Which methods (analytical or numerical) do you suggest to solve the following matrix diff. equation? And why? (Presumably analytical method)
$$ \textbf{M}\ddot{V}+\textbf{C}\dot{V}+\textbf{K}V=\textbf{P},$$ where matrices $\ddot{V}$, $\dot{V}$ and $V$ with respect to $t$ are as follows: $$ \mathbf{M} \begin{bmatrix} \ddot{V}(1,t) \\ \ddot{V}(2,t) \\ \vdots \\ \ddot{V}(n,t) \end{bmatrix} +\mathbf{C} \begin{bmatrix} \dot{V}(1,t) \\ \dot{V}(2,t) \\ \vdots \\ \dot{V}(n,t) \end{bmatrix} +\mathbf{K} \begin{bmatrix} V(1,t) \\ V(2,t) \\ \vdots \\ V(n,t) \end{bmatrix} = \mathbf{P}.$$
In here, the matrices $\textbf{M}$, $\textbf{C}$, $\textbf{K}$ and $\textbf{P}$ are as follows ($\textbf{M}$, $\textbf{C}$ and $\textbf{K}$ are $n \times n$ matrices, and $\textbf{V}$, $\textbf{P}$, $\textbf{F}$ are $n \times 1$ matrices)
$$\mathbf{P} = \frac{P}{\rho A} \begin{bmatrix} \sin\left(\dfrac{\pi vt}{l}\right) \\ \sin\left(\dfrac{2\pi vt}{l}\right) \\ \vdots \\ \sin\left(\dfrac{n\pi vt}{l}\right) \end{bmatrix} +\mathbf{F}(t), \quad\text{where } \mathbf{F}(t) = \begin{bmatrix} f_1(t) \\ f_2(t) \\ \vdots \\ f_n(t) \end{bmatrix}.$$
In here, $l$, $P$, $A$, $\rho$, $\alpha$, $v$, $N$, $E$ and $I$ are constants. It doesn't matter what are these constants or initial or boundary conditions for me. You can select appropriate values. The matrix $\textbf{F}(t)$ is ungiven. So solution $\textbf{V}$ will depend on $\textbf{F}$.