Proposition 10

Let g and h be two relatively prime polynomials such that $f = gh$. The subspace of solutions of (42) is the direct sum of the subspaces of solutions of the two equations

Now, by the Bezout identity (Alg., VII. 2, th. 1) there exist two polynomials $p(D)$ and $q(D)$ such that $p(D)g(D) + q(D)h(D) = 1$. For every solution $x$ of (42) one can write $x = y + z$, where $y = p(D)g(D)x$ and $z = q(D)h(D)x$; and then $h(D)y = p(D)(f(D)x) = 0$ and $g(D)z = q(D)(f(D)x) = 0$. On the other hand, if both $g(D)x = 0$ and $h(D)x = 0$, one deduces that

which completes the proof.

With the preceding notation one can write

and prop. 10, by induction on $q$, shows that the subspace of solutions of (42) is the direct sum of the subspaces of solutions of the $q$ equations

Now, for every complex number $r$ one has

so (43) is equivalent to

and thus has as solutions the functions $e^{r_j t} p_j(t)$, where $p_j$ runs through the set of polynomials of degree $\leq n_j - 1$; thus one recovers prop. 9 of IV, p. 194.

Assuming that the homogeneous equation (41) has been solved (that is, assuming that the characteristic roots have been found), one knows that the method of variation of parameters allows one to find the solutions of the inhomogeneous equation

where $b(t)$ is an arbitrary regulated function (IV, p. 182); we note that if $b(t)$ is indefinitely differentiable on an interval J then all the integrals of (45) are indefinitely differentiable on J. In the particular case $b(t) = e^{\alpha t} p(t)$, where $p$ is a polynomial (with complex coefficients) and $\alpha$ is an arbitrary complex number, one obtains an integral of (45) more simply by the following method. Put $x = e^{\alpha t} y$; the equation

can, by (44), be written

or, by Taylor’s formula applied to the polynomial $f(D)$,

Let $m$ be the degree of the polynomial $p(t) = \sum_{k=0}^m \lambda_k t^{m-k}$; if $f(\alpha) \neq 0$ (that is, if $\alpha$ is not a characteristic root), there exists one and only one polynomial $u(t) = \sum_{k=0}^m c_k t^{m-k}$ of degree $m$ which is a solution of (46), for the coefficients $c_k$ are determined by the system of linear equations

which clearly admits one and only one solution. If, on the other hand, $\alpha$ is a characteristic root, and $h$ is its multiplicity, the preceding calculation shows that there is one and only one polynomial of degree $m$ such that every solution of $D^h y = v(t)$ is an integral; in other words, every polynomial solution of (46) is then of degree $m + h$ ("resonance").