In many cases it is possible to rewrite a second-order differential equation in terms of two first-order differential equations. Consider again the case of Newton's second law in Eq. (3). If we define the position $x(t)=y^{(1)}(t)$ and the velocity $v(t)=y^{(2)}(t)$ as its derivative $$\begin{equation} \frac{dy^{(1)}(t)}{dt}=\frac{dx(t)}{dt}=y^{(2)}(t), \tag{7} \end{equation}$$ we can rewrite Newton's second law as two coupled first-order differential equations $$\begin{equation} m\frac{dy^{(2)}(t)}{dt}=-kx(t)=-ky^{(1)}(t), \tag{8} \end{equation}$$ and $$\begin{equation} \frac{dy^{(1)}(t)}{dt}=y^{(2)}(t). \tag{9} \end{equation}$$