Cent. Eur. J. Math. • 12(4) • 2014 • 648-657
DOI: 10.2478/s11533-013-0364-z
Central European Journal of Mathematics
On certain properties of linear iterative equations
Research Article
Jean-Claude Ndogmo1∗ , Fazal M. Mahomed2†
1 School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
2 Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics,
University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
Received 23 March 2013; accepted 1 October 2013
Abstract: An expression for the coefficients of a linear iterative equation in terms of the parameters of the source equation is
given both for equations in standard form and for equations in reduced normal form. The operator that generates
an iterative equation of a general order in reduced normal form is also obtained and some other properties of
iterative equations are established. An expression for the parameters of the source equation of the transformed
equation under equivalence transformations is obtained, and this gives rise to the derivation of important symmetry properties for iterative equations. The transformation mapping a given iterative equation to the canonical form
is obtained in terms of the simplest determining equation, and several examples of application are discussed.
MSC:
35A24, 65Q30, 65F10
Keywords: Linear iterative equation • Recurrence relations • Canonical form • Coefficients characterization • Normal form
© Versita Sp. z o.o.
1.
Introduction
It is well known [3] that linear ordinary differential equations (ODEs) of order one or two can all be reduced by a local
diffeomorphism of the (x, y)-plane to the canonical form y0 = 0 and y00 = 0, respectively, and that this is not the case for
equations of a general order n > 2. Lie [6] showed that a differential equation of a general order n > 2 is equivalent (by
a local diffeomorphism of the plane) to the equation y(n) = 0, which we shall henceforth refer to as the canonical form
of the linear equation, only if its symmetry algebra has the maximal dimension n + 4. In a much recent paper, Krause
and Michel [2] proved the converse of this statement and also showed that a linear ODE of order n > 2 has a symmetry
∗
†
E-mail: [email protected]
E-mail: [email protected]
648
Unauthenticated
Download Date | 6/16/17 9:08 AM
J.-C. Ndogmo, F.M. Mahomed
algebra of maximal dimension if and only if it is iterative. Linear iterative equations are the iterations Ψn y = 0 of a
linear first order equation, of the form
Ψy ≡ r(x)y0 + s(x)y = 0,
Ψn y = Ψ(n−1) Ψy,
n > 1.
(1)
In these iterations, the equation r(x)y0 + s(x)y = 0 is termed the source equation, while the functions r(x) and s(x)
are referred to as its parameters. Despite the unique symmetry properties of these iterative equations, many of their
properties are still not known. Mahomed and Leach [8] gave a listing of these equations for low orders not exceeding
eight. We extend this list to equations of a general order, in both the standard form and the associated reduced normal
form in terms of their parameters r and s, and determine the operator that generates the linear iterative equation
of any given order in reduced normal form. Other properties of iterative equations concerning their coefficients are
also established, and expressions for the parameters of the transformed equation under equivalence transformations are
obtained. From these expressions, some symmetry properties common to all iterative equations are derived, and a simple
characterization of the transformation that reduces the iterative equation to its canonical form is also obtained.
2.
Equations in standard form
By replacing the dependent variable y = y(x) by y+yp , where yp is a particular solution of the inhomogeneous equation,
we may assume without loss of generality that a linear iterative equation of a general order n has the form
Ψn y ≡ Kn0 y(n) + Kn1 y(n−1) + Kn2 y(n−2) + · · · + Knn−1 y0 + Knn y = 0.
2.1.
(2)
Ψ = d/dx + s
For the sake of clarity we first consider the case where the differential operator Ψ = rd/dx + s is much simpler, with
r ≡ r(x) = 1. It is clear that in this case, the operator Ψ leaves invariant the leading coefficient, and thus we have
Kn0 = 1 in this case. Using the formula for Ψn y in (1) gives the recurrence relations
1
Kn1 = Kn−1
+ s,
(3a)
d j−1
j−1
j
j−1
j
+ sKn−1 = Kn−1 + ΨKn−1 ,
K
Knj = Kn−1 +
dx n−1
d n−1
n−1
n−1
Knn =
K
+ sKn−1
= ΨKn−1
.
dx n−1
Setting
Kmj = 0
for
j<0
or
j > m,
and
j = 2, . . . , n − 1,
n > 2,
(3b)
(3c)
Kmj = 1
for
m = j = 0,
(4)
reduces the recurrence equations (3) to the single equation
j
j−1
Knj = Kn−1 + ΨKn−1 ,
0 ≤ j ≤ n,
n ≥ 1.
(5)
Solving the recurrence relations (5) together with the initial conditions
Ψy ≡ y0 + sy = y0 + K11 y,
Ψ2 y ≡ y00 + 2sy0 + s2 + s0 y = y00 + K21 y0 + K22 y
gives Kn1 = ns = n1 Ψ0 s, Knn = Ψn−1 s, for all n ≥ 1. Using these two equalities, and setting Ψ−1 f = 1 for every
j
function f = f(x), one readily sees by induction on j and n that Kn = nj Ψj−1 s, j = 0, . . . , n, n ≥ 1. We have thus
obtained the following result for the case where r = 1.
649
Unauthenticated
Download Date | 6/16/17 9:08 AM
On certain properties of linear iterative equations
Theorem 2.1.
When the operator Ψ that generates the iterative equation has the form Ψ = d/dx + s, that is, when it depends on the
j
single function s, the coefficients Kn of the iterative equation of a general order n are given by
Knj
n
=
Ψj−1 s,
j
j = 1, . . . , n,
(6)
and the iterative equation of a general order n is therefore given by
Ψn y = y(n) +
n X
n
j
j=1
Ψj−1 s y(n−j) .
Note however that (6) is also valid for j = 0.
2.2.
The source equation depends on both parameters
In this case we have Ψ = rd/dx + s, where the parameters r and s are given functions, and this is the most general
j
j
case. Using the formula for Ψn y given in (2) as well as the conventions for Kn set in (4) show that the coefficients Kn
of the iterative equation (2) satisfy the recurrence relations
j
j−1
Knj = rKn−1 + ΨKn−1 ,
0 ≤ j ≤ n,
n ≥ 1,
(7)
which naturally reduce to (5) for r = 1. Setting j = 0 or j = n in (7) readily gives by induction on n the identities
Kn0 = r n ,
Knn = Ψn−1 s,
n ≥ 1.
(8)
Applying (7) recursively and using the conventions set in (4) give a new recurrence relation
Knj =
n
X
j−1
r n−k ΨKk−1 ,
j = 0, . . . , n,
n ≥ 1.
(9)
k=j
j
Although (9) does not provide the required expression for Kn in terms of the parameters r and s, it represents an algorithm
j
for the computation of the coefficients Kn for all possible values of n and j. For instance, using (9) with j = 1, 2 gives
n 0
Kn1 = r n−1 ns +
r ,
2
n
n
3n − 5 02
Kn2 = r n−2
Ψs +
3sr 0 + rr 00 +
r
.
2
3
4
(10a)
(10b)
j
To obtain the general expression for Kn using a recurrence relation relating them, we rewrite (9) in the form
Knj =
n
X
j−1
r (k−j) ΨKn−k+j−1 .
(11)
k=j
Then, using (4) and (8), the following formulas are successively obtained:
Kn1 =
n
X
r k−1 Ψr n−k ,
(12a)
k=1
Kn2 =
n n−k
2 +1
X
X
r k2 −2 Ψ r k1 −1 Ψr n+1−(k1 +k2 ) ,
(12b)
k2 =2 k1 =1
Kn3 =
n n−k
3 +2
X
X
k3 =3 k2 =2
n−(k2 +k3 )+3
X
r k3 −3 Ψ r k2 −2 Ψ r k1 −1 Ψr n+3−(k1 +k2 +k3 ) .
(12c)
k1 =1
650
Unauthenticated
Download Date | 6/16/17 9:08 AM
J.-C. Ndogmo, F.M. Mahomed
j
Continuing this process with two more iterations by computing Kn4 and Kn5 , a clear pattern for the general coefficient Kn
emerges, and to write down this expression we introduce some notations. For n ≥ 1 and 0 ≤ i ≤ j ≤ n, set
βij =
j
X
ku ,
ku ∈ Z,
(13a)
u=i+1
j
i
Mi ≡ Mi (j) = n +
−
− βij ,
2
2
j
αj = n +
− β0j = M0 .
2
(13b)
(13c)
We have the following result for the general case, where Ψ = rd/dx + s.
Theorem 2.2.
j
In terms of the parameters r and s of the source equation, the general coefficient Kn of the iterative equation (2) has
the form
Mj−1
Mj
M2 M1
X
X
X
X
Knj =
...
r kj −j Ψ r kj−1 −(j−1) Ψ . . . Ψ r k1 −1 Ψr αj . . . ,
(14)
kj =j kj−1 =j−1
k2 =2 k1 =1
for n ≥ 1 and 1 ≤ j ≤ n, and where the expressions for βij , Mi , and αj are given by (13).
For the sake of clarity it would be useful to verify explicitly that (14) reduces indeed to (6) for r = 1.
Proposition 2.3.
Equation (14) reduces as expected to (6) for r = 1.
When r = 1, the general term in the summation (14) clearly reduces to Ψj · 1 = Ψj−1 s, and since this expression
does not depend on the running indices k1 , k2 , . . . , kj to prove the proposition, it suffices to show that the total number
Proof.
Pn,j =
Mj−1
Mj
X
X
kj =j kj−1 =j−1
...
M2 M1
X
X
1
k2 =2 k1 =1
of terms in this summation is precisely nj . For j = 1 and j = 2, it clearly follows from (12a) and (12b) respectively that
Pn,1 = n1 and Pn,2 = n2 . It also follows from (8) that Pn,0 and Pn,n also satisfy the required property for n ≥ 1. Let
v
n ≥ 2 and assume that Pv,j−1 = j−1
, for 1 ≤ v < n and 0 ≤ j − 1 ≤ v < n. Then it follows from (11) and the linearity
Pn n−k+j−1
of Ψ that Pn,j = k=j
. Setting n − k + j − 1 = q and j − 1 = m gives
j−1
Pn,j =
n−1 X
q
q=m
m
=
(n − 1) + 1
m+1
=
n
,
j
and this completes the proof by induction of the required property for Pn,j .
Although Proposition 2.3 gives a verification of the validity of the complicated formulas (14) at least for the simpler case
r = 1, these formulas can be slightly simplified, by a suitable change of variables. Indeed, if in (14) we set
(
kj − j = n − Pj ,
ki − i = Pi+1 − Pi − 1,
j ≥ 1,
i = 1, . . . , j − 1,
(15)
651
Unauthenticated
Download Date | 6/16/17 9:08 AM
On certain properties of linear iterative equations
this reduces the expression for Mi in (13) to
Mi = Pi+1 − 1.
(16)
αj = P1 − 1.
(17)
In particular, αj = Mo is reduced to
Since Mj = n for all j, thanks to (16), we may rewrite (15) as
k i − i = M i − Pi ,
i = 1, . . . , j.
(18)
Consequently, thanks to (16)–(18), and after renaming Pi , we may rewrite (14) in a slightly simplified form
Knj =
kj −1
n
X
X
k3 −1 k2 −1
...
kj =j kj−1 =j−1
3.
XX
r n−kj Ψ r kj −kj−1 −1 Ψ . . . r k3 −k2 −1 Ψ r k2 −k1 −1 Ψr k1 −1 . . . .
k2 =2 k1 =1
Equations in reduced normal form
By dividing through the general n-th order linear iterative equation Ψn y in (2) by Kn0 = r n , it can be put in the form
y(n) + Bn1 y(n−1) + · · · + Bnj y(n−j) + · · · + Bnn y = 0,
(19)
j
j
where Bn = Kn /r n . This is the standard form of the general linear iterative equation with leading coefficient one. It is
well known that (19) can be transformed to the normal form (in which the coefficient of y(n−1) has vanished) by a change
of the dependent variable of the form
Z x
1
1
B (v) dv .
(20)
y 7→ y exp
n x0 n
However, this amounts to the requirement that Bn1 = 0, i.e. that Kn1 = 0. Therefore, an n-th order linear equation in
reduced normal form is iterative (with source parameters r and s) if and only if it has the form
y(n) + A2n y(n−2) + · · · + Ajn y(n−j) + · · · + Ann y = 0,
(21a)
where
Ajn
j Kn ,
= n
r Kn1 =0
2 ≤ j ≤ n,
(21b)
j
and where Kn is given by (14). It follows from (10a) that setting Kn1 = 0 amounts to setting
1
s = − (n − 1)r 0 ,
2
(22)
and this shows why any iterative equation in normal form can be expressed in terms of the parameter r alone. Moreover,
j
j
thanks to (21b) the coefficients An inherit all of the characterization (14) obtained for the coefficients Kn of iterative
j
n
equations in standard form, and the corresponding expression for An has up to the factor 1/r , exactly the same form as
j
that for Kn because the parameter s does not appear explicitly in (14). In addition, using the same algorithm (e.g. (9))
j
j
obtained for the Kn , one can readily compute An for all n ≥ 2 and 2 ≤ j ≤ n. For instance, using the expression for Kn2
in (10b) together with (21b), one sees that
A2n =
n+1
A(r),
3
where
A(r) ≡ A22 =
r 02 − 2rr 00
.
4r 2
(23)
652
Unauthenticated
Download Date | 6/16/17 9:08 AM
J.-C. Ndogmo, F.M. Mahomed
Theorem 3.1.
Let the differential operator Φn be given by
1 n Φn = n Ψ .
r
Kn1 =0
Then the equation Φn y = 0 is exactly (21a), that is Φn generates the (most general) linear iterative equation of an
arbitrary order n in normal form.
Indeed, Ψn y generates the linear iterative equation of a general order n in standard form and (1/r n )Ψn
generates the same equation with leading coefficient 1, while setting Kn1 = 0 corresponds as already noted to reducing
the latter equation to its normal form.
Proof.
Differential equations are generally studied by first transforming them into their most simpler form, and the invertible
transformation (20) shows that it is always possible and rather easy to transform any given linear equation in standard
form into its normal form and vice-versa. Moreover, results obtained in this section show that in its normal form, the
linear iterative equation is completely determined by only one of the two parameters of the source equation, thanks to
condition (22). In addition, calculations done for lower order equations up to order eight in [8], and which can also be
done by an application of Theorem 3.1, show that in its normal form a linear iterative equation depends solely on the
coefficient A2n and its derivatives. For instance, for n = 3 or 4, if we set A23 = a3 and A24 = a4 , then iterative equations
of orders 3 and 4 take on respectively the forms
000
0
1
y + a3 y + a03 y = 0,
2
y
(4)
00
+ a4 y +
a04 y0
+
3 00
9 2
a +
a y = 0.
10 4 100 4
Thanks to a result obtained in this paper, namely (23), it follows that the coefficients of the linear iterative equation are
in fact functions of only the coefficient A22 of the second order source equation, and its derivatives. It should be noted
at this point that integrating a general linear iterative equation of the general order (21a) reduces to integrating the
corresponding second order source equation, as its general solution is a linear combination of the linearly independent
solutions
yk = uk v (n−1)−k ,
0 ≤ k ≤ n − 1,
(24)
where u and v are linearly independent solutions of the second order source equation (see for instance [2]).
4. Parameters of the transformed equation under equivalence transformations
We shall apply in this section some of the results we have obtained thus far to look at how the parameters r and s of the
source equation for a given iterative equation change under equivalence transformations. For convenience, but without
loss of generality, we may assume that the equation is in its reduced normal form (21), which also reduces the problem to
looking only at how the parameter r changes. Thus we suppose that (21) has source parameters r and s = −(n − 1)r 0 /2.
Recall that the equivalence transformations of (21) are given by invertible transformations of the form
x = f(z),
y = λ[f 0 (x)](n−1)/2 w,
(25)
where f is an arbitrary function and λ an arbitrary constant (see e.g. [9, 10, 12]). We let the transformed version of (21)
under (25) be of the form
w (n) + b2n w (n−2) + · · · + bjn w (n−j) + · · · + bnn w = 0,
(26)
where w = w(z). Since (26) also has maximal symmetry, it is iterative, and assuming that its first order source equation
has an expression of the form Rw 0 + Sw = 0, with S = −(n − 1)R 0 /2, we are interested in finding an expression for
653
Unauthenticated
Download Date | 6/16/17 9:08 AM
On certain properties of linear iterative equations
the parameter R, more generally in terms of the parameter r of the original equation and the parameters λ and f of the
equivalence transformation.
Given on one hand that two iterative equations of a general order of the form (21a) are identical if and only if their
coefficients A2n of the term of third highest order coincide, and on the basis of (23), to find a determining equation for R,
it will suffice to make use of the coefficient b22 . That is, it will suffice to equate the coefficient b22 = (R 02 − 2RR 00 )/4R 2
with the corresponding coefficient (of the term of order n − 2) in the transformed equation. However, to make it clearer
that these determining equations do not depend on the order of the equation, we find them explicitly for equations of
order 2, 3, 4. The corresponding determining equations for R take the form
1
3 002 1 0 (3)
2 04
A
f
−
f
+
f
f
f 02 2
4
2
1
b23 = 02 A23 f 04 − 3f 002 + 2f 0 f (3)
f
1 b24 = 02 2A24 f 04 − 15f 002 + 10f 0 f (3)
f
b22 =
for
n = 2,
(27a)
for
n = 3,
(27b)
for
n = 4,
(27c)
and, by using in these equations the substitutions
A2n ≡ A2n (x) =
n+1
A(r)(x),
3
b2n ≡ b2n (z) =
n+1
A(R)(z),
3
n = 2, 3, 4,
A(r)(x) =
r 0 (x)2 − 2r(x)r 00 (x)
, (28)
4r(x)2
we indeed see that the three equations in (27) all reduce to the same equation corresponding to the case n = 2, and
which has the form
R 02 − 2RR 00
1 r 0 (f)2 − 2r(f)r 00 (f) 04
002
0 (3)
=
f
−
3f
+
2f
f
.
(29)
R2
f 02
r 0 (f)2
Denoting by S(f)(z) = −3f 002 + 2f 0 f (3) /f 02 the Schwarzian derivative of the function f = f(z) with respect to the
argument z, and using (28), the determining equation (29) may also be put into the form
A(R)(z) = A(r)(f)f 02 +
1
S(f)(z).
2
(30)
In fact, multiplying each member of the latter equation by n+1
and using (28) clearly shows that a general expression
3
for the coefficient b2n of the term of third highest order in the transformed version of (21a) under (25) is given for any
order n by
1 n+1
2
2
02
S(f)(z).
(31)
bn (z) = An (f)f +
2
3
Example 4.1.
Suppose that equation (21a) has the canonical form y(n) = 0. Thus in this case A2n = 0, which amounts to having
r = (ax + b)2 for some constants a and b. If follows from (30) that the transformed version (26) of (21a) will be the same
equation if and only if the function f is chosen so that S(f)(z) = 0. In other words, since S(f)(z) = 0 if and only if f(z)
is a linear fractional transformation, it follows that the transformation (z, w) 7→ f(z), λ[f 0 (z)](n−1)/2 w , where f is a linear
fractional transformation is the most general symmetry transformation that leaves the equation y(n) = 0 invariant for all
possible values of n ≥ 1. In this case, the value of R = (pz + q)2 for some constants p and q is similar to that for r but
does not depends explicitly of the parameters of the transformation because the equation is left invariant.
Example 4.2.
Assume that in (30) we have f(z) = αz + β for some constants α and β and also that A(r)(x) = m/(ax + b)2 , for some
constants a =
6 0, b, and m. The latter equality is equivalent to assuming that
2
r(x) = λ1 (b + ax)E1 1 + λ2 (b + ax)E2 ,
where
E1 =
1−
√
a2 − 4m
a
√
and
E2 =
a2 − 4m
,
a
654
Unauthenticated
Download Date | 6/16/17 9:08 AM
J.-C. Ndogmo, F.M. Mahomed
and for some constants of integration λ1 and λ2 . With these values of f and r, the determining equation (30) for R
reduces to
M
,
(32)
A(R)(z) =
Az + B
with M = mα 2 , A = aα, and B = aβ + b. The corresponding value of R is thus
E3
2
R = γ1 [b + a(αz + β)] 1 + γ2 [b + a(αz+ β)]E4 ,
E3 =
1−
p
(a2 − 4m)α 2
,
aα
p
E4 =
(a2 − 4m)α 2
,
aα
for some constants of integration γ1 and γ2 . The expression for A(R)(z) in (32) also shows in particular that the
function f(z) = αz + β used here to transform the original equation will correspond to a symmetry transformation if and
only if it is the identity map.
Although it is known according to the already cited result of [2] that linear iterative equations can be reduced to the
canonical form y(n) = 0, a determining equation for the transformation that maps such an equation to this canonical form
is not known. The following result gives an answer to this problem.
Theorem 4.3.
A point transformation reduces a given iterative equation, which may be assumed without loss of generality to be of
the form (21a), to the canonical form w (n) (z) = 0 if and only if it is of the form (25), where f is the inverse of the
function z = h(x) satisfying
1
A22 (x) = S(h)(x).
(33)
2
Proof.
Since the equations involved in the transformations are all in normal form, it is clear that a point transformation
will reduce (21a) to w (n) (z) = 0 if and only if it is of the specified form (25), and with some appropriate function f. Thus
we only need to specify the condition satisfied by the function f in the latter transformation. If we let
z = h(x)
and
w=
1 0 (n−1)/2
[h (x)]
y
λ
(34)
be the transformations that map w (n) = 0 to (21a), then it follows from (31) (rewritten with the correct variables and
functions) that h satisfies (33). Conversely, if h satisfies (33), which on account of (28) is equivalent to A2n (x) =
n+1
S(h)(x)/2, then (31) shows again that the coefficient of the term of order n − 2 in the transformed version under (34)
3
is precisely A2n . This in turn shows that (34) transforms w (n) (z) = 0 into (21a), given that an iterative equation is
completely determined solely by the coefficient A2n of the term of order n − 2. To complete the proof of the theorem, we
only need to note that h is clearly locally invertible, and if we let f denote its inverse, then (25) is precisely the inverse
of (34), and thus it transforms (21a) back into the required canonical form.
For example, if we let the coefficient A22 = m/(ax + b)2 in (21a) for some constants a 6= 0, b, and m =
6 a2 /4 as in
Example 4.2, then the corresponding iterated equation of order 4 in normal form has the expression
9m(2a2 + m)y
20amy0
10my00
−
+
+ y(4) = 0.
4
3
(b + ax)
(b + ax)
(b + ax)2
(35)
On the other hand, the inverse f of the function h satisfying (33) for this value of A22 has expression
"
E5 #
1
1
k2
√
f(z) =
−b + E −1 −
,
a
k1 (z − k3 ) a2 − 4m
k1 5
√
with E5 = a/ a2 − 4m. One readily verifies that (25) with this value of f transforms (35) to the canonical form w (4) = 0.
655
Unauthenticated
Download Date | 6/16/17 9:08 AM
On certain properties of linear iterative equations
5.
Concluding remarks
It should be noted that the function f of Theorem 4.3 can also be obtained directly from (31) without having to first find
the function h in (33), and then its inverse. Indeed, it readily follows from (31) that
0 = A22 (f)f 02 +
1
S(f)(z).
2
(36)
In fact the latter equation (36) can also be obtained from (33) by substituting in that equation the inversion formula for
the Schwarzian derivative, namely the equality S(h)(f) = −S(f)(h)/(df/dh)2 . However, finding f by solving (36) could
often lead to a more complicated equation.
On the other hand, Theorem 4.3 shows that transforming an iterative equation of a given order n to its canonical form
reduces to solving (33), which can be seen as a second order equation by replacing in the latter equation the dependent
variable with its derivative. According to the well-known Lie linearization criteria [5, 11], (33) is linearizable, and is
thus equivalent to the second order source equation of the n-th order iterative equation at hand. In practice therefore,
finding the function f for the transformation that maps the iterative equation to its canonical form is essentially of the
same level of difficulty as solving the second order source equation. This is natural on the basis of (24) because the
solution of the reduced equation y(n) = 0 is essentially trivial.
It should also be noted that an equation of the form (33), or equivalently (36), is usually referred to as a Schwarz
equation [1, 4, 7, 13, 14]. As such, it is also a particular type of third-order Kummer–Schwarz equations [1, 7]. Schwarz
equations are known to possess some important geometric properties, and in particular their integration amounts to the
integration of certain Riccati equations, second-order Kummer–Schwarz equations, or harmonic oscillators [4, 7]. More
generally, it was recently shown that these equations can be studied through the so-called sl(2, R)-Lie systems [7], and
the latter fact can be employed for their integration or to analyze their solutions [1, 7].
Acknowledgements
We are indebted to an anonymous referee for helpful comments. The work of JCN and FMM was partly supported by
research grants from the NRF Incentive Support for Rated Researchers. JCN also acknowledges an NRF CSUR grant
and a Carnegie Transformation large research grant.
References
[1] Cariñena J.F., Grabowski J., de Lucas J., Superposition rules for higher-order systems and their applications,
J. Phys. A, 2012, 45(18), # 185202
[2] Krause J., Michel L., Equations différentielles linéaires d’ordre n > 2 ayant une algèbre de Lie de symétrie de
dimension n + 4, C. R. Acad. Sci. Paris, 1988, 307(18), 905–910
[3] Krause J., Michel L., Classification of the symmetries of ordinary differential equations, In: Group Theoretical
Methods in Physics, Moscow, June 4–9, 1990, Lecture Notes in Phys., 382, Springer, Berlin, 1991, 251–262
[4] Leach P.G.L., Andriopoulos K., The Ermakov equation: a commentary, Appl. Anal. Discrete Math., 2008, 2(2), 146–157
[5] Lie S., Classification und Integration von gewöhnlichen Differentialgleichungen zwischen xy, die eine Gruppe von
Transformationen gestatten. III, Archiv for Mathematik og Naturvidenskab, 1883, 8, 371–458
[6] Lie S., Classification und Integration von gewöhnlichen Differentialgleichungen zwischen xy, die eine Gruppe von
Transformationen gestetten, Math. Ann., 1888, 32(2), 213–281
[7] de Lucas J., Sardón C., On Lie systems and Kummer–Schwarz equations, J. Math. Phys., 2013, 54(3), # 033505
[8] Mahomed F.M., Leach P.G.L., Symmetry Lie algebras of nth order ordinary differential equations, J. Math. Anal.
Appl., 1990, 151(1), 80–107
656
Unauthenticated
Download Date | 6/16/17 9:08 AM
J.-C. Ndogmo, F.M. Mahomed
[9] Ndogmo J.C., Equivalence transformations of the Euler–Bernoulli equation, Nonlinear Anal. Real World Appl., 2012,
13(5), 2172–2177
[10] Ndogmo J.C., Some results on equivalence groups, J. Appl. Math., 2012, # 484805
[11] Schwarz F., Solving second order ordinary differential equations with maximal symmetry group, Computing, 1999,
62(1), 1–10
[12] Schwarz F., Equivalence classes, symmetries and solutions of linear third-order differential equations, Computing,
2002, 69(2), 141–162
[13] Sebbar A., Sebbar A., Eisenstein series and modular differential equations, Canad. Math. Bull., 2012, 55(2), 400–409
[14] Tsitskishvili A., Solution of the Schwarz differential equation, Mem. Differential Equations Math. Phys., 1997, 11,
129–156
657
Unauthenticated
Download Date | 6/16/17 9:08 AM