The input L is a differential operator. Denote V(L) as the solution space of L. eigenring computes a basis (a vector space) of the set of all operators r for which r(V(L)) is a subset of V(L). So r is an endomorphism of the solution space V(L). The characteristic polynomial of this map can be computed by the command endomorphism_charpoly(L,r).

For endomorphisms r, the product of L and r is divisible on the right by L. If the optional third argument is the equation verify=true then eigenring checks if the output satisfies this condition. This should not be necessary though.

The argument domain describes the differential algebra. If this argument is the list $\left[\mathrm{Dt}\,t\right]$ then the differential operators are notated with the symbols $\mathrm{Dt}$ and $t$. They are viewed as elements of the differential algebra C(t)[Dt] where $C$ is the field of constants.

If the argument domain is omitted then the differential specified by the environment variable _Envdiffopdomain will be used. If this environment variable is not set, then the argument domain may not be omitted.

These functions are part of the DEtools package, and so they can be used in the form eigenring(..) and endomorphism_charpoly(..) only after executing the command with(DEtools).  However, they can always be accessed through the long form of the command by using DEtools[eigenring](..) or DEtools[endomorphism_charpoly](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$A≔\left[\mathrm{Dx}\,x\right]$

$A≔\left[\mathrm{Dx}\,x\right]$

$L≔{\mathrm{Dx}}^4+2x{\mathrm{Dx}}^2+2\mathrm{Dx}+x^2-4$

$L≔{\mathrm{Dx}}^4+2x{\mathrm{Dx}}^2+x^2+2\mathrm{Dx}-4$

$v≔\mathrm{eigenring}\left(L\,A\right)$

$v≔\left[1\,{\mathrm{Dx}}^2+x\right]$

$$\begin{gathered} \mathbf{for}r\mathbf{in}v\mathbf{do} \\ \mathrm{print}\left('r'=r\right); \\ F≔\mathrm{factor}\left(\mathrm{endomorphism\_charpoly}\left(L\,r\,A\right)\right); \\ \mathbf{for}e\mathbf{in}\left\{\mathrm{solve}\left(F\,x\right)\right\}\mathbf{do} \\ \mathrm{print}\left(G=\mathrm{GCRD}\left(r-e\,L\,A\right)\right) \\ \mathbf{end}\mathbf{do} \\ \mathbf{end}\mathbf{do} \end{gathered}$$

$r=1$

$G={\mathrm{Dx}}^4+2x{\mathrm{Dx}}^2+2\mathrm{Dx}+x^2-4$

$r={\mathrm{Dx}}^2+x$

$G={\mathrm{Dx}}^2+x+2$

$G={\mathrm{Dx}}^2+x-2$

For a description of the method used, see:

van der Put, M., and Singer, M. F. Galois Theory of Linear Differential Equations, Vol. 328. Springer: 2003. An electronic version of this book is available at http://www4.ncsu.edu/~singer/ms_papers.html.

van Hoeij, M. "Rational Solutions of the Mixed Differential Equation and its Application to Factorization of Differential Operators." ISSAC '96 Proceedings. (1996): 219-225.