Variation of constants for n-th order ODE Given a basis $x_1(t),...,x_n(t), t\in(\alpha,\beta)$, of the vector space of all solutions of the homogeneous equation:
$$x^{(n)}(t)+a_{n-1}(t)x^{(n-1)}(t)+...a_1(t)x'(t)+a_0(t)x(t)=0$$
Suppose $x^*(t)=C_1(t)x_1(t)+...+C_n(t)x_n(t)$ is solution of the non-homogeneous equation:
$$x^{(n)}(t)+a_{n-1}(t)x^{(n-1)}(t)+...a_1(t)x'(t)+a_0(t)x(t)=f(t)$$
Prove that the derivatives of functions $C_1(t),...,C_n(t)$ satisfy the system of equations:
$$W(t)\cdot
\begin{pmatrix}
    C_{1}'(t)\\
    C_{2}'(t)\\
    \vdots\\
    C_{n-1}'(t)\\
    C_n'(t)\end{pmatrix}
=\begin{pmatrix}
    0\\
    0\\
    \vdots\\
    0\\
    f(t)\end{pmatrix}
$$
Where $W(t)$ is the Wronskian matrix of basis $x_1(t),...,x_n(t)$.
Edit:
I guess the solution has something to do with the fact that:
$$(W(t)\cdot\begin{pmatrix}
    C_{1}(t)\\
    C_{2}(t)\\
    \vdots\\
    C_{n-1}(t)\\
    C_n(t)\end{pmatrix})'-
W(t)'\cdot\begin{pmatrix}
    C_{1}(t)\\
    C_{2}(t)\\
    \vdots\\
    C_{n-1}(t)\\
    C_n(t)\end{pmatrix}
=
W(t)\cdot\begin{pmatrix}
    C_{1}'(t)\\
    C_{2}'(t)\\
    \vdots\\
    C_{n-1}'(t)\\
    C_n'(t)\end{pmatrix}$$
Edit: I also know that $det(W(t))\neq 0 $, therefore there exists a unique $\begin{pmatrix}
    B_{1}'(t)\\
    B_{2}'(t)\\
    \vdots\\
    B_{n-1}'(t)\\
    B_n'(t)\end{pmatrix}$ such that:
$$W(t)\cdot
\begin{pmatrix}
    B_{1}'(t)\\
    B_{2}'(t)\\
    \vdots\\
    B_{n-1}'(t)\\
    B_n'(t)\end{pmatrix}
=\begin{pmatrix}
    0\\
    0\\
    \vdots\\
    0\\
    f(t)\end{pmatrix}
$$
And I know this means $y^*(t)=B_1(t)x_1(t)+...+B_n(t)x_n(t)$ is solution of the non-homogeneous equation:
$$x^{(n)}(t)+a_{n-1}(t)x^{(n-1)}(t)+...a_1(t)x'(t)+a_0(t)x(t)=f(t)$$
 A: Note that the $C_i(t)$ are not uniquely defined by matching $x^*(t)$ only.
You have to match the derivatives as well. See below.
For brevity, I will omit the explicit dependency on $t$. Let
$$\begin{align}
    A &= \begin{bmatrix}
    0 & 1 & 0 & \cdots & 0
    \\\vdots & \ddots & \ddots & \ddots & \vdots
    \\\vdots & & \ddots & \ddots & 0
    \\0 & \cdots & \cdots & 0 & 1
    \\-a_0 & \cdots & \cdots & -a_{n-2} & -a_{n-1}
    \end{bmatrix}
&   W &= \begin{bmatrix}
    x_1 & \cdots & x_n
    \\  x_1' & \cdots & x_n'
    \\  \vdots & \ddots & \vdots
    \\  x_1^{(n-1)} & \cdots & x_n^{(n-1)}
    \end{bmatrix}
\\  X &= \begin{bmatrix}x\\x'\\\vdots\\x^{(n-1)}\end{bmatrix}
&   F &= \begin{bmatrix}0\\\vdots\\0\\f\end{bmatrix}
\end{align}$$
Suppose $X$ represents a solution of the inhomogeneous ODE.
Then we have
$$\begin{align}
    W' &= AW
&   X' &= AX + F
\end{align}$$
Since $W$ is invertible, there exists a unique vector of functions of $t$,
$$C = \begin{bmatrix}C_1\\\vdots\\C_n\end{bmatrix}
\quad\text{such that}\quad X = WC$$
And for that $C$ we find
$$F = X' - AX = WC' + W'C - AWC = WC' + AWC - AWC = WC'$$
as claimed.
