In my ODE class we are studying Sturm–Liouville theory and my professor mentioned that given a differential operator $$ Lu = \sum_{k = 0}^{n} a_n(t) \, u^{(n)}(t) $$ then for all $\lambda \in \mathbb{C}$ there is a non-trivial $u:[a, b] \to \mathbb{C}$ such that $Lu = \lambda u$.
However, I am not able to see why this is true. For example, even in the most basic case where $Lu = a_0(t)\,u(t)$ the we get
$$Lu = \lambda u \implies a_0(t)\,u = \lambda u \implies a_0(t) \equiv \lambda$$
But this means that we can't do this for arbitrary $L$. Am I missing or misinterpreting something?