Loewy decomposition explained

In the study of differential equations, the Loewy decomposition breaks every linear ordinary differential equation (ODE) into what are called largest completely reducible components. It was introduced by Alfred Loewy.^[1]

Solving differential equations is one of the most important subfields in mathematics. Of particular interest are solutions in closed form. Breaking ODEs into largest irreducible components, reduces the process of solving the original equation to solving irreducible equations of lowest possible order. This procedure is algorithmic, so that the best possible answer for solving a reducible equation is guaranteed. A detailed discussion may be found in.^[2]

Loewy's results have been extended to linear partial differential equations (PDEs) in two independent variables. In this way, algorithmic methods for solving large classes of linear PDEs have become available.

Decomposing linear ordinary differential equations

Let $D\; \backslash equiv\; \backslash frac$ denote the derivative with respect to the variable

x .A differential operator of order n is a polynomial of the form$$L\backslash equiv\; D^n\; +\; a\_1\; D^\; +\; \backslash cdots\; +\; a\_\; D\; +\; a\_n$$where the coefficients a _{i} , i = 1 , \ldots, n are from some function field, the base field of L . Usually it is the field of rational functions in the variable x , i.e. a _{i} \in \Q(x) . If y is an indeterminate with $\backslash frac\; \backslash neq\; 0$, Ly becomes a differential polynomial, and Ly = 0 is the differential equation corresponding to L .

An operator

L of order n is called reducible if it may be represented as the product of two operators L _{1} and L _{2} , both of order lower than n . Then one writes L = L _{1} L _{2} , i.e. juxtaposition means the operator product, it is defined by the rule D a _{i} = a _{i} D + a _{i'} ; L _{1} is called a left factor of L , L _{2} a right factor. By default, the coefficient domain of the factors is assumed to be the base field of L , possibly extended by some algebraic numbers, i.e. \bar\Q(x) is allowed. If an operator does not allow any right factor it is called irreducible.

For any two operators

L _{1} and L _{2} the least common left multiple \operatorname{Lclm}(L _{1,} L _{2)} is the operator of lowest order such that both L _{1} and L _{2} divide it from the right. The greatest common right divisior \operatorname{Gcrd}(L _{1,L} _{2)} is the operator of highest order that divides both L _{1} and L _{2} from the right. If an operator may be represented as \operatorname{Lclm} of irreducible operators it is called completely reducible. By definition, an irreducible operator is called completely reducible.

If an operator is not completely reducible, the

\operatorname{Lclm} of its irreducible right factors is divided out and the same procedure is repeated with the quotient. Due to the lowering of order in each step, this proceeding terminates after a finite number of iterations and the desired decomposition is obtained. Based on these considerations, Loewy obtained the following fundamental result.

The decomposition determined in this theorem is called the Loewy decomposition of

L . It provides a detailed description of the function space containing the solution of a reducible linear differential equation Ly = 0 .

For operators of fixed order the possible Loewy decompositions, differing by the number and the order of factors, may be listed explicitly; some of the factors may contain parameters. Each alternative is called a type of Loewy decomposition. The complete answer for

n = 2 is detailed in the following corollary to the above theorem.^[3]

Corollary 1Let

L be a second-order operator. Its possible Loewy decompositions are denoted by l L ^{2} _{0 ,} \ldots, l L ^{2} _{3} , they may be described as follows; l ^{(i)} and l ^{(i)} _{j} are irreducible operators of order i ; C is a constant.

The decomposition type of an operator is the decomposition

l L ^{2} _{i} with the highest value of i . An irreducible second-order operator is defined to have decomposition type l L ^{2} _{0} .

The decompositions

l L ^{2} _{0} , l L ^{2} _{2} and l L ^{2} _{3} are completely reducible.

If a decomposition of type

l L ^{2} _{i} , i = 1 , 2 or 3 has been obtained for asecond-order equation Ly = 0 , a fundamental system may be given explicitly.

Corollary 2Let

L be a second-order differential operator, $D\; \backslash equiv\; \backslash frac$, y a differential indeterminate, and a _{i} \in \Q(x) . Define $\backslash varepsilon\_i(x)\; \backslash equiv\; \backslash exp$ for i = 1 , 2 and $\backslash varepsilon(x,C)\; \backslash equiv\; \backslash exp$, C is a parameter; the barred quantities \bar{C} and \bar{\bar{C}} are arbitrary numbers, \bar{C} ≠ \bar{\bar{C}} . For the three nontrivial decompositions of Corollary 1 the following elements y _{1} and y _{2} of a fundamental system are obtained.

$$\backslash mathcal\; L^2\_1:\; Ly\; =\; (D\; +\; a\_2)(D\; +\; a\_1)y\; =\; 0;$$$$y\_1=\backslash varepsilon\_1(x),\; \backslash quad\; y\_2\; =\; \backslash varepsilon\_1(x)\; \backslash int\; \backslash frac\backslash ,dx.$$$$\backslash mathcal\; L^2\_2\; :\; Ly\; =\; \backslash operatorname(D\; +\; a\_2,\; D\; +\; a\_1)y\; =\; 0;$$$$y\_i\; =\; \backslash varepsilon\_i(x);$$

a _{1} is not equivalent to a _{2} .

$$\backslash mathcal\; L^2\_3\; :\; Ly\; =\; \backslash operatorname(D\; +\; a(C))\; y\; =\; 0;$$$$y\_1\; =\; \backslash varepsilon(x,\; \backslash bar)$$$$y\_2\; =\; \backslash varepsilon(x,\; \backslash bar).$$

Here two rational functions

p, q \in \Q(x) are called equivalent if there exists another rational function r \in \Q(x) such that $$p\; -\; q\; =\; \backslash frac.$$

There remains the question how to obtain a factorization for a given equation or operator. It turns out that for linear ode's finding the factors comes down to determining rational solutions of Riccati equations or linear ode's; both may be determined algorithmically. The two examples below show how the above corollary is applied.

Example 1Equation 2.201 from Kamke's collection.^[4] has the

l L ^{2} _{2} decomposition$$y$$ + \left(2 + \frac\right)y' - \fracy = \operatorname\left(D + \frac - \frac, D+2+\frac - \frac\right) y = 0 .

The coefficients $a\_1\; =\; 2+\backslash frac\; -\; \backslash frac$ and $a\_2\; =\; \backslash frac\; -\; \backslash frac$ are rational solutions of the Riccati equation $a\text{'}\; -\; a^2\; +\; \backslash left(2\; +\; \backslash frac\backslash right)\; +\; \backslash frac\; =\; 0$, they yield the fundamental system$$y\_1\; =\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac,$$$$y\_2\; =\; \backslash frac\; +\; \backslash frace^.$$

Example 2An equation with a type

l L ^{2} _{3} decomposition is$$y$$ - \fracy = \operatorname\left(D+\frac - \frac\right)y =0.

The coefficient of the first-order factor is the rational solution of $a\text{'}\; -\; a^2\; +\; \backslash frac\; =\; 0$. Upon integration the fundamental system $y\_1\; =\; x^3$ and $y\_2\; =\; \backslash frac$ for

C = 0 and C \to inf ty respectively is obtained.

These results show that factorization provides an algorithmic scheme for solving reducible linear ode's. Whenever an equation of order 2 factorizes according to one of the types defined above the elements of a fundamental system are explicitly known, i.e. factorization is equivalent to solving it.

A similar scheme may be set up for linear ode's of any order, although the number of alternatives grows considerably with the order; for order

n =3 the answer is given in full detail in.

If an equation is irreducible it may occur that its Galois group is nontrivial, then algebraic solutions may exist.^[5] If the Galois group is trivial it may be possible to express the solutions in terms of special function like e.g. Bessel or Legendre functions, see ^[6] or.^[7]

Basic facts from differential algebra

In order to generalize Loewy's result to linear PDEs it is necessary to apply the more general setting of differential algebra. Therefore, a few basic concepts that are required for this purpose are given next.

A field

l F is called a differential field if it is equipped with a derivation operator. An operator \delta on a field l F is called a derivation operator if \delta(a + b) = \delta(a) + \delta(b) and \delta(ab) = \delta(a)b + a\delta(b) for all elements a,b \in l F . A field with a single derivation operator is called an ordinary differential field; if there is a finite set containing several commuting derivation operators the field is called a partial differential field.

Here differential operators with derivatives $\backslash partial\_x\; =\; \backslash frac$ and $\backslash partial\_y\; =\; \backslash frac$ with coefficients from some differential field are considered. Its elements have the form $\backslash sum\_\; r\_(x,y)\; \backslash partial\_x^i\; \backslash partial\_y^j$; almost all coefficients

r _{i,j} are zero. The coefficient field is called the base field. If constructive and algorithmic methods are the main issue it is \Q(x,y) . The respective ring of differential operators is denoted by l D = \Q(x,y)[\partial _{x,\partial} _{y]} or l D = l F[\partial _{x,\partial} _{y]} . The ring l D is non-commutative, $\backslash partial\_x\; a\; =\; a\backslash partial\_x\; +\; \backslash frac$ and similarly for the other variables; a is from the base field.

For an operator $L\; =\; \backslash sum\_\; r\_(x,y)\; \backslash partial\_x^i\; \backslash partial\_y^j$ of order

n the symbol of L is the homogeneous algebraic polynomial $\backslash operatorname(L)\; \backslash equiv\; \backslash sum\_\; r\_(x,y)\; X^i\; Y^j$ where X and Y algebraic indeterminates.

Let

I be a left ideal which is generated by l _{i} \in l D , i = 1 , \ldots, p . Then one writes I = \langle l _{1,} \ldots, l _{p\rangle} . Because right ideals are not considered here, sometimes I is simply called an ideal.

The relation between left ideals in

l D and systems of linear PDEs is established as follows. The elements l _{i\inlD} are applied to a single differential indeterminate z . In this way the ideal I = \langle l _{1,} l _{2,} \ldots \rangle corresponds to the system of PDEs l _{1z} = 0 , l _{2z} = 0 , \ldots for the single function z .

The generators of an ideal are highly non-unique; its members may be transformed in infinitely many ways by taking linear combinations of them or its derivatives without changing the ideal. Therefore, M. Janet^[8] introduced a normal form for systems of linear PDEs (see Janet basis).^[9] They are the differential analog to Gröbner bases of commutative algebra (which were originally introduced by Bruno Buchberger);^[10] therefore they are also sometimes called differential Gröbner basis.

In order to generate a Janet basis, a ranking of derivatives must be defined. It is a total ordering such that for any derivatives

\delta , \delta _{1} and \delta _{2} , and any derivation operator \theta the relations \delta \preceq \theta \delta , and \delta _{1} \preceq \delta _{2 →} \delta \delta _{1} \preceq \delta \delta _{2} are valid. Here graded lexicographic term orderings grlex are applied. For partial derivatives of a single function their definition is analogous to the monomial orderings in commutative algebra. The S-pairs in commutative algebra correspond to the integrability conditions.

If it is assured that the generators

l _{1,} \ldots, l _{p} of an ideal I form a Janet basis the notation I = { \ langle \ langle} l _{1,} \ldots, l _{p{\rangle\rangle}} is applied.

Example 3Consider the ideal$$I=\backslash Big\backslash langle\; l\_1\; \backslash equiv\; \backslash partial\_\; -\; \backslash frac\; \backslash partial\_x\; -\; \backslash frac\backslash partial\_y,\; \backslash ;\; l\_2\; \backslash equiv\; \backslash partial\_\; +\; \backslash frac\; \backslash partial\_y,\; \backslash ;\; l\_3\; \backslash equiv\; \backslash partial\_\; +\; \backslash frac\; \backslash partial\_y\; \backslash Big\backslash rangle$$in

grlex term order with x \succ y . Its generators are autoreduced. If the integrability condition$$l\_\; =\; l\_-l\_\; =\; \backslash frac\; \backslash partial\_\; +\; \backslash frac\; \backslash partial\_$$is reduced with respect to I , the new generator \partial _{y} is obtained. Adding it to the generators and performing all possible reductions, the given ideal is represented as $I\; =\; \backslash left\backslash langle\backslash left\backslash langle\backslash partial\_\; -\; \backslash frac\; \backslash partial\_x,\; \backslash partial\_y\; \backslash right\backslash rangle\backslash right\backslash rangle$.Its generators are autoreduced and the single integrability condition is satisfied, i.e. they form a Janet basis.

Given any ideal

I it may occur that it is properly contained in some larger ideal J with coefficients in the base field of I ; then J is called a divisor of I . In general, a divisor in a ring of partial differential operators need not be principal.

The greatest common right divisor (Gcrd) or sum of two ideals

I and J is the smallest ideal with the property that both I and J are contained in it. If they have the representation I\equiv\langle f _{1,\ldots,f} _{p\rangle} and J\equiv\langle g _{1,\ldots,g} _{q\rangle,} f _{i} , g _{j\inlD} for all i and j , the sum is generated by the union of the generators of I and J . The solution space of the equations corresponding to \operatorname{Gcrd}(I,J) is the intersection of the solution spaces of its arguments.

The least common left multiple (Lclm) or left intersection of two ideals

I and J is the largest ideal with the property that it is contained both in I and J . The solution space of \operatorname{Lclm}(I,J)z =0 is the smallest space containing the solution spaces of its arguments.

A special kind of divisor is the so-called Laplace divisor of a given operator

L , page 34. It is defined as follows.

DefinitionLet

L be a partial differential operator in the plane; define$$\backslash mathfrak\; l\_m\backslash equiv\backslash partial\_\; +\; a\_\backslash partial\_\; +\; \backslash dots\; +\; a\_1\backslash partial\_x\; +\; a\_0$$and$$\backslash mathfrak\; k\_n\backslash equiv\backslash partial\_\; +\; b\_\backslash partial\_\; +\; \backslash dots\; +\; b\_1\backslash partial\_y\; +\; b\_0$$be ordinary differential operators with respect to x or y ; a _{i,} b _{i} \in \Q(x,y) for all i; m and n are natural numbers not less than 2. Assume the coefficients a _{i} , i =0 , \ldots, m -1 are such that L and ak l _{m} form a Janet basis. If m is the smallest integer with this property then L _{x ^{m}} (L)\equiv{\langle\langle} L, ak l _{m{\rangle\rangle}} is called a Laplace divisor of L . Similarly, if b _{j} , j =0 , \ldots, n -1 are such that L and ak k _{n} form a Janet basis and n is minimal, then L _{y ^{n}} (L)\equiv{\langle\langle} L, ak k _{n{\rangle\rangle}} is also called a Laplace divisor of L .

In order for a Laplace divisor to exist the coeffients of an operator

L must obey certain constraints. An algorithm for determining an upper bound for a Laplace divisor is not known at present, therefore in general the existence of a Laplace divisor may be undecidable.

Decomposing second-order linear partial differential equations in the plane

Applying the above concepts Loewy's theory may be generalized to linear PDEs. Here it is applied to individual linear PDEs of second order in the plane with coordinates

x and y , and the principal ideals generated by the corresponding operators.

Second-order equations have been considered extensively in the literature of the 19th century,.^{[11] [12]} Usually equations with leading derivatives

\partial _{xx} or \partial _{xy} are distinguished. Their general solutions contain not only constants but undetermined functions of varying numbers of arguments; determining them is part of the solution procedure. For equations with leading derivative \partial _{xx} Loewy's results may be generalized as follows.

Theorem 2Let the differential operator

L be defined by$$L\; \backslash equiv\; \backslash partial\_\; +\; A\_1\; \backslash partial\_\; +\; A\_2\; \backslash partial\_\; +\; A\_3\; \backslash partial\_x\; +\; A\_4\; \backslash partial\_y\; +\; A\_5$$ where A _{i} \in \Q(x,y) for all i .

Let

l _{i} \equiv \partial _{x} + a _{i} \partial _{y} + b _{i} for i = 1 and i = 2 , and l(\Phi) \equiv \partial _{x} + a\partial _{y} + b(\Phi) be first-order operators with a _{i,} b _{i,} a\in\Q(x,y) ; \Phi is an undetermined function of a single argument. Then L has a Loewy decomposition according to one of the following types.l L ^{1 :} _{xx} L = l _{2l} _{1;} l L ^{2 :} _{xx} L = \operatorname{Lclm}(l _{2,l} _{1);} l L ^{3 :} _{xx} L = \operatorname{Lclm}(l(\Phi)).

The decomposition type of an operator

L is the decomposition l L ^{i} _{xx} with the highest value of i . If L does not have any first-order factor in the base field, its decomposition type is defined to be l L ^{0} _{xx} . Decompositions l L ^{0} _{xx} , l L ^{2} _{xx} and l L ^{3} _{xx} are completely reducible.

In order to apply this result for solving any given differential equation involving the operator

L the question arises whether its first-order factors may be determined algorithmically. The subsequent corollary provides the answer for factors with coefficients either in the base field or a universal field extension.

Corollary 3In general, first-order right factors of a linear pde in the base field cannot be determined algorithmically. If the symbol polynomial is separable any factor may be determined. If it has a double root in general it is not possible to determine the right factors in the base field. The existence of factors in a universal field, i.e. absolute irreducibility, may always be decided.

The above theorem may be applied for solving reducible equations in closed form. Because there are only principal divisors involved the answer is similar as for ordinary second-order equations.

Proposition 1Let a reducible second-order equation $$Lz\; \backslash equiv\; z\_\; +\; A\_1\; z\_\; +\; A\_2\; z\_\; +\; A\_3\; z\_x\; +\; A\_4\; z\_y\; +\; A\_5\; z\; =\; 0$$ where

A _{1,} \ldots, A _{5} \in \Q(x,y) .

Define

l _{i} \equiv \partial _{x} + a _{i} \partial _{y} + b _{i} , a _{i,} b _{i} \in \Q(x,y) for i = 1 , 2 ; \varphi _{i(x,y)} = const is a rational first integral of (dy)/(dx) = a _{i(x,y)} ; \bar{y} \equiv \varphi _{i(x,y)} and the inverse y = \psi _{i(x,\bar{y})} ; both \varphi _{i} and \psi _{i} are assumed to exist. Furthermore, define$$\backslash mathcal\; E\_i(x,y)\; \backslash equiv\; \backslash left.\backslash exp\backslash left(-\backslash int\; b\_i(x,y)\backslash big|\_dx\backslash right)\backslash right|\_$$ for i = 1 , 2 .

A differential fundamental system has the following structure for the various decompositions into first-order components.

$$\backslash mathcal\; L^1\_:\; z\_1(x,y)=\backslash mathcal\; E\_1(x,y)F\_1(\backslash varphi\_1),$$$$z\_2(x,y)=\backslash mathcal\; E\_1(x,y)\backslash fracF\_2\; \backslash big(\backslash varphi\_2(x,y)\backslash big)\backslash big|\_dx\; \backslash Big|\_;$$$$\backslash mathcal\; L^2\_:\; z\_i(x,y)\; =\; \backslash mathcal\; E\_i(x,y)\; F\_i\backslash big(\backslash varphi\_i(x,y)\backslash big),\; i=1,2;$$$$\backslash mathcal\; L^3\_:\; z\_i(x,y)\; =\; \backslash mathcal\; E\_i(x,y)\; F\_i\backslash big(\backslash varphi(x,y)\backslash big),i=1,2.$$

The

F _{i} are undetermined functions of a single argument; \varphi ,\varphi _{1} and \varphi _{2} are rational in all arguments; \psi _{1} is assumedto exist. In general \varphi _{1 ≠ \varphi} _{2} , they are determinedby the coefficients A _{1} , A _{2} and A _{3} of the given equation.

A typical example of a linear pde where factorization applies is an equation that has been discussed by Forsyth,^[13] vol. VI, page 16,

Example 5 (Forsyth 1906)Consider the differential equation $z\_\; -\; z\_\; +\; \backslash frac\; z\_x\; =\; 0$. Upon factorization the representation$$Lz\; \backslash equiv\; l\_2\; l\_1\; z\; =\; \backslash left(\backslash partial\_x\; +\; \backslash partial\_y\; +\; \backslash frac\backslash right)\; \backslash left(\backslash partial\_x\; -\; \backslash partial\_y\; +\; \backslash frac\; \backslash right)z\; =\; 0$$is obtained. There follows$$\backslash varphi\_1(x,y)=x+y,\backslash psi\_1(x,y)=\backslash bar-x,\; \backslash mathcal\; E\_1(x,y)\; =\; \backslash exp,$$$$\backslash varphi\_2(x,y)=x-y,\; \backslash psi\_2(x,y)\; =\; x-\backslash bar,\; \backslash mathcal\; E\_2(x,y)\; =\; -\backslash frac.$$

Consequently, a differential fundamental system is

$$z\_1(x,y)\; =\; \backslash expF(x\; +\; y),$$$$z\_2(x,y)\; =\; \backslash frac\; \backslash exp\; \backslash int\backslash expG(2x-\backslash bar)dx\; \backslash Big|\_.$$

F and G are undetermined functions.

If the only second-order derivative of an operator is

\partial _{xy} , its possible decompositionsinvolving only principal divisors may be described as follows.

Theorem 3Let the differential operator

L be defined by$$L\; \backslash equiv\; \backslash partial\_\; +\; A\_1\; \backslash partial\_x\; +\; A\_2\; \backslash partial\_y\; +\; A\_3$$ where A _{i} \in \Q(x,y) for all i .

Let

l \equiv \partial _{x} + A _{2} and k \equiv \partial _{y} + A _{1} are first-order operators. L has Loewy decompositions involving first-order principal divisors of the following form.l L ^{1 :} _{xy} L = kl; l L ^{2 :} _{xy} L = lk; l L ^{3 :} _{xy} L = \operatorname{Lclm}(k,l).

The decomposition type of an operator

L is the decomposition l L ^{i} _{xy} with highest value of i . The decomposition of type l L ^{3} _{xy} is completely reducible

In addition there are five more possible decomposition types involving non-principal Laplace divisors as shown next.

Theorem 4Let the differential operator

L be defined by$$L\backslash equiv\backslash partial\_\; +\; A\_1\backslash partial\_x\; +\; A\_2\backslash partial\_y\; +\; A\_3$$ where A _{i} \in \Q(x,y) for all i .L _{x ^{m}} (L) and L _{y ^{n}} (L) as well as ak l _{m} and ak k _{n} are defined above; furthermore l\equiv \partial _{x+a} , k\equiv\partial _{y+b} , a,b\in\Q(x,y) . L has Loewy decompositions involving Laplace divisors according to one of the following types; m and n obey m,n\geq 2 .

$$\backslash mathcal\; L\_^4:\; L=\; \backslash operatorname\; \backslash left(\backslash mathbb\_(L),\; \backslash mathbb\_(L)\backslash right);$$$$\backslash mathcal\; L\_^7:\; L=\; \backslash operatorname\backslash big(k,\backslash mathbb\_(L)\backslash big);$$$$\backslash mathcal\; L\_^8:\; L=\; \backslash operatorname\backslash big(l,\backslash mathbb\_(L)\backslash big).$$

If

L does not have a first order right factor and it may be shown that a Laplace divisor does not exist its decomposition type is defined to be l L ^{0} _{xy} . The decompositions l L ^{0} _{xy} , l L ^{4} _{xy} , l L ^{7} _{xy} and l L ^{8} _{xy} are completely reducible.

An equation that does not allow a decomposition involving principal divisors but is completely reducible with respect to non-principal Laplace divisors of type

l L ^{4} _{xy} has been considered by Forsyth.

Example 6 (Forsyth 1906) Define$$L\; \backslash equiv\; \backslash partial\_\; +\; \backslash frac\; \backslash partial\_xv-\; \backslash frac\; \backslash partial\_y\; -\; \backslash frac$$generating the principal ideal

\langle L\rangle . A first-order factor does not exist. However, there are Laplace divisors$$\backslash mathbb\; L\_(L)\backslash equiv\; \backslash partial\_-\backslash frac\backslash partial\_x+\backslash frac,L$$ and $$\backslash mathbb\; L\_(L)\backslash equiv\; L,\backslash partial\_+\backslash frac\backslash partial\_y+\backslash frac.$$

The ideal generated by

L has the representation \langle L\rangle = \operatorname{Lclm} ( L _{x ^{2}} (L), L _{y ^{2}} (L) ) , i.e. it is completely reducible; its decomposition type is l L ^{4} _{xy} . Therefore, the equation Lz = 0 has the differential fundamental system$$z\_1(x,y)\; =\; 2(x-y)F(y)\; +\; (x-y)^2\; F\text{'}(y)$$ and $$z\_2(x,y)\; =\; 2(y-x)G(x)\; +\; (y-x)^2\; G\text{'}(x).$$

Decomposing linear PDEs of order higher than 2

It turns out that operators of higher order have more complicated decompositions and there are more alternatives, many of them in terms of non-principal divisors. The solutions of the corresponding equations get more complex. For equations of order three in the plane a fairly complete answer may be found in. A typical example of a third-order equation that is also of historical interest is due to Blumberg.^[14]

Example 7 (Blumberg 1912)In his dissertation Blumberg considered the third order operator

$$L\; \backslash equiv\; \backslash partial\_\; +\; x\backslash partial\_\; +\; 2\backslash partial\_\; +\; 2(x+1)\backslash partial\_\; +\; \backslash partial\_x\; +\; (x+2)\; \backslash partial\_y.$$

It allows the two first-order factors

l _{1\equiv\partial} _{x+1} and l _{2\equiv\partial} _{x+x\partial} _{y} . Their intersection is not principal; defining

$$L\_1\; \backslash equiv\; \backslash partial\_\; -\; x^2\; \backslash partial\_\; +\; 3\backslash partial\_\; +\; (2x\; +\; 3)\backslash partial\_\; -\; x^2\; \backslash partial\_\; +\; 2\backslash partial\_x\; +\; (2x\; +\; 3)\; \backslash partial\_y$$$$L\_2\; \backslash equiv\; \backslash partial\_\; +\; x\; \backslash partial\_\; -\; \backslash frac\; \backslash partial\_\; -\; \backslash frac\; \backslash partial\_\; +\; x\; \backslash partial\_yy\; -\; \backslash frac\backslash partial\_x\; -\; \backslash left(1+\backslash frac\backslash right)\backslash partial\_y.$$

it may be written as

\operatorname{Lclm}(l _{2,} l _{1)} = {\langle\langle}L _{1,} L _{2{\rangle\rangle}} . Consequently, the Loewy decomposition of Blumbergs's operator is

It yields the following differential fundamental system for the differential equation

Lz = 0 .z _{1 (x,y) = F(y - (1)/(2)} x ^{2)} ,z ^{- x} _{2 (x,y) = G(y)e} ,z _{3(x,y)=} \int xe ^{-x} H\left(\bar{y} + (1)/(2) x ^{2\right)} dx | _{\bar{y} = y - (1)/(2) x ^{2}} F,G and H are an undetermined functions.

Factorizations and Loewy decompositions turned out to be an extremely useful method for determining solutions of linear differential equations in closed form, both for ordinary and partial equations. It should be possible to generalize these methods to equations of higher order, equations in more variables and system of differential equations.

Notes and References

1. Loewy . A. . 1906 . Über vollständig reduzible lineare homogene Differentialgleichungen . 10.1007/bf01448417 . Mathematische Annalen . 62 . 89–117 . 121139339 .
2. , F.Schwarz, Loewy Decomposition of Linear Differential Equations, Springer, 2012
3. Schwarz . F. . 2013 . Loewy Decomposition of linear Differential Equations . Bulletin of Mathematical Sciences . 3 . 19–71 . 10.1007/s13373-012-0026-7 . free .
4. E. Kamke, Differentialgleichungen I. Gewoehnliche Differentialgleichungen, Akademische Verlagsgesellschaft, Leipzig, 1964
5. M. van der Put, M.Singer, Galois theory of linear differential equations, Grundlehren der Math. Wiss. 328, Springer, 2003
6. M.Bronstein, S.Lafaille, Solutions of linear ordinary differential equations in terms of special functions, Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation; T.Mora, ed., ACM, New York, 2002, pp. 23–28
7. F. Schwarz, Algorithmic Lie Theory for Solving Ordinary Differential Equations, CRC Press, 2007, page 39
8. Janet . M. . 1920 . Les systemes d'equations aux derivees partielles . Journal de Mathématiques . 83 . 65–123 .
9. Janet Bases for Symmetry Groups, in: Gröbner Bases and Applications Lecture Notes Series 251, London Mathematical Society, 1998, pages 221–234, B. Buchberger and F. Winkler, Edts.
10. Buchberger . B. . 1970 . Ein algorithmisches Kriterium fuer die Loesbarkeit eines algebraischen Gleichungssystems . Aequ. Math. . 4 . 3. 374–383 . 10.1007/bf01844169. 189834323 .
11. E. Darboux, Leçons sur la théorie générale des surfaces, vol. II, Chelsea Publishing Company, New York, 1972
12. [Édouard Goursat]
13. A.R.Forsyth, Theory of Differential Equations, vol. I,...,VI, Cambridge, At the University Press, 1906
14. H.Blumberg, Ueber algebraische Eigenschaften von linearen homogenen Differentialausdruecken, Inaugural-Dissertation, Goettingen, 1912