# Inverse Problem to finding Homogenous Solution of Differential Operator: Given a function $f$ find an Operator that has $f$ in its kernel

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).