If a particle of mass $m$ moves in the $x$-$y$ plane, then its equations of motion are
$m\frac{d^2x}{dt^2}=f(t,x,y)\space \space \text{and} \space \space m\frac{d^2y}{dt^2}=g(t,x,y).$
Here $f$ and $g$ represent the $x$ and $y$ components, respectively, of the force acting on the particle. Replace this system of two second-order equations by an equivalent system of four first order equations of the form:
$y_1'=f_1(x,y_1,...,y_n)$
$y_2'=f_2(x,y_1,...,y_n)$
$y_n'=f_n(x,y_1,...,y_n)$
I understand how replace a differential equation by an equivalent system of first order equations when the differential looks something like
$xy''-x^2y'-x^3y=0$
An equivalent system is:
$y_0'=y_1$ $y_1'=x^2y_0+xy_1$
Therefore, I need someone to point me in the right direction for my question stated at the top.