In the book of Differential equations: A higher system approach by Hubbard and West, at page 36, it is given that
the essential size of a DE is the smallest number $n$ such that the DE can be transformed into an autonomous first order DE in $\mathbb{R}^n$
And as an example, it is given that
$$x'' = x^2 - y \\ y'' = y-x$$ can be expressed as $$x' = w \\ y' = v \\ w' = x^2 - y \\ v' = y-x$$ so the essential size of this system is $4$.
And as another example,
$$x' = x+ y \\ y' = y-x^2 \\ z' = 1$$ and stated that the this system has essential size $2$.
However, I do not understand how what is the difference between the first example and the second so that the former is of size 4 whereas the ladder is 2. I mean, how do we determine the essential size of a differential equation ?
For this system $$x' = x+ y \\ y' = y-x^2 \\ z' = 1$$ The equation for $z$
$$z' = 1$$
is independent of the other two variables and could be solved by itself.
Since we are looking for the minimum number of autonomous first order equations, the first two equations would be sufficient.