I am reading an introductory differential equations text where the author makes a claim that I feel should be obvious, but I cannot prove to myself.
The author proposes a first-order, autonomous differential equation of the form $x' = f(x)$, where solution $x$ is a function of real variable $t$, possessing two equilibrium points, $x_l$ and $x_r$, such that $f'(x_l)>0$ and $f'(x_r)<0$. I assume these are derivatives w/ respect to $x$.
Because of the sign of their derivatives, he says $x_l$ is a source and $x_r$ is a sink.
Would someone please provide a hint on how to prove the previous statement? I've tried working along the lines of $$\frac{f(y)-f(x_l)}{y-x_l}=\frac{x'(y)-x'(x_l)}{y-x_l}=\frac{x'(y)}{y-x_l}>0$$ for all $y$ in some neighborhood of $x_l$, but now I'm unsure of how to work $t$ into the picture. The mixture of derivatives w.r.t $x$ and $t$ and the function $x$ being treated as a variable makes my head spin and blocks me from any good intuition. Apart from help with this specific problem, advice on how to think about/approach these problems would be welcome.
Thanks in advance,
Paul
Assuming that $x_0$ is an equilibrium point then $f(x_0) = 0$ and near $x_0$ the system can be described as
$$ \frac{d}{dt}(x_0+\delta(t)) = f(x_0)+f'(x_0)\delta(t) + O(\delta^2(t)) $$
or
$$ \dot\delta = f'(x_0)\delta $$
now if $f'(x_0) = -k$ with $k > 0$ then
$$ \dot\delta + k\delta = 0\Rightarrow \delta = C_0e^{-k t} $$
showing that $x_0$ is asymptotically stable (sink). Analogously if $f'(x_0) > 0$ the point is unstable (source).