Show that a power series $f(x) = \sum_{n=0}^{\infty} c_n x^n \in \mathbf{Q}\llbracket x \rrbracket$ is a solution of a differential equation if and only if there exist an integer $d \geq 0$ and polynomials not all zero $S_0, \ldots, S_d \in \mathbf{Z}[x]$ such that: for all $n \geq 0$,
$$S_0(n) c_n + \cdots + S_d(n) c_{n+d} = 0.$$