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In the study of [[differential equation]]s, the '''Loewy decomposition''' breaks every linear [[ordinary differential equation]] (ODE) into what are called largest completely reducible components. It was introduced by [[Alfred Loewy]].<ref name="Loewy">A. Loewy, Ueber vollstaendig reduzible lineare homogene Differentialgleichungen, Mathematische Annalen, '''56''', pages 89–117 (1906)</ref>

Solving [[differential equation]]s is one of the most important subfields in [[mathematics]]. Of particular interest are solutions in [[closed-form expression|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.<ref name="Schwarz2012">, F.Schwarz, Loewy Decomposition of Linear Differential Equations, Springer, 2012</ref>

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

== Decomposing linear ordinary differential equations ==
Let <math>D\equiv\frac{d}{dx}</math> denote the derivative w.r.t. the variable <math>x</math>. 
A differential operator of order <math>n</math> is a polynomial of the form

: <math>L\equiv D^n+a_1D^{n-1}+\cdots +a_{n-1}D+a_n</math>

where the coefficients <math>a_i</math>, <math>i=1,\ldots,n</math> are from some function field, the
''base field'' of <math>L</math>. Usually it is the field of rational functions in the variable 
<math>x</math>, i.e. <math>a_i\in{\mathbb Q}(x)</math>. If <math>y</math> is an indeterminate with 
<math>\frac{dy}{dx}\neq 0</math>, <math>Ly</math> becomes a differential polynomial, and <math>Ly=0</math> is 
the differential equation corresponding to <math>L</math>.

An operator <math>L</math> of order <math>n</math> is called ''reducible'' if it may be represented as the 
product of two operators <math>L_1</math> and <math>L_2</math>, both of order lower than <math>n</math>. Then one writes 
<math>L=L_1L_2</math>, i.e. juxtaposition means the operator product, it is defined by the rule
<math>Da_i=a_iD+a_i'</math>; <math>L_1</math> is called a left factor of <math>L</math>, <math>L_2</math> a right factor. By 
default, the coefficient domain of the factors is assumed to be the base field of <math>L</math>, 
possibly extended by some algebraic numbers, i.e. <math>{\bar{\mathbb Q}}(x)</math> is allowed. If an operator does not allow any
right factor it is called ''irreducible''.

For any two operators <math>L_1</math> and <math>L_2</math> the ''least common left multiple''
<math>Lclm(L_1,L_2)</math> is the operator of lowest order such that both <math>L_1</math> and <math>L_2</math> divide it 
from the right. The ''greatest common right divisior'' <math>Gcrd(L_1,L_2)</math> is the operator
of highest order that divides both <math>L_1</math> and <math>L_2</math> from the right. If an operator may be
represented as <math>Lclm</math> 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 <math>Lclm</math> 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 <ref name="Loewy" /> obtained the following fundamental result.

'''Theorem 1''' (Loewy 1906)
Let <math>D=\frac{d}{dx}</math> be a derivative and <math>a_i\in{\mathbb Q(x)}</math>. A differential operator

: <math>L\equiv D^n+a_1D^{n-1}+\cdots +a_{n-1}D+a_n</math>

of order <math>n</math> may be written uniquely as the product of completely reducible 
factors <math>L^{(d_k)}_k</math> of maximal order <math>d_k</math> over <math>{\mathbb Q}(x)</math> in the 
form

: <math>L=L_m^{(d_m)}L_{m-1}^{(d_{m-1})}\ldots L_1^{(d_1)}</math>

with <math>d_1+\ldots+d_m=n</math>. The factors <math>L^{(d_k)}_k</math> are unique. Any factor <math>L^{(d_k)}_k</math>,
<math>k=1,\ldots,m</math> may be written as

: <math>L^{(d_k)}_k=Lclm(l^{(e_1)}_{j_1},l^{(e_2)}_{j_2},\ldots,l^{(e_k)}_{j_k})</math>

with <math>e_1+e_2+\ldots+e_k=d_k</math>; <math>l^{(e_i)}_{j_i}</math> for <math>i=1,\ldots,k</math>, denotes 
an irreducible operator of order <math>e_i</math> over <math>{\mathbb Q}(x)</math>.

The decomposition determined in this theorem is called the ''Loewy decomposition'' of  <math>L</math>. It provides a detailed description of the function space containing the solution of a reducible linear differential equation <math>Ly=0</math>.

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 <math>n=2</math> is detailed in the following corollary to the above theorem.<ref name="Schwarz2013">F. Schwarz, Loewy Decomposition of linear 
Differential Equations, Bulletin of Mathematical Sciences, 3, page 19–71 (2013); 
http://link.springer.com/article/10.1007/s13373-012-0026-7</ref>

'''Corollary 1'''
Let <math>L</math> be a second-order operator. Its possible Loewy decompositions are denoted by
<math>{\mathcal L}^2_0,\ldots{\mathcal L}^2_3</math>, they  may be described as follows; <math>l^{(i)}</math> 
and <math>l^{(i)}_j</math> are irreducible operators of order <math>i</math>; <math>C</math> is a constant.

: <math> {\mathcal L}^2_1: L=l^{(1)}_2l^{(1)}_1; </math> &nbsp; &nbsp;
: <math> {\mathcal L}^2_2: L=Lclm(l^{(1)}_2,l^{(1)}_1); </math> &nbsp;&nbsp;
: <math> {\mathcal L}^2_3: L=Lclm(l^{(1)}(C)). </math>

The decomposition type of an operator is the decomposition <math>{\mathcal L}^2_i</math> with the highest value 
of <math>i</math>. An irreducible second-order operator is defined to have decomposition type <math>{\mathcal L}^2_0</math>.

The decompositions <math>{\mathcal L}^2_0</math>, <math>{\mathcal L}^2_2</math> and <math>{\mathcal L}^2_3</math> are completely reducible.

If a decomposition of type <math>{\mathcal L}^2_i</math>, <math>i=1,2</math> or <math>3</math> has been obtained for a
second-order equation <math>Ly=0</math>, a fundamental system may be given explicitly.

'''Corollary 2'''
Let <math>L</math> be a second-order differential operator, <math>D\equiv\frac{d}{dx}</math>, 
<math>y</math> a differential indeterminate, and <math>a_i\in{\mathbb Q}(x)</math>. Define 
<math>\varepsilon_i(x)\equiv\exp{(-\int a_i dx)}</math> for <math>i=1,2</math> and 
<math>\varepsilon(x,C)\equiv\exp{(-\int a(C)dx)}</math>, <math>C</math> is a parameter; the barred 
quantities <math>\bar{C}</math> and <math>\bar{\bar{C}}</math> are arbitrary numbers, 
<math>\bar{C}\neq\bar{\bar{C}}</math>. For the three nontrivial decompositions of 
Corollary 1 the following elements <math>y_1</math> and <math>y_2</math> of
a fundamental system are obtained.

: <math>{\mathcal L}^2_1</math>:  <math>Ly=(D+a_2)(D+a_1)y=0</math>; &nbsp;&nbsp;
<math>y_1=\varepsilon_1(x),</math>

: <math>y_2=\varepsilon_1(x)\int\frac{\varepsilon_2(x)}{\varepsilon_1(x)}\,dx.</math>

: <math>{\mathcal L}^2_2</math>: <math>Ly=Lclm(D+a_2,D+a_1)y=0;</math>

: <math>y_i=\varepsilon_i(x);</math>

<math>a_1</math> is not equivalent to <math>a_2</math>.

: <math>{\mathcal L}^2_3</math>: <math>Ly=Lclm(D+a(C))y=0;</math>

: <math>y_1=\varepsilon(x,\bar{C})</math>}

: <math>y_2=\varepsilon(x,\bar{\bar{C}}).</math>

Here two rational functions <math>p,q\in{\mathbb Q}(x)</math> are called ''equivalent'' 
if there exists another rational function <math>r\in{\mathbb Q}(x)</math> such that 
:<math>p-q=\frac{r'}{r}</math>.

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 1'''
Equation 2.201 from Kamke's collection.<ref name="Kamke1964">E. Kamke, Differentialgleichungen I. 
Gewoehnliche Differentialgleichungen, Akademische Verlagsgesellschaft, Leipzig, 1964</ref>
has the <math>{\mathcal L}^2_2</math> decomposition

<math>y''+(2+\frac{1}{x})y'-\frac{4}{x^{2}}y=
Lclm\Big(D+\frac{2}{x}-\frac{2x-2}{x^{2}-2x+ {\frac{3}{2}}}, 
D+2+\frac{2}{x}-\frac{1}{x+{\frac{3}{2}}}\Big)y=0. </math>

The coefficients <math>a_1=2+\frac{2}{x}-\frac{1}{x+\frac{3}{2}}</math> and
<math>a_2=\frac{2}{x}-\frac{2x-2}{x^2-2x+\frac{3}{2}}</math> are rational solutions of the Riccati
equation <math>a'-a^2+\big(2+\frac{1}{x}\big)+\frac{4}{x^2}=0</math>, they yield the fundamental system

: <math>y_1=\frac{2}{3}-\frac{4}{3x}+\frac{1}{x^2},</math>
: <math>y_2=\frac{2}{x}+\frac{3}{x^2}e^{-2x}.</math>

'''Example 2'''
An equation with a type <math>{\mathcal L}^2_3</math> decomposition is

: <math>y''-\frac{6}{x^2}y=Lclm\big(D+\frac{2}{x}-\frac{5x^4}{x^5+C}\big)y=0.</math>

The coefficient of the first-order factor is the rational solution of 
<math>a'-a^2+\frac{6}{x^2}=0</math>. Upon integration the fundamental system <math>y_1=x^3</math> and
<math>y_2=\frac{1}{x^2}</math> for <math>C=0</math> and <math>C\rightarrow\infty</math> 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 <math>n=3</math> the answer is given in full detail in.<ref name="Schwarz2012" />

If an equation is irreducible it may occur that its Galois group is nontrivial, then 
algebraic solutions may exist.<ref name="PutSi">M. van der Put, M.Singer, Galois theory of linear differential equations}, Grundlehren der Math. Wiss. '''328''', Springer, 2003</ref> 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 <ref name="BronsteinLafaille">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</ref> or.<ref name="Schwarz2007">F. Schwarz, Algorithmic Lie Theory for Solving Ordinary Differential Equations, CRC Press, 2007, page 39</ref>

== Basic facts from differential algebra ==

In order to generalize Loewy's result to linear pde's 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 <math>{\mathcal F}</math> is called a ''differential field'' if it is equipped with a 
''derivation operator''. An operator <math>\delta</math> on a field <math>{\mathcal F}</math> is called a 
derivation operator if <math>\delta(a+b)=\delta(a)+\delta(b)</math> and 
<math>\delta(ab)=\delta(a)b+a\delta(b)</math> for all elements <math>a,b\in{\mathcal F}</math>. 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 <math>\partial_x=\frac{\partial}{\partial x}</math> and
<math>\partial_y=\frac{\partial}{\partial y}</math> with coefficients from some differential field 
are considered. Its elements have the form <math>\sum_{i,j} r_{i,j}(x,y)\partial_x^i\partial_y^j</math>; almost 
all coefficients <math>r_{i,j}</math> are zero. The coefficient field is called the 
''base field''. If constructive and algorithmic methods are the main issue it is 
<math>{\mathbb Q}(x,y)</math>. The respective ring of differential operators is denoted by 
<math>{\mathcal D}={\mathbb Q}(x,y)[\partial_x,\partial_y]</math> or 
<math>{\mathcal D}={\mathcal F}[\partial_x,\partial_y]</math>. The ring <math>{\mathcal D}</math> is non-commutative, 
<math>\partial_xa=a\partial_x+\frac{\partial a}{\partial x}</math> and similarly for the other 
variables; <math>a</math> is from the base field.

For an operator <math>L=\sum_{i+j\leq n}r_{i,j}(x,y)\partial_x^i\partial_y^j</math> of order <math>n</math> the 
''symbol of L'' is the homogeneous algebraic polynomial <math>symb(L)\equiv\sum_{i+j=n}r_{i,j}(x,y)X^iY^j</math>
where <math>X</math> and <math>Y</math> algebraic indeterminates.

Let <math>I</math> be a left ideal which is generated by <math>l_i\in{\mathcal D}</math>, 
<math>i=1,\ldots,p</math>. Then one writes <math>I=\langle l_1,\ldots,l_p\rangle</math>. Because right ideals 
are not considered here, sometimes <math>I</math> is simply called an ideal.

The relation between left ideals in <math>{\mathcal D}</math> and systems of linear pde's is 
established as follows. The elements <math>l_i\in{\mathcal D}</math> are applied to a single
differential indeterminate <math>z</math>. In this way the ideal <math>I=\langle l_1,l_2,\ldots\rangle</math> 
corresponds to the system of pde's <math>l_1z=0</math>, <math>l_2z=0,\ldots</math> for the single function <math>z</math>.

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 <ref name="Janet">M.Janet, Les systemes d'equations aux 
derivees partielles, Journal de mathematiques '''83''' (1920), 65–123</ref> introduced a normal form for 
systems of linear pde's that has been baptized ''[[Janet basis]]''.<ref name="Schwarz1998">Janet Bases for Symmetry Groups, in: Groebner Bases and Applications Lecture Notes Series 251, London Mathematcial Society, 1998,pages 221–234, B. Buchberger and F. Winkler, Edts.</ref>
They are the differential analog to [[Groebner bases]] of commutative  algebra, originally they have been introduced by [[Bruno Buchberger]];<ref name="Buchberger">B.Buchberger, Ein algorithmisches Kriterium fuer
die Loesbarkeit eines algebraischen Gleichungssystems, Aequ. Math. '''4''', 374–383(1970)</ref> therefore they are also called ''differential Groebner 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 <math>\delta</math>, <math>\delta_1</math> and <math>\delta_2</math>, and 
any derivation operator <math>\theta</math> the relations <math>\delta\preceq\theta\delta</math>, and 
<math>\delta_1\preceq\delta_2\rightarrow\delta\delta_1\preceq\delta\delta_2</math> are valid. Here
graded lexicographic term orderings <math>grlex</math> 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 <math>l_1,\ldots,l_p</math> of an ideal <math>I</math> form a Janet basis the notation 
<math>I={\big\langle\big\langle} l_1,\ldots,l_p{\big\rangle\big\rangle}</math> is applied.

'''Example 3'''
Consider the ideal

  <math>I=\Big\langle l_1\equiv\partial_{xx}-\frac{1}{x}\partial_x-\frac{y}{x(x+y)}\partial_y,</math>&nbsp;
<math> l_2\equiv\partial_{xy}+\frac{1}{x+y}\partial_y,</math>&nbsp;
<math>l_3\equiv\partial_{yy}+\frac{1}{x+y}\partial_y\Big\rangle
</math>

in <math>grlex</math> term order with <math>x\succ y</math>. Its generators are autoreduced. If the 
integrability condition

 <math>l_{1,y}=l_{2,x}-l_{2,y}=\frac{y+2x}{x(x+y)}\partial_{xy}+\frac{y}{x(x+y)}\partial_{yy}</math>

is reduced w.r.t. to <math>I</math>, the new generator <math>\partial_y</math> is obtained. Adding it to the 
generators and performing all possible reductions, the given ideal is represented as
<math>I={\Big\langle\Big\langle}\partial_{xx}-\frac{1}{x}\partial_x,\partial_y{\Big\rangle\Big\rangle}</math>. 
Its generators are autoreduced and the single integrability condition is satisfied, i.e. they form a Janet basis.

Given any ideal <math>I</math> it may occur that it is properly contained in some larger ideal <math>J</math> 
with coefficients in the base field of <math>I</math>; then <math>J</math> is called a ''divisor'' of <math>I</math>. 
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 <math>I</math> and <math>J</math>
is the smallest ideal  with the property that both <math>I</math> and <math>J</math> are contained in it. 
If they have the representation
<math>I\equiv\langle f_1,\ldots,f_p\rangle</math> and
<math>J\equiv\langle g_1,\ldots,g_q\rangle, </math>
<math>f_i</math>, <math>g_j\in{\mathcal D}</math> for all <math>i</math> and <math>j</math>, 
the sum is generated by the union of the generators of <math>I</math> and <math>J</math>. The solution space 
of the equations corresponding to <math>Gcrd(I,J)</math> is the intersection of the solution spaces
of its arguments.

The ''least common left multiple (Lclm)'' or ''left intersection'' of two ideals <math>I</math>
and <math>J</math> is the largest ideal with the property that it is contained both in <math>I</math> and <math>J</math>.
The solution space of <math>Lclm(I,J)z=0</math> 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
<math>L</math>,<ref name="Schwarz2012" /> page 34. It is defined as follows.

'''Definition'''
Let <math>L</math> be a partial differential operator in the plane; define

 <math>{\mathfrak l}_m\equiv\partial_{x^m}+a_{m-1}\partial_{x^{m-1}}+\ldots+a_1\partial_x+a_0</math> and

 <math>{\mathfrak k}_n\equiv\partial_{y^n}+b_{n-1}\partial_{y^{n-1}}+\ldots+b_1\partial_y+b_0</math>

be ordinary differential operators w.r.t. <math>x</math> or <math>y</math>; 
<math>a_i,b_i\in{\mathbb Q}(x,y)</math> for all i; <math>m</math> and <math>n</math> are natural numbers not 
less than 2. Assume the coefficients <math>a_i</math>, <math>i=0,\ldots,m-1</math> are such that <math>L</math> 
and <math>{\mathfrak l}_m</math> form a Janet basis. If <math>m</math> is the smallest integer with this 
property then <math>{\mathbb L}_{x^m}(L)\equiv{\langle\langle} L,{\mathfrak l}_m{\rangle\rangle}</math> 
is called a ''Laplace divisor'' of <math>L</math>. Similarly, if <math>b_j</math>, <math>j=0,\ldots,n-1</math> are 
such that <math>L</math> and <math>{\mathfrak k}_n</math> form a Janet basis and <math>n</math> is minimal, then 
<math>{\mathbb L}_{y^n}(L)\equiv{\langle\langle} L,{\mathfrak k}_n{\rangle\rangle}</math> 
is also called a ''Laplace divisor'' of <math>L</math>.

In order for a Laplace divisor to exist the coeffients of an operator <math>L</math> must obey
certain constraints.<ref name="Schwarz2013" /> 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 pde's. Here it 
is applied to individual linear pde's of second order in the plane with coordinates <math>x</math> 
and <math>y</math>, and the principal ideals generated by the corresponding operators.

Second-order equations have been considered extensively in the literature of the 19th 
century,.<ref name="Darboux">E.Darboux, Lecons sur la theorie generale des surfaces, 
vol II, Chelsea Publishing Company, New York 1972</ref><ref name="Goursat">E.~Goursat, Lecon sur l'integration des 
equation aux de}rivees partielles, vol. I and II, A.Hermann, Paris 1898</ref> Usually equations with leading derivatives <math>\partial_{xx}</math> or <math>\partial_{xy}</math> 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 <math>\partial_{xx}</math> Loewy's results may be generalized as follows.

'''Theorem 2'''
Let the differential operator <math>L</math> be defined by

   <math>L\equiv\partial_{xx}+A_1\partial_{xy}+A_2\partial_{yy}+A_3\partial_x+A_4\partial_y+A_5</math> where <math>A_i\in{\mathbb Q}(x,y)</math> for all <math>i</math>.

Let <math>l_i\equiv\partial_x+a_i\partial_y+b_i</math> for <math>i=1</math> 
and <math>i=2</math>, and <math>l(\Phi)\equiv\partial_x+a\partial_y+b(\Phi)</math> be first-order operators with 
<math>a_i,b_i,a\in{\mathbb Q}(x,y)</math>; <math>\Phi</math> is an undetermined function of a single argument.
Then <math>L</math> has a Loewy decomposition according to one of the following types.

<math>{\mathcal L}_{xx}^1:  L=l_2l_1;</math> &nbsp;
<math>{\mathcal L}_{xx}^2:  L=Lclm(l_2,l_1);</math> &nbsp;
<math>{\mathcal L}_{xx}^3: L=Lclm(l(\Phi)). </math>

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

In order to apply this result for solving any given differential equation 
involving the operator <math>L</math> 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 3'''
In 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 1'''
Let a reducible second-order equation

  <math>Lz\equiv z_{xx}+A_1z_{xy}+A_2z_{yy}+A_3z_x+A_4z_y+A_5z=0</math> where <math>A_1,\ldots,A_5\in{\mathbb Q}(x,y)</math>.

Define <math>l_i\equiv\partial_x+a_i\partial_y+b_i</math>, 
<math>a_i,b_i\in{\mathbb Q}(x,y)</math> for <math>i=1,2</math>; 
<math>\varphi_i(x,y)=const</math> is a rational first integral of 
<math>\frac{dy}{dx}=a_i(x,y)</math>; <math>\bar{y}\equiv\varphi_i(x,y)</math> and the inverse 
<math>y=\psi_i(x,\bar{y})</math>; both <math>\varphi_i</math> and <math>\psi_i</math> are assumed to exist. 
Furthermore define

<math>{\mathcal E}_i(x,y)\equiv\exp\Big(-{\displaystyle\int} b_i(x,y)\big|_{y=\psi_i(x,\bar{y})}dx\Big)\Big|_{\bar{y}=\varphi_i(x,y)}</math> for <math>i=1,2</math>.

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

<math>{\mathcal L}^1_{xx}: z_1(x,y)={\mathcal E}_1(x,y)F_1(\varphi_1)</math>,
<math> z_2(x,y)={\mathcal E}_1(x,y){\displaystyle\int}\frac{{\mathcal E}_2(x,y)}{{\mathcal E}_1(x,y)}F_2
       \big(\varphi_2(x,y)\big)\big|_{y=\psi_1(x,\bar{y})}dx
       \Big|_{\bar{y}=\varphi_1(x,y)};</math>

<math>{\mathcal L}^2_{xx}: z_i(x,y)={\mathcal E}_i(x,y)F_i\big(\varphi_i(x,y)\big), i=1,2; </math>

<math>{\mathcal L}^3_{xx}: z_i(x,y)={\mathcal E}_i(x,y)F_i\big(\varphi(x,y)\big),i=1,2.</math>

The <math>F_i</math> are undetermined functions of a single argument; <math>\varphi</math>,
<math>\varphi_1</math> and <math>\varphi_2</math> are rational in all arguments; <math>\psi_1</math> is assumed
to exist. In general <math>\varphi_1\neq\varphi_2</math>, they are determined
by the coefficients <math>A_1</math>, <math>A_2</math> and <math>A_3</math> of the given equation.

A typical example of a linear pde where factorization applies is an equation that has been discussed by Forsyth
,<ref name="Forsyth">A.R.Forsyth, Theory of Differential Equations, vol. I,...,VI, Cambridge, At the University Press, 1906</ref>
vol. VI, page 16,

'''Example 5''' (Forsyth 1906)}
Consider the differential equation
<math>z_{xx}-z_{yy}+\frac{4}{x+y}z_x=0</math>. Upon factorization the representation

<math>
Lz\equiv l_2l_1z=\Big(\partial_x+\partial_y+\frac{2}{x+y}\Big)\Big(\partial_x-\partial_y+\frac{2}{x+y}\Big)z=0</math>
is obtained. There follows

<math>
\varphi_1(x,y)=x+y,\psi_1(x,y)=\bar{y}-x,
 {\mathcal E}_1(x,y)=\exp{\Big(\frac{2y}{x+y}\Big)}</math>,

<math>
\varphi_2(x,y)=x-y,\psi_2(x,y)=x-\bar{y},
 {\mathcal E}_2(x,y)=-\frac{1}{x+y}.</math>

Consequently a differential fundamental system is

<math>z_1(x,y)=\exp{\Big(\frac{2y}{x+y}\Big)}F(x+y),</math>
<math>z_2(x,y)=\frac{1}{x+y}\exp{\Big(\frac{2y}{x+y}\Big)}
   {\displaystyle\int}\exp{\Big(\frac{2x-\bar{y}}{\bar{y}}\Big)}G(2x-\bar{y})dx
   \Big|_{\bar{y}=x+y}.
</math>

<math>F</math> and <math>G</math> are undetermined functions.

If the only second-order derivative of an operator is <math>\partial_{xy}</math>, its possible decompositions
involving only principal divisors may be described as follows.

'''Theorem 3'''
Let the differential operator <math>L</math> be defined by

<math>L\equiv\partial_{xy}+A_1\partial_x+A_2\partial_y+A_3</math> where <math>A_i\in{\mathbb Q}(x,y)</math> for all <math>i</math>.

Let <math>l\equiv\partial_x+A_2</math> and 
<math>k\equiv\partial_y+A_1</math> are first-order operators. <math>L</math> has Loewy decompositions involving 
first-order principal divisors of the following form.

<math>{\mathcal L}_{xy}^1: L=kl;</math> &nbsp; 
<math>{\mathcal L}_{xy}^2: L=lk;</math> &nbsp;
<math>{\mathcal L}_{xy}^3: L=Lclm(k,l).</math>

The decomposition type of an operator <math>L</math> is the decomposition <math>{\mathcal L}_{xy}^i</math> with highest value of
<math>i</math>. The decomposition of type <math>{\mathcal L}_{xy}^3</math> is completely reducible

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

'''Theorem 4'''
Let the differential operator <math>L</math> be defined by

<math>L\equiv\partial_{xy}+A_1\partial_x+A_2\partial_y+A_3</math> where <math>A_i\in{\mathbb Q}(x,y)</math> for all <math>i</math>.

<math>\mathbb{L}_{x^m}(L)</math> and 
<math>\mathbb{L}_{y^n}(L)</math> as well as <math>{\mathfrak l}_m</math> and <math>{\mathfrak k}_n</math> are defined above; 
furthermore <math>l\equiv \partial_x+a</math>, <math>k\equiv\partial_y+b</math>,
<math>a,b\in{\mathbb Q}(x,y)</math>. <math>L</math> has Loewy decompsitions involving Laplace divisors 
according to one of the following types; <math>m</math> and <math>n</math> obey <math>m,n\geq 2</math>.

<math>{\mathcal L}_{xy}^4: L=Lclm\big(\mathbb{L}_{x^m}(L),\mathbb{L}_{y^n}(L)\big);</math>

<math>{\mathcal L}_{xy}^5: L=Exquo\big(L,\mathbb{L}_{x^m}(L)\big)\mathbb{L}_{x^m}(L)=
    \left(\begin{array}{cc}  1  &  0\\ 0  &  \partial_y+A_1\end{array}\right)
    \left(\begin{array}{c} L\\ {\mathfrak l}_m\end{array}\right);</math>

<math>{\mathcal L}_{xy}^6: L=Exquo\big(L,\mathbb{L}_{y^n}(L)\big)\mathbb{L}_{y^n}(L)=
    \left(\begin{array}{cc}  1  &  0\\ 0  & \partial_x+A_2\end{array}\right)
    \left(\begin{array}{c} L\\ {\mathfrak k}_n\end{array}\right);</math>

<math>{\mathcal L}_{xy}^7: L=Lclm\big(k,\mathbb{L}_{x^m}(L)\big);</math>
<math>{\mathcal L}_{xy}^8: L=Lclm\big(l,\mathbb{L}_{y^n}(L)\big).</math>

If <math>L</math> 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 <math>{\mathcal L}_{xy}^0</math>. The 
decompositions <math>{\mathcal L}_{xy}^0</math>, <math>{\mathcal L}_{xy}^4</math>, <math>{\mathcal L}_{xy}^7</math> and 
<math>{\mathcal L}_{xy}^8</math> are completely reducible.

An equation that does not allow a decomposition involving principal divisors but 
is completely reducible w.r.t. non-principal Laplace divisors of type <math>{\mathcal L}_{xy}^4</math> has been
considered by Forsyth.

'''Example 6''' (Forsyth 1906) Define

<math>L\equiv\partial_{xy}+\frac{2}{x-y}\partial_x-\frac{2}{x-y}\partial_y-\frac{4}{(x-y)^2}</math>

generating the principal ideal <math>\langle L\rangle</math>. A first-order factor does not exist. However, there are 
Laplace divisors

 <math>{\mathbb L}_{x^2}(L)\equiv{\Big\langle\Big\langle}
     \partial_{xx}-\frac{2}{x-y}\partial_x+\frac{2}{(x-y)^2},L{\Big\rangle\Big\rangle}</math> and <math>{\mathbb L}_{y^2}(L)\equiv{\Big\langle\Big\langle} L,\partial_{yy}+\frac{2}{x-y}\partial_y+\frac{2}{(x-y)^2}{\Big\rangle\Big\rangle}.</math>

The ideal generated by <math>L</math> has the representation
<math>\langle L\rangle=Lclm\big({\mathbb L}_{x^2}(L),{\mathbb L}_{y^2}(L)\big)</math>, 
i.e. it is completely reducible; its decomposition type is 
<math>{\mathcal L}^4_{xy}</math>. Therefore the equation <math>Lz=0</math> has the the differential fundamental system

 <math>z_1(x,y)=2(x-y)F(y)+(x-y)^2F'(y)</math> and <math> z_2(x,y)=2(y-x)G(x)+(y-x)^2G'(x)</math>.

== Decomposing linear pde's 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.<ref name="Schwarz2012" /> A typical example of a third-order equation that is also of historical interest is due to Blumberg
.<ref name="Blumberg">H.Blumberg, Ueber algebraische Eigenschaften von linearen homogenen Differentialausdruecken, 
Inaugural-Dissertation, Goettingen, 1912</ref>

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

<math>L\equiv\partial_{xxx}+x\partial_{xxy}+2\partial_{xx}+2(x+1)\partial_{xy}+\partial_x+(x+2)\partial_y</math>.

It allows the two first-order factors <math>l_1\equiv\partial_x+1</math> and <math>l_2\equiv\partial_x+x\partial_y</math>. Their
intersection is not principal; defining

<math>L_1\equiv\partial_{xxx}-x^2\partial_{xyy}+3\partial_{xx}+(2x+3)\partial_{xy}-x^2\partial_{yy}+2\partial_x+(2x+3)\partial_y</math>

<math>L_2\equiv\partial_{xxy}+x\partial_{xyy}-\frac{1}{x}\partial_{xx}-\frac{1}{x}\partial_{xy}+x\partial_yy
  -\frac{1}{x}\partial_x-\big(1+\frac{1}{x}\big)\partial_y{\big\rangle\big\rangle}.
</math>

it may be written as <math>Lclm(l_2,l_1)={\langle\langle}L_1,L_2{\rangle\rangle} </math>.
Consequently the Loewy decomposition of Blumbergs's operator is

<math>
L=\left(\begin{array}{cc} 1 & x \\ 0 & \partial_x+1+\frac{1}{x}\end{array}\right)
  \left(\begin{array}{c}L_1\\L_2\end{array}\right).
</math>

It yields the following differential fundamental system for the differential equation <math>Lz=0</math>.

<math>z_1(x,y)=F(y-\frac{1}{2}x^2)</math>,&nbsp; 
<math>z_2(x,y)=G(y)e^{-x}</math>, &nbsp;
<math>z_3(x,y)={\displaystyle\int} xe^{-x}H\big(\bar{y}+\frac{1}{2}x^2\big)
      dx\Big|_{\bar{y}=y-\frac{1}{2}x^2}</math>

<math>F,G</math> and <math>H</math> 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.

== References ==

<references/>

[[Category:Differential equations]]