## Method of undetermined coefficients with the use of matrices - reinventing the wheel?

The subject are LDE's with constant coefficients of the following form:

$\sum_{i=0}^{N} a _{i}y ^{\left(i\right)} = e^{kx}\left( W_{m}\left(x\right)\sin \left(kx\right)+ V_{n}\left(x\right)\cos \left(kx\right)\right)$,

$W_{m}$ and $V_{n}$ being polynomials of given degrees.

1) I choose the $max\{m,n\}$, let it be $n$.
2) I write down the following functions:

$\varphi_{n}\left(x\right) = x ^{n}e ^{kx}\sin \left(\beta x\right)\\$
$\phi_{n}\left(x\right)=x^{n}e^{kx}\cos \left(\beta x\right)$

every such function satisfies the conditions below:

$\varphi_{n}'=k\varphi_{n}+\beta \phi_{n}+n\varphi_{n-1}$
$\phi_{n}'=k\phi_{n}-\beta \varphi_{n}+n\phi_{n-1}$

3) I put down both the source term and the solution function using a base $\mathbb{B}=\left[\varphi_{n} \ \phi_{n} \ \varphi_{n-1} \ \phi_{n-1} \ ... \ \varphi_{0} \ \phi_{0}]^{T}$, so that the right side of the equation becomes $C\mathbb B$, and the solution becomes $A\mathbb B$.

4) I make two additional matrices:

$\mathbb{R}^{2}_{2} \ni G:= \begin{bmatrix} k&\beta\\-\beta &k\end{bmatrix}$

$\mathbb{R}^{2n+2}_{2n+2} \ni S:= \begin{bmatrix} G&nI&0&...&0&0&0\\0&G&(n-1)I&...&0&0&0\\0&0&G&...&0&0&0\\...&...&...&...&.. .&...&...\\0&0&0&...&G&2I&0\\0&0&0&...&0&G&I\\0&0& 0&...&0&0&G \end{bmatrix}$,

therefore

$y^{(n)}=AS^{n}\mathbb B.$

5) Finally I rewrite the initial equation:

$\sum_{i=0}^{N} a _{i}y ^{(i)} = a _{N}AS^{N}\mathbb{B} + a _{N-1}AS^{N-1}\mathbb{B} +...+ a _{0}A\mathbb{B} = A \left( \sum_{i=0}^{N} a_{i}S^{i} \right) \mathbb{B}= C\mathbb{B}$

eventually giving $A\sum_{i=0}^{N} a_{i}S^{i} = C$, which can be solved for $A$.

Has such a method been described somewhere, or is it something new?