I am teaching a class on differential equations lately, and in the course of trying to invent problems for a midterm, I got to wondering how to generate an ordinary differential equation having prescribed functions $f_i$ as its solutions$^1$. I came up with one solution, which I will post as an answer below (in the spirit of sharing knowledge), but I welcome other solutions too. Here's the statement of the problem:
Given smooth functions $\{f_i:\mathbb{R}\rightarrow\mathbb{C} \; |\; i=1,\dots,n\}$, how can we construct a (possibly non-linear) differential operator $L$ such that $L f_i=0$ for all $i$?
- I found this useful in particular for illustrating the method of finding a second solution to a second order ODE by way of the Wronskian.
This procedure generates a linear differential operator having the desired functions as solutions. Let $x$ be the independent variable, and define $L_1 = \frac{d}{dx} - \frac{f_1'}{f_1}$. Note that $L_1 f_1 = 0$. Inductively define $$L_i = \left( \frac{d}{dx} - \frac{\frac{d}{dx}L_{i-1} f_i}{L_{i-1} f_i}\right) L_{i-1}$$
If $L_{i-1} f_j=0$ for all $j<i$, then $L_i f_j=0$ also. And the definition of $L_i$ is designed to make $L_i f_i=0$ also. So $L_n$ will be a differential operator with the desired properties.