Let K be the field of formal Laurent series with coefficients
from the Galois field
Fq.
We study some classes of equations with Carlitz derivatives for
Fq-linear functions on subsets of K, which
are the natural function field counterparts of linear ordinary
differential equations with a regular singularity. In particular,
an analog of the equation for the power function, the Fuchs and
Euler type equations, and Thakur's hypergeometric equation are
considered. Some properties of the above equations are similar to
the classical case while others are different. For example, a
simple model equation shows a possibility of existence of a
non-trivial continuous locally analytic
Fq-linear solution
which vanishes on an open neighbourhood of the initial point.