Proposition 8

The set of integrals of the homogeneous linear equation (37), defined on $J$, is a vector space of dimension $n$ over the field $\mathbf{C}$.

Given any $n$ integrals $u_j$ ($1 \leq j \leq n$) of equation (37) one calls the Wronskian of this system of integrals the determinant (with respect to the canonical basis of $\mathbf{C}^n$) of the corresponding $n$ integrals $\mathbf{u}_j$ of the equation $d\mathbf{x}/dt = A(t).\mathbf{x}$, that is to say, the function

For the $n$ integrals $u_j$ to be linearly independent it is necessary and sufficient that $W(t) \neq 0$ on $J$; moreover, it suffices for this that $W(t_0) \neq 0$ at only one $t_0$ of $J$ (IV, p. 184, prop. 4); further, one has (IV, p. 184, prop. 5)

(38)

We identify the resolvent $C(t, t_0)$ of equation (35) with its matrix with respect to the canonical basis of $\mathbf{C}^n$; the columns $\mathbf{v}_j(t, t_0)$ ($1 \leq j \leq n$) of this matrix are then $n$ linearly independent integrals

of the homogeneous equation $d\mathbf{x}/dt = A(t).\mathbf{x}$ which correspond to $n$ linearly independent integrals $v_j(t, t_0)$ of equation (37) such that

(Kronecker delta) for $1 \leq j \leq n, 1 \leq k \leq n$ (on agreeing to put $D^0 v_j = v_j$). It results in particular that the method of variation of parameters (IV, p. 182) applied to equation (35) here gives as a particular integral of (33), equal to 0 together with its first $n-1$ derivatives at the point $t_0$, the function

(39)

In the particular case of the equation $D^n x = b(t)$ the formula (39) again gives the formula expressing the $n^{th}$ primitive of the regulated function $b(t)$ which vanishes with its first $n - 1$ derivatives at the point $t_0$, namely

(II, p. 62, formula (19)): the integral of $D^n x = 0$ which vanishes together with its first $n - 2$ derivatives at the point $t_0$, and whose $n - 1$th derivative is equal to 1 there, is actually the polynomial $(t-t_0)^{n-1}/(n-1)!$.