  We investigate various properties of p-adic differential equations which have
as a solution an analytic function of the form $F_k (x) = \sum_{n\geq 0} n! P_k
(n) x^n$, where $P_k (n) = n^k + C_{k-1} n^{k-1} + ...+ C_0$ is a polynomial in
n with $C_i\in Z$ (in a more general case $C_i\in Q$ or $C_i\in C_p$). For some
special classes of $P_k (n)$, as well as for the general case, the existence of
the corresponding linear differential equations of the first- and second-order
for $F_k (x)$, is shown. In some cases such equations are constructed. For the
second-order differential equations there is no other analytic solution of the
form $\sum a_n x^n$. Due to the fact that the corresponding inhomogeneous
first-order differential equation exists one can construct infinitely many
inhomogeneous second-order equations with the same analytic solution. Relation
to some rational sums with the Bernoulli numbers and to $F_k (x)$ for some
$x\in Z$ is considered. Some of these differential equations can be related to
p-adic dynamics and p-adic information theory.
