The four coupled differential equations can then be discretized using Euler's method as (with step length $h$) $$v_{x,i+1}=v_{x,i}-h\frac{4\pi^2}{r_i^3}x_i,$$ $$x_{i+1}=x_i+hv_{x,i},$$ $$v_{y,i+1}=v_{y,i}-h\frac{4\pi^2}{r_i^3}y_i,$$ $$y_{i+1}=y_i+hv_{y,i},$$