I'm taking a course in PDE's, and I'm trying to understand an example from the course literature, and would like some help.
The example is the following
Example 1: (baby wave equation)
Consider the second order ODE $$f''(t)=-w^2f(t)+a(t)$$
where $a$ is a given function, $w \in \mathbb R$ is a given constant. A way to solve this is to write it a s a $2 \times 2$ system of first order ODE's and solve it by diagonalisation.
Let $u : \mathbb R \to \mathbb R^2$ be the vector valued function $u(t) = \begin{bmatrix}f(t) \\ f'(t)\end {bmatrix}$
where it satisfies $u'(t)=\begin{bmatrix} 0 & 1 \\ -w^2 & 0 \end{bmatrix}u(t)+\begin{bmatrix} 0 \\ a(t) \end{bmatrix}$.
Then the example goes on to diagonalize the matrix $\begin{bmatrix} 0 & 1 \\ -w^2 & 0 \end{bmatrix}$ and solve it.
My question is, why do we choose $u(t)$ to be what is is? It does not make sense to me what they chosen $u(t)$ to be.