Proposition 9

Let $r_j$ ($1 \leq j \leq q$) be the distinct roots of the characteristic equation (40), and let $n_j$ be the multiplicity of the root $r_j$ ($1 \leq j \leq q$). Then the $n$ functions $t^k e^{r_j t}$ ($1 \leq k \leq n_j,\ 1 \leq j \leq q$) are linearly independent integrals of the homogeneous equation (41).

One can prove this result directly in the following way. It follows from equation (41) that the $n^{th}$ derivative of every integral of this equation is differentiable on $\mathbf{R}$, from which one deduces immediately, by induction on the integer $m > n$, that every integral of (41) admits a derivative of order $m$, that is, is indefinitely differentiable on $\mathbf{R}$. Let $\mathscr{D}$ be the (non-topological) vector space over $\mathbf{C}$ of indefinitely differentiable complex-valued functions on $\mathbf{R}$; the map $x \mapsto Dx$ is an endomorphism of this space, and equation (41) can be written

where $f(D) = D^n - a_1 D^{n-1} - \ldots - a_{n-1} D - a_n$ (Alg., IV, p. 4).