Could the Riemann zeta function be a solution for a known differential equation?

When posed properly, a long-standing open problem, but in the form you ask:

Robert A. Van Gorder, MR 3276353 Does the Riemann zeta function satisfy a differential equation?, J. Number Theory 147 (2015), 778--788.

The Riemann zeta function is "hypertranscendental" in the sense shown HERE

It is not the solution $y(x)$ of a differential equation of the form $$ F(x,y,y',y'',\dots,y^{(n)})=0 $$ where $F$ is a polynomial (with constant coefficients).

The fact that $\zeta$ satisfy no algebraic differential equation is due to its famous relation with the Gamma function which was proved by Hölder not to satisfy such an equation.

Detailed answer can be found here with five (commented and linked) references.

On the other hand, this function is linked with many other transcendental special functions like polylogarithms which satisfy Fuschian non commutative differential equations.