Chapter 7.1, Problem 24P

Prove that, for the system $\frac{dx}{dt}=F\left(x,y\right),\frac{dy}{dt}=G\left(x,y\right)$, there is at most one trajectory passing through a given point $\left(x_0,y_0\right)$.

Hint: Let $C_0$ be the trajectory generated by the solution $x={\varphi }_0\left(t\right),y={\psi }_0\left(t\right)$ wit ${\varphi }_0\left(t_0\right)=x_0,{\psi }_0\left(t_0\right)=y_0$, and let $C_1$ be the trajectory generated by the solution $x={\varphi }_1\left(t\right),y={\psi }_1\left(t\right)$ with ${\varphi }_1\left(t_1\right)=x_0,{\psi }_1\left(t_1\right)=y_0$. Use the fact that the system is autonomous, and also the existence and uniqueness theorem, to show that $C_0$ and $C_1$ are the same.