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](https://mathoverflow.net/questions/352683/non-polynomial-splines-a-non-linear-problem).)



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


