Formal solutions for homogeneous linear ordinary differential systems with polynomial coefficients

The procedure FormalSolution is an implementation in Maple 2015 of an algorithm to construct a basis of the space of formal exponential-logarithmic solutions for systems of differential equations with polynomial coefficients. The algorithm is based on the use of resolving sequences.

The details can be found in the preliminary version of the paper S.A.Abramov, M.Petkovšek, A.A.Ryabenko. Resolving sequences of operators for linear ordinary differential and difference systems.

A system can be represented in several forms.

The matrix form of a full-rank system:

An(x) diff(y(x),x\$n) + … + A1(x) diff(y(x),x) + A0(x) y(x) = 0,

where

• Ai(x) are matrices whose entries are polynomial in x;
• y(x) is a column vector of unknown functions.

The normal form of a first order system:

y'(x) = A(x) y(x),

where

• A(x) is a matrix whose entries are rational functions in x.

Source

LDSformalsols.mpl - the Maple code of the procedure (implemented by A.A.Ryabenko).

Paper_Examples_FS.mw - the Maple session file with examples from the preliminary version of the paper.

Paper_Examples_FS.pdf - the pdf copy of that Maple session. 