I have concluded to the following results:
An homogeneous linear differential equation in the ring $\mathbb{C}[x]$ has a solution if at least one root of the characteristic equation is equal to $0$.
An homogeneous linear differential equation in the ring $\mathbb{C}[x, e^{\lambda x} \mid \lambda \in \mathbb{C}]$ has always a solution.
So, is there an algorithm that, given an equation $Dy=0$ and inequations $D_i y\neq 0$, determines whether the system $\displaystyle{Dy=0 \wedge D_i y\ne 0}$ has non-zero solutions in the above rings?
How can we can check whether the system $\displaystyle{Dy=0 \wedge D_i y\ne 0}$ has non-zero solutions in each ring or not?
$D$ and $D_i$, $i=1, \dots , n$ are differential operators.
Let $D = P(d/dx)$ and $D_i = P_i(d/dx)$ where $P$ and $P_i$ are polynomials (the characteristic polynomials of the differential operators).
In order for $D u = 0$ to have a nontrivial solution of the form $g(x) \exp(\lambda x)$ where $g(x)$ is a polynomial, we need $\lambda$ to be a root of $P$. If it is a root of multiplicity $m$, then the solutions of this form are $g(x) \exp(\lambda x)$ where $g$ is any polynomial of degree at most $m$.
Thus the general solution of $D u = 0$ is $\sum_j g_j(x) \exp(\lambda_j x)$ where the sum is over all roots $\lambda_j$ of $P$, and each $g_j$ is a polynomial of degree at most $m_j$, where $m_j$ is the multiplicity of $\lambda_i$ as a root of $P$.
In order for such a solution (with all allowed coefficients nonzero) to also be a solution of $D_i u = 0$, what is needed is that each $\lambda_j$ is also a root of $P_i$, with multiplicity at least $m_j$. Of course, if that happens, $P$ must divide $P_i$. Thus the condition for there to be a solution of $Du = 0$ that is not a solution of any $D_i u = 0$ is that $P$ does not divide any of the $P_i$.
In order for there to be a solution of $Du = 0$ in $\mathbb C[x]$ that is not a solution of $D_i u = 0$, what is needed is that the multiplicity of $0$ as a root of $P$ is greater than its multiplicity as a root of $P_i$. Thus the condition for there to be a solution of $Du = 0$ in $\mathbb C[x]$ that is not a solution of any $D_i u = 0$ is that the multiplicity of $0$ as a root of $P$ is greater than the maximum of its multiplicities as a root of the $P_i$.