the $(t,x)$-plane, ${\rm C}^\omega(\Omega)$ is the space of real analytic functions on $\Omega$. ${\rm C}^p_x(\Omega)$ with $p\in\mathbb N$ denotes the subspace of functions from ${\rm C}(\Omega)$ that admit derivatives with respect to $x$ up to order $p$, and these derivatives are continuous on $\Omega$. Analogously, ${\rm C}^\omega_x(\Omega)$ denotes the subspace of functions from ${\rm C}(\Omega)$ that are real analytic with respect to $x$ and whose derivatives with respect to $x$ are continuous on $\Omega$. ${{\rm DO}}(\Omega)$, ${{\rm DO}}^\infty(\Omega)$, ${{\rm DO}}^\omega_x(\Omega)$ and ${{\rm DO}}^\omega(\Omega)$ denote the sets of linear differential operators in $x$ (hence of the form ) with coefficients from ${\rm C}(\Omega)$, ${\rm C}^\infty(\Omega)$, ${\rm C}^\omega_x(\Omega)$ and ${\rm C}^\omega(\Omega)$, respectively, and whose leading coefficients do not vanish on the entire (open) set $\Omega$. ${{\rm DO}}_1(\Omega)$, ${{\rm DO}}^\infty_1(\Omega)$, ${{\rm DO}}^\omega_{x,1}(\Omega)$ and ${{\rm DO}}^\omega_1(\Omega)$ are, respectively, the subsets of operators from ${{\rm DO}}(\Omega)$, ${{\rm DO}}^\infty(\Omega)$, ${{\rm DO}}^\omega_x(\Omega)$ and ${{\rm DO}}^\omega(\Omega)$ whose leading coefficients are equal to one. The notation ${\mathop{\rm ord}\nolimits}P$ and ${\mathop{\rm lcoef}\nolimits}P$ is used for the order and the leading coefficient of the operator $P\in{{\rm DO}}(\Omega)$, respectively. If ${\mathop{\rm ord}\nolimits}P=p$, then we view the operator $P$ as a map from ${\rm C}^p_x(\Omega)$ to ${\rm C}(\Omega)$, and thus (classical) solutions of 