Show that there exists a $k>o$ such that solutions of this system of differential equations never cross the line $y = kx$. For the system: 
$\frac{d x}{d t} = -y$
$\frac{d y}{d t} = x(1-x) - Ay$
Where $A \geq 2$. 
I want to show that there exists a $k > 0$ such that $(x(t), y(t))$ cannot cross through the line $\{ y = k x, x > 0 \}$ when we start below this line.
I started with showing that for the line $y = k x$ we have that $\frac{d y}{d t} = k \frac{d x}{ d t} = -k y$, and now I wanted to show that if we start below this line, we have a lower $\frac{dy}{dt}$ so we can never cross the line.
But I have been unsuccesful.
Any ideas? Thanks.

Hint:
You need to prove that there exists $k$ such that on the line $y=kx$ the inequality $
\frac{y'(t)}{x'(t)}\le k$ holds.
