We can rewrite $$ \frac{dv_x^E}{dt}=-\frac{GM_{\odot}}{r^3}x_E-\frac{GM_J}{r_{EJ}^3}(x_E-x_J). $$ to $$ \frac{dv_x^E}{dt}=-\frac{4\pi^2}{r^3}x_E-\frac{4\pi^2M_J/M_{\odot}}{r_{EJ}^3}(x_E-x_J), $$ where we used $$ GM_J = GM_{\odot}\left(\frac{M_J}{M_{\odot}}\right)=4\pi^2 \frac{M_J}{M_{\odot}}. $$ Similarly, for the velocity in \( y \)-direction we have $$ \frac{dv_y^E}{dt}=-\frac{4\pi^2}{r^3}y_E-\frac{4\pi^2M_J/M_{\odot}}{r_{EJ}^3}(y_E-y_J). $$ Similar expressions apply for Jupiter. The equations for \( x \) and \( y \) derivatives are unchanged. This equations are similar for all other planets and as we will see later, it will be convenient to object orient this part when we program the full solar system.