Suppose I have a coupled system of ODEs of the following form $$\ddot{y}_1(t)=F_1(y_1(t),y_2(t),\dot{y}_1(t), \dot{y}_2(t))$$ $$ \dddot{y}_2(t)=F_2(y_1(t),y_2(t),\dot{y}_1(t),\dot{y}_2(t)) $$
Now the highest order in $y_1$ is 2 while the highest order in $y_2$ is 3. What do you classify this system as? From what I know this would be a coupled system of third order. In general, if given an $n^{th}$ order equation we can reduce it to an $n-$dimensional system of $1^{st}$ order equations. Then in this case I can turn the first equation into a 2-dimensional system of first order equations and the second equation into a 3-dimensional system of first order equations, so effectively I can reduce this to a 5-dimensional system of first order equations. Is this correct for this system?
Also if someone has a reference I can look at for systems of ODEs (mostly nonlinear) with an asymmetry in the highest order of derivatives in different equations, I would be very grateful.
Letting $y_1=u_1, y_1'=u_2, y_2=u_3, y_2'=u_4$, and $y_2''=u_5$ leads to the system
$u_1'=u_2 (=y_1')$
$u_2'=F_1(u_1, u_3, u_2, u_4) (=y_1'')$
$u_3'= u_4 (=y_2')$
$u_4'=u_5 (=y_2'')$
$u_5= F_2(u_1, u_3, u_2, u_4) (=y_2''')$