We know that every differential equation is equivalent to a first-order system. I am trying...

Consider your example. In order to make this a second-order equation in $x$, you want to solve $\dot{x}=f(x,y)$ for $y$ as a function of $x$ and $\dot{x}$, say $y=a(x,\dot{x})$. This may or may not be possible, (and in most cases even if it is possible in principle it can't be done in closed form). Then we get
$\ddot{x}=f_1(x,y)\dot{x}+f_2(x,y)\dot{y}=f_1(x,a(x,\dot{x}))\dot{x}+f_2(x,a(x,\dot{x}))g(x,a(x,\dot{x}))$
where $f_1$ and $f_2$ are the partial derivatives of $f$.