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Putnam 1969 A5

Source: Putnam 1969

March 31, 2022
Putnamfunctiondifferential equation

Problem Statement

Let u(t)u(t) be a continuous function in the system of differential equations dxdt=−2y+u(t),      dydt=−2x+u(t).\frac{dx}{dt} =-2y +u(t),\;\;\; \frac{dy}{dt}=-2x+ u(t). Show that, regardless of the choice of u(t)u(t), the solution of the system which satisfies x=x0,y=y0x=x_0 , y=y_0 at t=0t=0 will never pass through (0,0)(0, 0) unless x0=y0.x_0 =y_0. When x0=y0x_0 =y_0 , show that, for any positive value t0t_0 of tt, it is possible to choose u(t)u(t) so the solution is equal to (0,0)(0,0) when t=t0.t=t_0 .
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