Conversion of a differential equation to a system of first-order differential equations

Statement

Consider a differential equation with independent variable $x$ and dependent variable $y$ given as follows:

$F(x,y,y',\dots ,y^{(k)})=0$

We can convert this to a system of first-order differential equations with $k$ variables and $k$ equations as follows.

Take the variables to be $y,y_{1},y_{2},\dots ,y_{k-1}$ where $y_{i}$ is the variable that we will eventually set as $y^{(i)}$. The system has $k-1$ first-order differential equations of the form:

$y'=y_{1},y_{1}'=y_{2},\dots ,y_{k-2}'=y_{k-1}$

and one more first-order differential equation:

$F(x,y,y_{1},\dots ,y_{k-1},y_{k-1}')=0$

Note that, with the exception of the last differential equation, all the other equations are autonomous (no reference to the dependent variable), first-order (the highest derivative order is one) and linear.