Consider vector second order differential equations system:
$$\mathbf{y}'' + a\mathbf{y}'= \mathbf{b} + \mathbf{A}\mathbf{y}$$ with initial conditions $\mathbf{y} = \mathbf{y}' = \mathbf{0} $, $\mathbf{A} \in \mathbb{R}^{n \times n}$, $\mathbf{y} \in \mathbb{R}^{n }$ and $a$ is a scaler. I am not sure how to start with this, I read somewhere that its solution might include bassel function but I am not sure.
if $\mathbf y = \begin{bmatrix} y_1\\y_2\end{bmatrix}$
And $A = \begin{bmatrix} a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}$
We can say let $\mathbf u = \begin{bmatrix} y_1'\\ y_2'\\y_1\\y_2\end{bmatrix}$
$\mathbf u' = \begin{bmatrix} y_1''\\y_2''\\ y_1'\\ y_2'\end{bmatrix} = \begin{bmatrix} -ay_1'+ a_{11} y_1 + a_{12}y_2\\ -ay_2' + a_{21} y_1 + a_{22} y_2\\ y_1'\\ y_2' \end{bmatrix}+\mathbf b\\ \mathbf u' = \begin{bmatrix} -a& 0 & a_{11}&a_{12}\\ 0&-a & a_{21}&a_{22}\\1&0&0&0\\0&1&0&0\end{bmatrix}\mathbf u + \mathbf b $
The eigenvalues of this matrix will then point you to the solutions of the homogeneous part of the solution.