Given a function $f_\alpha(t)$, where $f$ is smooth defined on a compact interval, I want to find an differential Operator $D(\alpha)_t$, such that $$D(\alpha)_t f_\alpha(t)=0.$$
You may imagine $\alpha$ to be fixed.
$f$ does not have many other properties, it is not periodic, does not come from the solution of some ODE or anything.
A systematic numerical procedure would be satisfactory, such as determining the coefficients $d_i$ of a (non unique perhaps) finite linear combination of derivatives: $$D_t = \sum_i d_i (\partial_t)^i$$.
Of course, I cannot even prove that such an operator always exists (or is unique up to scaling. Uniqueness does not matter, but I would guess that if a constructive procedure exists, then it always leads to a unique operator. I do not care about uniqueness.) and if it doesn't, it would be nice to give some characterization when it does (or not).
