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.