Putnam 1969 A5

Let $u(t)$ be a continuous function in the system of differential equations $$\frac{dx}{dt} =-2y +u(t),\;\; \frac{dy}{dt}=-2x+ u(t).$$ Show that, regardless of the choice of $u(t)$ , the solution of the system which satisfies $x=x_0 , y=y_0$ at $t=0$ will never pass through $(0, 0)$ unless $x_0 =y_0.$ When $x_0 =y_0$ , show that, for any positive value $t_0$ of $t$ , it is possible to choose $u(t)$ so the solution is equal to $(0,0)$ when $t=t_0 .$