Question: What are the methods for solving $2$nd order inhomogeneous Systems of Differential Equations with Variable Coefficients? (Analytical or numerical methods) $$\textbf{Y}''(t)+\textbf{A}(t)\textbf{Y}'(t)+\textbf{B}(t)\textbf{Y}(t)=\textbf{F}(t),$$ where $\textbf{A}$, $\textbf{B}$, $\textbf{F}$ and $\textbf{Y}$ are $\boldsymbol{n \times n}$ matrices with respect to $t$ (where $t$ is nondimensional parameter.)
Or, I know that we can reduce the second order system of DE to a first order system of DE as follows:
$$\textbf{C}(t)\textbf{Y}'(t)+\textbf{D}(t)\textbf{Y}(t)=\textbf{G}(t)$$ (If you share some examples, notes, books, etc., I can solve the question myself.)
In the first-order form, if $\Phi(t)$ is a fundamental matrix (a nonsingular matrix solution of $C(t) \Phi'(t) + D(t) \Phi(t) = 0$), then $$Y(t) = \Phi(t) \left(c + \int_{t_0}^t \Phi(s)^{-1} G(s)\; ds\right)$$ where $c$ is an arbitrary constant vector.