What is the vector field associated with this ODE $\ddot{x}+p(t)\dot{x}+q(t)x=0$? What is the divergence of the vector field?
My attempt:
Let $y=\dot{x}$
$\ddot{x}+p(t)\dot{x}+q(t)x=0\ \Rightarrow \frac{d}{dt}\begin{pmatrix} x\\y \end{pmatrix}=\begin{pmatrix}\ y\\ -p(t)y-q(t)x\end{pmatrix}=F$
Then $\nabla\ \cdot F=\frac{\partial y}{\partial x}+\frac{\partial (-py-qx)}{\partial y}=-p(t)$.
Is this correct?