I have one more question regarding this problem. What are the generalized Eigenspaces of the differential operator $D$?
So far I was able to show that $p_\ell(t)=t^\ell e^{\lambda t}$ are the generalized eigenvectors of $D$ since $$(D-\lambda\text{id})^n(p_\ell(t))=\begin{cases}\frac{\ell !}{(\ell-n)!}p_{\ell -n}(t), & n\geq \ell\\0, & n>\ell\end{cases}$$ Assuming I did my calculations right, this would mean that $p_\ell(t)$ is a generalized eigenvector of rank $\ell+1$. So the generalized Eigenspace would be $$GE(\lambda, D)=\{f\in C^\infty(\mathbb R,\mathbb C):(D-\lambda\text{id})^n(f)=0\}=\\=\dots=?$$
The last dots are where I'm not sure about. Which of the $p_\ell$ exactly do I take here?
A guess would be: $GE(\lambda, D)=\{\sum_{k=0}^n a_kt^ke^{\lambda t}:a_k\in\mathbb C\}$