#
Decomposing a 4'th order linear differential equation as a symmetric
product.

In this paper useful formulas are given for solving fourth order linear
differential equations. The case that is handled is when the equation
has solutions which are products of solutions of second order equations.
These formulas are found as well as proven by computations with a computer
algebra system. From a mathematical point of view this may not seem
interesting, however, there is one intriguing feature about these
formulas: It turns out that, starting with an equation of order 4,
to determine if it is a symmetric product of 2nd order equations one has
to solve an equation of order 3, i.e. one less than the order of the
equation we started with, which is unusual.
The explanation that the order should be 3 is at follows: If we take
the example L = Dx^4 and then compute all {L1,L2} with order 2,
L = symmetric_product(L1, L2), and coeff(L1, Dx^1) = coeff(L2, Dx^1)
then there are infinitely many possible {L1,L2}, more precisely: they are
parametrized by 3 homogeneous parameters, i.e. parametrized by points
in the projective plane P^2 = P(C^3). So we can expect that the equation
to be solved to find {L1,L2} has a 3-dimensional vector space of solutions
and thus will be a linear ode with order 3. Click here for more
details on this.
Download this paper as a dvi file,
as a Maple worksheet, or as a
pdf file.
Click here for a reference on symmetric products
and here for the implementation.
Here is another Maple computation on a similar topic, namely
The two highest coefficients
of a symmetric power of a second order operator.
Additional results on this topic can be found in
thesis of Axelle Person.