This is a problem I stuck in Berkeley Problems in Mathematics, fall 1983:
Let $x(t)=(x_{1}(t)...x_{n}(t))$ be a differentiable function from $\mathbb{R}$ to $\mathbb{R}^{n}$. It satisfies a differential equation of the form $x'(t)=f(x(t))$
where $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is a continuous function. Assuming that $f$ satisfies the condition $\langle f(y),y\rangle\le |y|^{2}$ derive an inequality showing that $|x(t)|$ grows at most exponentially.
I am thinking about decompoising $f(y)$ into $y^{T}$ and $y$ directions, but this does not allow me to integrate $x'(t)$ from $t_{0}$ to $t$. If I can show $|f(y)|\le K|y|$ then the inequality would be immediate; but this is false as $f(y)$ can be orthogonal to $y$ at every point $y=x(t)$ and have arbitrarily large norm. So I do not know how to solve this in an elegant way. I can solve it in the most radical case that $x'(t)\cdot x(t)=0\forall t$ and $x'(t)=K(t)x(t)$, but I do not know how to deal with arbitrarily curves.