We can rewrite these two equations $$F_{x}=-\frac{GM_{\odot}M_E}{r^2}\cos{(\theta)}=-\frac{GM_{\odot}M_E}{r^3}x,$$ and $$F_{y}=-\frac{GM_{\odot}M_E}{r^2}\sin{(\theta)}=-\frac{GM_{\odot}M_E}{r^3}y,$$ as four first-order coupled differential equations $$\frac{dv_x}{dt}=-\frac{GM_{\odot}}{r^3}x,$$ $$\frac{dx}{dt}=v_x,$$ $$\frac{dv_y}{dt}=-\frac{GM_{\odot}}{r^3}y,$$ $$\frac{dy}{dt}=v_y.$$