# 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.