Completeness of eigenfunctions of higher order differential equation

Question

I have a third order linear differential equation, with a free parameter, and boundary conditions that depend on that parameter. I don't think it is possible to obtain an analytic solution, but I would like to know if the eigenfunction, i.e. the family of solutions that correspond to different values of the free parameter, are complete. As a side note, is there any example for an equation whose eigenfunction are not complete?

Answer 1

For your side note: if I understood what you mean correctly, here is a differential equation whose corresponding eigenfunction is not "complete":

$$ y' = y; \\ \text{Boundary condition: } y(0) = \lambda $$

Surely, we cannot express a general continuous function as superpositions of $\lambda e^x$.

If you mean an eigenfunction in the sense that $L(f) = \lambda f$, then things become different. It is necessary, at least, to consider complex eigenvalues.