I need to solve the following system of ODEs:
\begin{cases} \dot{x}=2Dp_x(t)+\frac{v_0(x(t)-x_1)}{\sqrt{(x(t)-x_1)^2+(y(t)-y_1)^2}}+F \\ \dot{y}=2Dp_y(t)+\frac{v_0(y(t)-y_1)}{\sqrt{(x(t)-x_1)^2+(y(t)-y_1)^2}} \\ \dot{p_x}=-\frac{v_0(y(t)-y_1)}{(x(t)-x_1)^2+(y(t)-y_1)^2}\biggr(\frac{(y(t)-y_1)p_x(t)-(x(t)-x_1)p_y(t)}{\sqrt{(x(t)-x_1)^2+(y(t)-y_1)^2}}\biggr) \\ \dot{p_y}=\frac{v_0(x(t)-x_1)}{(x(t)-x_1)^2+(y(t)-y_1)^2}\biggr(\frac{(y(t)-y_1)p_x(t)-(x(t)-x_1)p_y(t)}{\sqrt{(x(t)-x_1)^2+(y(t)-y_1)^2}}\biggr) \end{cases} given these boundary conditions: \begin{cases} x(0)=x_0 \\ y(0)=y_0 \\ x(t_1)=x_1 \\ y(t_1)=y_1 \end{cases} where $D,F,v_0$ are all constants.
I have the feeling this system can be further symplified and was hoping to find an analytical solution to it. Any sort of suggestion on how to proceed is more than welcome.
Thanks!