  We study some classes of equations with Carlitz derivatives for $F_q$-linear
functions, 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 $F_q$-linear solution which vanishes on
an open neighbourhood of the initial point. Part I: J. Number Theory, 83
(2000), 137-154.
