This procedure is algorithmic, so that the best possible answer for solving a reducible equation is guaranteed.
[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.
By default, the coefficient domain of the factors is assumed to be the base field of
Due to the lowering of order in each step, this proceeding terminates after a finite number of iterations and the desired decomposition is obtained.
It provides a detailed description of the function space containing the solution of a reducible linear differential equation
These results show that factorization provides an algorithmic scheme for solving reducible linear ode's.
[2] If an equation is irreducible it may occur that its Galois group is nontrivial, then algebraic solutions may exist.
In order to generate a Janet basis, a ranking of derivatives must be defined.
For partial derivatives of a single function their definition is analogous to the monomial orderings in commutative algebra.
Adding it to the generators and performing all possible reductions, the given ideal is represented as
Its generators are autoreduced and the single integrability condition is satisfied, i.e. they form a Janet basis.
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
does not have any first-order factor in the base field, its decomposition type is defined to be
In order to apply this result for solving any given differential equation involving the operator
Corollary 3 In general, first-order right factors of a linear pde in the base field cannot be determined algorithmically.
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.
A differential fundamental system has the following structure for the various decompositions into first-order components.
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
has Loewy decompositions involving first-order principal divisors of the following form.
is completely reducible In addition there are five more possible decomposition types involving non-principal Laplace divisors as shown next.
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
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.
For equations of order three in the plane a fairly complete answer may be found in.
[2] A typical example of a third-order equation that is also of historical interest is due to Blumberg.