Chapter 7.1, Problem 25P

Prove that if a trajectory starts at a noncritical point of the system $\frac{dx}{dt}=F\left(x,y\right),\frac{dy}{dt}=G\left(x,y\right)$, then it cannot reach a critical point $\left(x_0,y_0\right)$ in a finite length of time.

Hint: Assume the contrary, that is, assume that the solution $x=\varphi \left(t\right),y=\psi \left(t\right)$ satisfies $\varphi \left(a\right)=x_0,\psi \left(a\right)=y_0$. Then use that fact that $x=x_0,y=y_0$ is a solution of the given system satisfying the initial condition $x=x_0,y=y_0$ at $t=a$.