functions, A1 , B1 and A2 , B2 , that define two independent The output for equations in A2 is a one dimensional array with a pair of functions
F (t, x), G(t, x) such that by using (6) the equation in A2 becomes XT T = 0.
[3] Muriel C and Romero J L, IMA J. Appl. Math. 66 (2001), 111–125.
[4] Duarte L G S, Moreira I C and Santos F C, J. Phys. A: Math. Gen. 27 (1994),
L739–L743.
[5] Olver P J, 1986 Applications of Lie Groups to Differential Equations (New-York:
Springer)
[6] Chandrasekar V K, Senthilvelan M and Lakshmanan M, J. Phys. A: Math. Gen.
39 (2006), L69–L76
[7] Euler N and Euler M, J. Nonlinear Math. Phys. 11 (2004), 399–421.
[8] Euler N, Wolf T, Leach P G L and Euler M, Acta Appl. Math. 76 (2003), 89–115.
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