Some Results on Equivalence Groups

Abstract

The comparison of two common types of equivalence groups of differential equations is discussed, and it is shown that one type can be identified with a subgroup of the other type, and a case where the two groups are isomorphic is exhibited. A result on the determination of the finite transformations of the infinitesimal generator of the larger group, which is useful for the determination of the invariant functions of the differential equation, is also given. In addition, the Levidecomposition of the Lie algebra associated with the larger group is found; the Levi factor of which is shown to be equal, up to a constant factor, to the Lie algebra associated with the smaller group.

1. Introduction

An invertible point transformation that maps every element in a family ℱ of differential equations of a specified form into the same family is commonly referred to as an equivalence transformation of the equation [1–3]. Elements of the family ℱ are generally labeled by a set of arbitrary functions, and the set of all equivalence transformations forms, in general, an infinite dimensional Lie group called the equivalence group of ℱ. One type of equivalence transformations usually considered [1, 4, 5] is that in which the arbitrary functions are also transformed. More specifically, if we denote by 𝐴=(𝐴₁,…,𝐴_𝑚) the arbitrary functions specifying the family element in ℱ, then for given independent variables 𝑥=(𝑥¹,…,𝑥^𝑝) and dependent variable 𝑦, this type of equivalence transformations takes the form(),(),(),𝑥=𝜑𝑧,𝑤,𝐵𝑦=𝜓𝑧,𝑤,𝐵𝐴=𝜁𝑧,𝑤,𝐵 (1.1a)(1.1b)(1.1c)where 𝑧=(𝑧¹,…,𝑧^𝑝) is the new set of independent variables, 𝑤=𝑤(𝑧) is the new dependent variable, and 𝐵=(𝐵₁,…,𝐵_𝑚) represents the new set of arbitrary functions. The original arbitrary functions 𝐴_𝑖 may be functions of 𝑥, 𝑦, and the derivatives of 𝑦 up to a certain order, although quite often they arise naturally as functions of 𝑥 alone, and for the equivalence transformations of the type (1.1a), (1.1b), and (1.1c), the corresponding equivalence group that we denote by 𝐺_𝑆 is simply the symmetry pseudogroup of the equation, in which the arbitrary functions are also considered as additional dependent variables.

The other type of equivalence transformations commonly considered [2, 6–8] involves only the ordinary variables of the equation, that is, the independent and the dependent variables, and thus with the notation already introduced, it consists of point transformations of the form(),().𝑥=𝜑𝑧,𝑤𝑦=𝜓𝑧,𝑤 (1.2a)(1.2b)If we let 𝐺 denote the resulting equivalence group, then it follows from a result of Lie [9] that 𝐺 induces another group of transformations 𝐺_𝑐 acting on the arbitrary functions of the equation. The invariants of the group 𝐺_𝑐 are what are referred to as the invariants of the family ℱ of differential equations, and they play a crucial role in the classification and integrability of differential equations [1, 6, 10–14].

In the recent scientific literature, there has been a great deal of interest for finding infinitesimal methods for the determination of invariant functions of differential equations [2, 7, 15–17]. Some of these methods consist in finding the infinitesimal (generic) generator 𝑋 of 𝐺_𝑆, and then using it in one way or another [7, 18] to obtain the infinitesimal generator 𝑋⁰ of 𝐺_𝑐, which gives the determining equations for the invariant functions. Most of these methods remain computationally demanding and in some cases quite inefficient, perhaps just because the connection between the three groups 𝐺, 𝐺_𝑐, and 𝐺_𝑆 does not seem to have been fully investigated.

We therefore present in this paper a comparison of the groups 𝐺 and 𝐺_𝑆 and show in particular that 𝐺 can be identified with a subgroup of 𝐺_𝑆, and we exhibit a case where the two groups are isomorphic. We also show that the generator 𝑋 of 𝐺_𝑆 admits a simple linear decomposition of the form 𝑋=𝑋¹+𝑋², where 𝑋¹ is an operator uniquely associated with 𝐺, and we also give a very simple and systematic method for extracting 𝑋¹ from 𝑋. This decomposition also turns out to be intimately associated with the Lie algebraic structure of the equation, as we show that 𝑋¹ and 𝑋² each generate a Lie algebra, the two of which are closely related to the components of the Levi decomposition of the Lie algebra of 𝐺_𝑆.

2. The Relationship between 𝐺 and 𝐺_𝑆

We will call type I the equivalence transformations of the form (1.2a) and (1.2b) and type II those of the form (1.1a), (1.1b), and (1.1c), whose equivalence groups we have denoted by 𝐺 and 𝐺_𝑆, respectively. When the coordinates system in which a vector field is expressed is clearly understood, it will be represented only by its components, so that a vector field 𝜔=𝜉𝜕_𝑥+𝜂𝜕_𝑦+𝜙𝜕_𝐴 (2.1) will be represented simply by 𝜔={𝜉,𝜂,𝜙}. On the other hand, for a vector 𝑎=(𝑎¹,…,𝑎^𝑛) representing a subset of coordinates, the notation 𝑓𝜕_𝑎 will mean 𝑓𝜕_𝑎=𝑓¹𝜕_𝑎1+⋯+𝑓^𝑛𝜕_𝑎𝑛,𝑓=𝑓¹,…,𝑓^𝑛. (2.2) Hence, with the notation introduced in the previous section, we may represent the generator 𝑋 of 𝐺_𝑆 as 𝑋=𝜉,𝜂,𝜙≡𝜉𝜕_𝑥+𝜂𝜕_𝑦+𝜙𝜕_𝐴. (2.3) Let 𝑉={𝜉,𝜂} be the projection of this generator into the (𝑥,𝑦)-space, and let 𝑉⁰={𝜉⁰,𝜂⁰} be the infinitesimal generator of 𝐺. Elements of ℱ may be thought of as differential equations of the form Δ𝑥,𝑦_(𝑛);𝐴₁,…,𝐴_𝑚=0, (2.4) where 𝑦_(𝑛) denotes 𝑦 and all its derivatives up to the order 𝑛. We have the following result.

Theorem 2.1. (a) The group 𝐺 can be identified with a subgroup of 𝐺_𝑆.
(b) The component functions 𝜉⁰ and 𝜂⁰ are particular values of the functions 𝜉 and 𝜂, respectively.

Proof. Suppose that the action of 𝐺_𝑐 induced by that of 𝐺 on the arbitrary functions of the equation is given by the transformations 𝐴_𝑖=𝛾_𝑖𝑧,𝑤,𝐵₁,…,𝐵_𝑚,𝑖=1,…,𝑚. (2.5) Then, since (1.2a) and (1.2b) leave the equation invariant except for the arbitrary functions, by also viewing the functions 𝐴_𝑖 as dependent variables, (1.2a) and (1.2b) together with (2.5) constitute a symmetry transformation of the equation. This is more easily seen if we consider the inverse transformations of (1.2a) and (1.2b) which may be put in the form𝜑,𝜓.𝑧=𝑥,𝑦𝑤=𝑥,𝑦 (2.6a)(2.6b)
If we now denote by 𝐵_𝑖=𝛾𝑥,𝑦,𝐴₁,…,𝐴_𝑚,𝑖=1,…,𝑚 (2.7) the resulting arbitrary functions in the transformed equation, it follows that in terms of the new set of variables 𝑧, 𝑤, and 𝐵_𝑖, any element of ℱ is locally invariant under (2.6a), (2.6b), and (2.7), and this proves the first part of the theorem. The second part of the theorem is an immediate consequence of the first part, for we can associate with any element (𝜑,𝜓) of 𝐺 a triplet (𝜑,𝜓,𝛾) in 𝐺_𝑆, where 𝛾 is the action in (2.5) induced by (𝜑,𝜓) on the arbitrary functions of the equation. The result thus follows by first recalling that 𝐺_𝑆 has generic generator 𝑋={𝜉,𝜂,𝜙} and by considering the infinitesimal counterpart of the finite transformations (𝜑,𝜓,𝛾), which must be of the form {𝜉⁰,𝜂⁰,𝜁⁰} for a certain function 𝜁⁰.

On the basis of Theorem 2.1, it is clear that one can obtain the generator 𝑉⁰={𝜉⁰,𝜂⁰} of 𝐺 by imposing on the projection 𝑉={𝜉,𝜂} of 𝑋 the set of minimum conditions Ω that reduces it to the infinitesimal generator of the equivalence group 𝐺 of ℱ, so that 𝑉⁰=𝑉_|Ω. It was also observed (see [7]) that in case 𝐴 is the function of 𝑥 alone, if we let 𝜙⁰ denote the resulting value of 𝜙 when these minimum conditions are also imposed on 𝑋={𝜉,𝜂,𝜙}, then the generator 𝑋⁰ of 𝐺_𝑐 can be obtained by setting 𝑋⁰={𝜉⁰,𝜙⁰}. However, the problem that arises is that of finding the simplest and most systematic way of extracting 𝑋¹=𝑋_|Ω={𝜉⁰,𝜂⁰,𝜙⁰} from 𝑋={𝜉,𝜂,𝜙}.

To begin with, we note that the coefficient 𝜙⁰ is an 𝑚-component vector that depends in general on (𝑝+1)+𝑚 variables, and finding its corresponding finite transformations by integrating the vector field {𝜉⁰,𝜂⁰,𝜙⁰} can be a very complicated task. Fortunately, once the finite transformations of the generator 𝑉⁰ of 𝐺 which are easier to find are known, we can easily obtain those associated with 𝜙⁰ using the following result.

Lemma 2.2. The finite transformations associated with the component 𝜙⁰ of 𝑋¹={𝜉⁰,𝜂⁰,𝜙⁰} are precisely given by the action (2.7) of 𝐺_𝑐 induced by that of (2.6a) and (2.6b).

Proof. Since 𝑋¹=𝑋_|Ω, where Ω is the set of minimum conditions to be imposed on 𝑉={𝜉,𝜂} to reduce it into an infinitesimal generator 𝑉⁰={𝜉⁰,𝜂⁰} of 𝐺, it first follows that once the finite transformations (2.6a) and (2.6b) corresponding to 𝑉⁰ are applied to the equation, the resulting equation is invariant, except for the expressions of the arbitrary functions which are now given by (2.7). Thus if (𝑧,𝑤,𝑏) are the new variables generated by the symmetry operator 𝑋¹, where 𝑏=(𝑏₁,…,𝑏_𝑚), then the only way to have an invariant equation is to set 𝑏_𝑖=𝛾𝑥,𝑦,𝐴₁,…,𝐴_𝑚,𝑖=1,…,𝑚, (2.8) where 𝛾 is the same function appearing in (2.7), and this readily proves the lemma.

3. Case of the General Third Order Linear ODE

We will look more closely at the connection between the two operators 𝑋 and 𝑋¹ by considering the case of the family of third-order linear ordinary differential equations (ODEs) of the form 𝑦⁽³⁾+𝑎¹(𝑥)𝑦^′+𝑎⁰(𝑥)𝑦=0, (3.1) which is said to be in its normal reduced form. Here, the arbitrary functions 𝐴_𝑖 of the previous section are simply the coefficients 𝑎^𝑗 of the equation. This form of the equation is in no way restricted, for any general linear third order ODE can be transformed into (3.1) by a simple change of the dependent variable [8, 16]. If we consider the arbitrary functions 𝑎^𝑗 as additional dependent variables, then by applying known procedures for finding Lie point symmetries [19–21], the infinitesimal generator 𝑋 of the symmetry group 𝐺_𝑆 in the coordinates system (𝑥,𝑦,𝑎¹,𝑎⁰) is found to be of the form𝑋=𝑓,𝑘₁+𝑓^′𝑦+𝑔,−2𝑎¹𝑓^′+𝑓⁽³⁾,𝐶₄, (3.2a) where 𝐶₄1=−𝑦𝑎⁰𝑔+𝑎¹𝑔^′+𝑔⁽³⁾−3𝑎⁰𝑓^′+𝑎¹𝑓^′′+𝑓⁽⁴⁾, (3.2b)and where 𝑓 and 𝑔 are arbitrary functions of 𝑥. The projection of 𝑋 in the (𝑥,𝑦)-space is therefore 𝑉=𝑓,𝑘₁+𝑓^′𝑦+𝑔, (3.3) and a simple observation of this expression shows that due to the homogeneity of (3.1), (3.3) may represent an infinitesimal generator of the equivalence group 𝐺 only if 𝑔=0. A search for the one-parameter subgroup exp(𝑡𝑊), satisfying exp(𝑡𝑊)(𝑥,𝑦)=(𝑧,𝑤) and generated by the resulting reduced vector field 𝑊={𝑓,(𝑘₁+𝑓^′)𝑦}, readily gives ̇(𝑧),̇𝑘𝑧=𝑓𝑤=₁+𝑓^′(𝑧)𝑤, (3.4) where 𝑧̇=𝑑𝑧,̇𝑑𝑡𝑤=𝑑𝑤.𝑑𝑡 (3.5) Integrating these last two equations while taking into account the initial conditions gives𝐽(𝑧)=𝑡+𝐽(𝑥),𝑤=𝑒^𝑘1𝑡𝑓(𝑧)𝑓(𝑥)𝑦, (3.6a)(3.6b) where 𝐽(𝑧)=𝑑𝑧𝑓(𝑧). (3.6c)Differentiating both sides of (3.6a) with respect to 𝑥 shows that 𝑑𝑧/𝑑𝑥=𝑓(𝑧)/𝑓(𝑥). Thus, if we assume that 𝑧 is explicitly given by 𝑧=𝐹_𝑡(𝑥)≡𝐹(𝑥), (3.7) for some function 𝐹, then this leads to𝑧=𝐹(𝑥),𝑤=𝑒^𝑘1𝑡𝐹^′(𝑥)𝑦, (3.8a)(3.8b)and we thus recover the well-known equivalence transformation [6, 8, 14] of (3.1). Therefore, the condition 𝑔=0 is the necessary and sufficient condition for the vector 𝑉 in (3.3) to represent the infinitesimal generator of 𝐺. In other words, the set Ω of necessary and sufficient conditions to be imposed on 𝑋 to obtain 𝑋¹ is reduced in this case to setting 𝑔=0. More explicitly, we have𝑉⁰=𝑓,𝑘₁+𝑓^′𝑦≡𝜉⁰,𝜂⁰𝜙₀,=−2𝑎¹𝑓^′+𝑓^′′′,−3𝑎⁰𝑓^′+𝑎¹𝑓^′′+𝑓⁽⁴⁾𝑋₁,=𝑓,𝑘₁+𝑓^′𝑦,−2𝑎¹𝑓^′+𝑓^′′′,−3𝑎⁰𝑓^′+𝑎¹𝑓^′′+𝑓⁽⁴⁾, (3.9a)(3.9b)(3.9c)where 𝑋¹=𝑋_|𝑔=0.

We would now like to derive some results on the algebraic structure of 𝐿_𝑆, the Lie algebra of the group 𝐺_𝑆 related to (3.1), and its connection with that for the corresponding group 𝐺. Thus, for any generator 𝑋 of 𝐺_𝑆, set 𝑋²=𝑋−𝑋¹, where 𝑋¹=𝑋_|𝑔=0 is given by (3.9c), while 𝑋² takes the form 𝑋²=0,𝑔,0,−1𝑦𝑎⁰𝑔+𝑎¹𝑔^′+𝑔⁽³⁾. (3.10)

Since 𝑋¹ depends on 𝑓 and 𝑘₁ while 𝑋² depends on 𝑔, we set 𝑋¹𝑓,𝑘₁=𝑓,𝑘₁+𝑓^′𝑦,−2𝑎¹𝑓^′+𝑓^′′′,−3𝑎⁰𝑓^′+𝑎¹𝑓^′′+𝑓⁽⁴⁾𝑋₂,𝑔=0,𝑔,0,−1𝑦𝑎⁰𝑔+𝑎¹𝑔^′+𝑔⁽³⁾, (3.11a)(3.11b)for any arbitrary functions 𝑓 and 𝑔 and arbitrary constant 𝑘₁. Let 𝐿₀, 𝐿₁, and 𝐿₂ be the vector spaces generated by 𝑋¹(𝑓,0), 𝑋¹(𝑓,𝑘₁), and 𝑋²(𝑔), respectively, where 𝑓, 𝑔, and 𝑘₁ are viewed as parameters. Let 𝐿_𝑆,0=𝐿₀∔𝐿₂ (3.12) be the subspace of the Lie algebra 𝐿_𝑆=𝐿₁∔𝐿₂ of 𝐺_𝑆. We note that 𝐿_𝑆,0 is obtained from 𝐿_𝑆 simply by setting 𝑘₁=0 in the generator 𝑋¹(𝑓,𝑘₁) of 𝐺_𝑆, which according to (3.8b) amounts to ignoring the constant factor 𝜆=𝑒^𝑘1𝑡 in the transformation of the dependent variable under 𝐺. Moreover, we have dim𝐿_𝑆,0=dim𝐿_𝑆−1, while 𝐿_𝑆 itself is infinite dimensional, in general.

Theorem 3.1. (a) The vector spaces 𝐿₀, 𝐿₁, and 𝐿₂ are all Lie subalgebras of 𝐿_𝑆.
(b) 𝐿₀ and 𝐿₂ are the components of the Levi decomposition of the Lie algebra 𝐿_𝑆,0, that is, 𝐿₀,𝐿₂⊂𝐿₂, (3.13) and 𝐿₂ is a solvable ideal while 𝐿₀ is semisimple.

Proof. A computation of the commutation relations of the vector fields shows that𝑋¹𝑓₁,𝑘₁,𝑋¹𝑓₂,𝑘₂=𝑋¹−𝑓₂𝑓^′₁+𝑓₁𝑓^′₂𝑋,0,₂𝑔₁,𝑋²𝑔₂𝑋=0,₁𝑓₁,𝑘₁,𝑋²𝑔₁=𝑋²𝑓₁𝑔^′₁−𝑔₁𝑘₁+𝑓^′₁, (3.14a)(3.14b)(3.14c)where the 𝑓_𝑗 and 𝑔_𝑗 are arbitrary functions, while the 𝑘_𝑗 are arbitrary constants. Consequently, it readily follows from (3.14a) and (3.14b) that 𝐿₁ and 𝐿₂ are Lie subalgebras of 𝐿_𝑆, while setting 𝑘₁=𝑘₂=0 in (3.14a) shows that 𝐿₀ is also a Lie subalgebra, and this proves the first part of the theorem. Moreover, it follows from the commutation relations (3.14a), (3.14b), and (3.14c) that [𝐿_𝑆,𝐿_𝑆]⊂𝐿_𝑆,0, and hence that 𝐿_𝑆,0 is an ideal of 𝐿_𝑆, while (3.14b) and (3.14c) show that 𝐿₂ is an abelian ideal in 𝐿_𝑆, and in particular in 𝐿_𝑆,0. Thus, we are only left with showing that 𝐿₀ is a semisimple subalgebra of 𝐿_𝑆,0. Clearly, [𝐿₀,𝐿₀]≠0, and if 𝐿₀ had a proper ideal 𝐴, then for a given nonzero operator 𝑋¹(𝐻,0) in 𝐴, all operators 𝑋¹(−𝑓𝐻^′+𝑓^′𝐻,0) would be in 𝐴 for all possible functions 𝑓. However, since for every function ℎ of 𝑥 the equation −𝑓𝐻^′+𝑓^′𝐻=ℎ (3.15) admits a solution in 𝑓, it follows that 𝐴 would be equal to 𝐿₀. This contradiction shows that 𝐿₀ has no proper ideal and is therefore a simple subalgebra of 𝐿_𝑆,0.

Note that part (b) of Theorem 3.1 can also be interpreted as stating that up to a constant factor, 𝑋¹ and 𝑋² generate the components of the Levi decomposition of 𝐿_𝑆. The theorem therefore shows that the decomposition 𝑋=𝑋¹+𝑋² is not fortuitous, as it is intimately associated with the the Levi decomposition of 𝐿_𝑆, and this decomposition is unique up to isomorphism for any given Lie algebra.

Although we have stated the results of this theorem only for the general linear third order equation (3.1) in its normal reduced form, these results can certainly be extended to the general linear ODE 𝑦^(𝑛)+𝑎^𝑛−1𝑦^(𝑛−1)+𝑎^𝑛−2𝑦^(𝑛−2)+⋯+𝑎⁰𝑦=0 (3.16) of an arbitrary order 𝑛≥3. We first note that if we write the infinitesimal generator 𝑋 of the symmetry group 𝐺_𝑆 of this equation in the form 𝑋=𝜉,𝜂,𝜙≡𝜉𝜕_𝑥+𝜂𝜕_𝑦+𝜙𝜕_𝐴, (3.17) where 𝐴={𝑎^𝑛−1,𝑎^𝑛−2,…,𝑎⁰} is the set of all arbitrary functions, then on account of the linearity of the equation, we must have 𝜂=ℎ𝑦+𝑔 (3.18) for some arbitrary functions ℎ and 𝑔. Now, let again 𝑋¹={𝜉⁰,𝜂⁰,𝜙⁰} and 𝑋² be given by 𝑋¹=𝑋_|𝑔=0,𝑋²=𝑋−𝑋¹, (3.19) and set 𝑋⁰={𝜉⁰,𝜙⁰}. We have shown in another recent paper [16] that 𝑋⁰ thus obtained using 𝑔=0 as the minimum set of conditions is the infinitesimal generator of the group 𝐺_𝑐 for 𝑛=3,4,5. This should certainly also hold for the linear equation (3.16) of a general order, and we thus propose the following.

Conjecture 3.2. For the general linear ODE (3.16), 𝑋⁰={𝜉,𝜙}_|𝑔=0 is the infinitesimal generator of 𝐺_𝑐, where 𝑋={𝜉,𝜂,𝜙} is the generator of 𝐺_𝑆.

As already noted, it has been proved [7] that for any family ℱ of (linear or nonlinear) differential equations of any order in which the arbitrary functions depend on the independent variables alone, if 𝑋¹={𝜉⁰,𝜂⁰,𝜙⁰} is obtained by setting 𝑋¹=𝑋_|Ω for some set Ω of minimum conditions that reduce 𝑉={𝜉,𝜂} into a generator of 𝐺, then 𝑋⁰={𝜉⁰,𝜙⁰} is the generator of 𝐺_𝑐. However, the difficulty lies in finding the set Ω of minimum conditions, and we have proved that for (3.1), Ω is given by {𝑔=0} and extended this as a conjecture for a general linear homogeneous ODE.

Moreover, calculations done for equations of low order up to five suggest that all subalgebras appearing in Theorem 3.1 can also be defined in a similar way for the general linear equation (3.16) and that all the results of the theorem also hold for this general equation.

We now wish to pay some attention to the converse of part (a) of Theorem 2.1 which states that for any given family ℱ of differential equations, the group 𝐺 can be identified with a subgroup of 𝐺_𝑆. From the proof of that theorem, it appears that the symmetry group 𝐺_𝑆 is much larger in general, because there are symmetry transformations that do not arise from type I equivalence transformations. A simple example of such a symmetry is given by the term 𝑋² appearing in (3.10) of the symmetry generator of (3.1). Indeed, by construction, its projection 𝑋^2,0={0,𝑔} in the (𝑥,𝑦)-space does not match any particular form of the generic infinitesimal generator 𝑉⁰={𝑓,(𝑘₁+𝑓^′)𝑦} of 𝐺, where 𝑓 is an arbitrary function and 𝑘₁ an arbitrary constant.

Nevertheless, although (3.1) gives an example in which the inclusion 𝐺⊂𝐺_𝑆 is strict, there are equations for which the two groups are isomorphic. Such an equation is given by the nonhomogeneous version of (3.1) which may be put in the form 𝑦⁽³⁾+𝑎¹(𝑥)𝑦^′+𝑎⁰(𝑥)𝑦+𝑟(𝑥)=0, (3.20) where 𝑟 is also an arbitrary function, in addition to 𝑎¹ and 𝑎⁰. The linearity of this equation forces its equivalence transformations to be of the form 𝑥=𝑓(𝑧),𝑦=ℎ(𝑧)𝑤+𝑔(𝑧), (3.21) and the latter change of variables transforms (3.20) into an equation of the form 𝑤^′′′+𝐵₂𝑤^′′+𝐵₁𝑤^′+𝐵₀𝑤+𝐵₋₁=0, (3.22) where the 𝐵_𝑗, for 𝑗=−1,…,2 are functions of 𝑧 and 𝐵₂ℎ=3^′ℎ−𝑓^′′𝑓^′. (3.23) The required vanishing of 𝐵₂ shows that the necessary and sufficient condition for (3.21) to represent an equivalence transformation of (3.20) is to have ℎ=𝜆𝑓^′ for some arbitrary constant 𝜆. The equivalence transformations of (3.20) are therefore given by 𝑥=𝑓(𝑧),𝑦=𝜆𝑓^′(𝑧)𝑤+𝑔(𝑧). (3.24) On the other hand, the generator 𝑋 of the symmetry group 𝐺_𝑆 of the nonhomogeneous equation (3.20) in the coordinates system (𝑥,𝑦,𝑎¹,𝑎⁰,𝑟) is found to be of the form𝑋=𝐽,𝑘₁+𝐽^′𝑦+𝑃,−2𝑎¹𝐽^′+𝐽^′′′,𝐶₃,𝜙⁴, (3.25a) where 𝐶₃=−1𝑦𝑎⁰𝑃+𝜙⁴+2𝑟𝐽^′−𝑟𝑘₁+𝑎¹𝑃^′+𝑃⁽³⁾−3𝑎⁰𝐽^′+𝑎¹𝐽^′′+𝐽⁽⁴⁾, (3.25b)and where 𝐽 and 𝑃 are arbitrary functions of 𝑥 and 𝑘₁ is an arbitrary constant, while 𝜙⁴ is an arbitrary function of 𝑥,𝑦,𝑎¹,𝑎⁰, and 𝑟. Thus, 𝑋 has projection 𝑉={𝐽,(𝑘₁+𝐽^′)𝑦+𝑃} on (𝑥,𝑦)-space and this is exactly the infinitesimal transformation of (3.24). Consequently, the minimum set Ω of conditions to be imposed on 𝑉 to reduce it into the infinitesimal generator 𝑉⁰={𝜉⁰,𝜂⁰} of 𝐺 is void in this case, and hence 𝑋=𝑋¹=𝜉⁰,𝜂⁰,𝜙⁰. (3.26) It thus follows from Lemma 2.2 that the finite transformations associated with 𝑋 are given precisely by (3.24), together with the corresponding induced transformations of the arbitrary functions 𝑎¹,𝑎⁰, and 𝑟. Consequently, to each symmetry transformation 𝑋 in 𝐺_𝑆, there corresponds a unique equivalence transformation in 𝐺 and vice versa. We have thus proved the following results.

Proposition 3.3. For the nonhomogeneous equation (3.20), the groups 𝐺 and 𝐺_𝑆 are isomorphic.

This proposition should certainly also hold for the nonhomogeneous version of the general linear equation (3.16) of an arbitrary order 𝑛. In such cases, invariants of the differential equation are determined simply by searching the symmetry generator 𝑋 of 𝐺_𝑆, which must satisfy (3.26) and then solving the resulting system of linear first-order partial differential equations (PDEs) resulting from the determining equation of the form 𝑋^0,𝑚⋅𝐹=0, (3.27) where 𝑋^0,𝑚 is the generator 𝑋⁰={𝜉⁰,𝜙⁰} of 𝐺_𝑐 prolonged to the desired order 𝑚 of the unknown invariants 𝐹.

4. Concluding Remarks

Because type I equivalence group 𝐺 can be identified with a subgroup of type II equivalence group 𝐺_𝑆, every function invariant under 𝐺_𝑆 must be invariant under 𝐺, and hence 𝐺 has much more invariant functions than 𝐺_𝑆, and functions invariant under 𝐺 are naturally much easier to find than those invariant under 𝐺_𝑆. If we consider for instance the third order linear equation (3.1), it is well known [6] that its first nontrivial invariant function is given by the third order differential invariant 4Ψ=−9𝑎¹𝜇²+7𝜇^′2−6𝜇𝜇^′′3𝜇⁸, (4.1) where 𝜇(𝑥)=−2𝑎⁰+𝑎^1′, while at order four [16] it has two differential invariants, Ψ₁Ψ=Ψ,₂=−118𝜇⁴216𝑎⁰⁴−324𝑎⁰³𝑎^1′+18𝑎⁰²9𝑎^1′2+2𝑎¹𝜇^′+9𝜇²𝜇⁽³⁾+−118𝜇⁴𝜇^′28𝜇^′2+9𝑎^1′𝑎¹𝑎^1′−4𝜇^′′−9𝑎⁰3𝑎^1′3+4𝑎¹𝑎^1′𝜇^′−8𝜇^′𝜇^′′. (4.2)

It can be verified on the other hand that 𝐺_𝑆 has no nontrivial differential invariants up to the order four.

Acknowledgment

This publication was made possible in part by a grant from the NRF CSUR program of South Africa.