I didn't see this covered in this article and I don't remember. How were the coefficients for these differential equations calculated? Also, how useful were t

I didn't see this covered in this article and I don't remember. How were the coefficients for these differential equations calculated? Also, how useful were these? Today, I think most people use vector field plots because differential equations are so complicated. And I can't think of an easy way to get from the output of this machine to a vector field.