Differential equation satisfied by linear combinations of eigenfunctions of linear differential operator

Question

Let $D$ be a linear differential operator on $\mathcal{C}^\infty(\mathbb{R})$, and let $\mathcal{E}_\lambda=\{f\in\mathcal{C}^\infty(\mathbb{R})|Df=\lambda f\}$ be the space of eigenfunctions of $D$ to the eigenvalue $\lambda$. It is easy to see that $\bigcup_{\lambda}\mathcal{E}_\lambda$ can be characterized by the non-linear ODE $(Df)'f-f'Df=0$. Is there a similar non-linear ODE satisfied by linear combinations of the eigenfunctions, i.e. can $\mathrm{span}\bigcup_{\lambda}\mathcal{E}_\lambda$ be characterized by an ODE? (This is loosely related to this question.)

Answer 1

If $f$ is a linear combination of at most $N$ eigenfunctions, then $f$,$Df$,$D^2f$,...,$D^Nf$ are linearly dependent. Hence $W(f,Df,...,D^Nf)=0$, where $W$ denotes the Wronskian.

Answer 2

The characterization of $\mathrm{span}\bigcup_\lambda\mathcal{E}_\lambda$ asked for in the question is generally not possible. Consider e.g. $D=\frac{\rm d^2}{{\rm d}x^2}$. Then already $\mathrm{span}\bigcup_{\lambda=-n^2,n\in\mathbb{Z}}\mathcal{E}_\lambda$ is dense in $L^2([-\pi;\pi])$, and there is no hope to find a continuous $F$ such that $F(f)=0$ for all $f\in \mathrm{span}\bigcup_{\lambda=-n^2,n\in\mathbb{Z}}\mathcal{E}_\lambda$, but $F(f)\not=0$ for other $f\in L^2([-\pi;\pi])$. Adding in the other eigenfunctions of $D$ can make the issue only worse, not better.

The question I should have asked is: Let $D$ and $\mathcal{E}_\lambda$ be as above, and let $N\in\mathbb{N}$. Is there some differential equation satisfied by all those $f\in\mathrm{span}\bigcup_\lambda\mathcal{E}_\lambda$ that can be written as linear combinations of elements of at most $N$ different $\mathcal{E}_\lambda$? That question doesn't look quite so hopeless.