  In 1900, at the international congress of mathematicians, Hilbert claimed
that the Riemann zeta function $\zeta(s)$ is not the solution of any algebraic
ordinary differential equations on its region of analyticity. In 2015, Van
Gorder considered the question of whether $\zeta(s)$ satisfies a non-algebraic
differential equation and showed that it formally satisfies an infinite order
linear differential equation. Recently, Prado and Klinger-Logan extended Van
Gorder's result to show that the Hurwitz zeta function $\zeta(s,a)$ is also
formally satisfies a similar differential equation
\begin{equation*}\label{HurDE}
  T\left[\zeta (s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}.
\end{equation*} But unfortunately in the same paper they proved that the
operator $T$ applied to Hurwitz zeta function $\zeta(s,a)$ does not converge at
any point in the complex plane $\mathbb{C}$.
  In this paper, by defining $T_{p}^{a}$, a $p$-adic analogue of Van Gorder's
operator $T,$ we establish an analogue of Prado and Klinger-Logan's
differential equation satisfied by $\zeta_{p,E}(s,a)$ which is the $p$-adic
analogue of the Hurwitz-type Euler zeta functions \begin{equation*}\label{HEZ}
\zeta_E(s,a)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+a)^s}. \end{equation*} In
contrast with the complex case, due to the non-archimedean property, the
operator $T_{p}^{a}$ applied to the $p$-adic Hurwitz-type Euler zeta function
$\zeta_{p,E}(s,a)$ is convergent $p$-adically in the area of
$s\in\mathbb{Z}_{p}$ with $s\neq 1$ and $a\in K$ with $|a|_{p}>1,$ where $K$ is
any finite extension of $\mathbb{Q}_{p}$ with ramification index over
$\mathbb{Q}_{p}$ less than $p-1.$

Number Theory;Classical Analysis and ODEs