Prove that ODE solution is unstable using unstablitity definition

Question

There is a ODE $\dot x=-2x, \dot y=3y$ . Prove using definition that at this point $x=0, y=0$ it is unstable. I am not sure how to interpret the solution of this ODE and tell anything about lines behavior $x=A\cdot e^{-2t} $ and $y=B\cdot e^{3t}$

Answer 1

Assume the origin is stable. That means that it is possible to find a disk of radius $\delta$ centered at the origin such that the solutions with initial data on that disk will remain in a disk of radius, say, $1$. Now choose your initial condition $x(0)=0$, $y(0)=\delta/2$. This point is clearly in the disk of radius $\delta$. However the solution is $x(t)=0$, $y(t)=\delta e^{3t}/2$ which eventually (as $t\to\infty$) leaves the disk of radius $1$. Contradiction.