Let $a=y-x+1$, then $\dot{x}=a$ and $\dot{a}=-rx^2+x-1$, so $\ddot{x}=-rx^2+x-1$.

So perhaps it is possible to accidentally bumb into a system of the form $\ddot{x}=f(x)$.

How to tell this from your integral: naively trying to put it into a simpler form by completing the square seemed to work pretty well.

How to tell this from your differential equation: Compute $\ddot{x}$ and $\ddot{y}$ and see if you can write them in terms of only $x$ or only $y$.

Let $a=y-x+1$, then $\dot{x}=a$ and $\dot{a}=-rx^2+x-1$, so $\ddot{x}=-rx^2+x-1$.

So perhaps it is possible to accidentally bump into a system of the form $\ddot{x}=f(x)$.

How to tell this from your integral: naively trying to put it into a simpler form by completing the square seemed to work pretty well.

How to tell this from your differential equation: Compute $\ddot{x}$ and $\ddot{y}$ and see if you can write them in terms of only $x$ or only $y$.