Show that on $\mathbb R^2$ the vector field $y^2\frac{\partial}{\partial x}+x^2\frac{\partial}{\partial y}$ is not complete

Question

How to prove that on $\mathbb R^2$ the vector field $$y^2\frac \partial{\partial x}+x^2\frac \partial{\partial y}$$ is not complete?

I tried to solve the simultaneous ODE, but I realized that it is much complicated...

Answer 1

Note that this vector fields is tangent to the line $x=y$. This enables to compute the orbit of the point $(1,1)$.

The solution which starts at $t=0$ at the point $(1,1)$ is $x(t)={1\over 1-t}=y(t)$. It cannot be defined for all $t \in \mathbb{R}$, as it goes to $\infty$ in a finite time ($t=1$).