Every differential equation is equivalent to a first-order system. Does the converse hold?

Question

We know that every differential equation is equivalent to a first-order system. I am trying to prove or disprove the converse. For example in $\mathbb{R}^2$, if we have a system $\dot{x}=f(x,y)$, $\dot{y}=g(x,y)$. Can we always convert it to one differential equation (for example, only in terms of $x$)? Under what condition, this is possible? Thank you, in advance, for your response!

Answer 1

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 $f$.