How to get $y_1$ and $y_2$?

Question

I am not able to see how to get the $y_1$ and $y_2$ (highlighted below). I hope someone could help out with this!

Answer 1

You have $$x''=f_1\\y''=f_2$$Now you name $x'=y_1$ and $y'=y_2$. Then you have $$x'=y_1\\y_1'=x''=f_1\\y'=y_2\\y_2'=y''=f_2$$So if $\vec y=(x,y_1,y,y_2)$, then $\vec y'=(x',y_1',y',y_2')=(y_1,f_1,y_2,f_2)$.

Answer 2

The proposed naming is very confusing. For a 4 component vector $y$ I would expect the components to be indexed as $\vec y = (y_1,y_2,y_3,y_4)$. Then you would get $\vec y'=(y_2, f_1, y_4, f_2)$. Or give totally different names, $x'=v_x, y'=v_y$ so that then the derivatives are $(v_x,f_1,v_y,f_2)$.

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In the first convention to avoid using $y$ twice as variable name let's call the big state vector $\vec u$. Then $u_1=x$, $u_2=x'$, $u_3=y$, $u_4=y'$ and thus the derivatives of the components of $\vec u$ are \begin{align} u_1'=x'=u_2,\\ u_2'=x''=f_1,\\ u_3'=y'=u_4,\\ u_4'=y''=f_2. \end{align}