FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 1, PAGES 3-21

**On singular points of solutions of linear differential systems with
polynomial coefficients**

S. A. Abramov

D. E. Khmelnov

Abstract

View as HTML
View as gif image

We consider systems of linear ordinary differential equations
containing $m$ unknown functions of
a single variable $x$.
The coefficients of the systems are polynomials over
a field $k$ of
characteristic $0$.
Each of the systems consists of $m$ equations
independent over $k[x,d/dx]$.
The equations are of arbitrary orders.
We propose a computer algebra algorithm that, given
a system $S$ of this form, constructs
a polynomial $d(x)$Î k[x]\{0} such that if
$S$ possesses
a solution in $\$\; \backslash bar\; k((x-\backslash alpha\; ))^m\; \$$ for some $\$\; \backslash alpha\; \backslash in\; \backslash bar\; k\; \$$ and
a component of this solution has a nonzero polar part, then
$d($a)=0.
In the case where $k$Í **C** and
$S$ possesses an
analytic solution having a singularity of an arbitrary type (not
necessarily a pole) at $$a, the equality
$d($a)=0 is also satisfied.

Location: http://mech.math.msu.su/~fpm/eng/k1112/k111/k11101h.htm

Last modified: January 31, 2012