> [!NOTE] Definition (Stationary Point of First Order Autonomous Scalar ODE) > Let $x'(t)=f(x(t))$with $f:\mathbb{R}\to \mathbb{R},$ $t\in (\alpha ,\beta)\subset \mathbb{R},$ be a first order [[Autonomous Scalar Ordinary Differential Equation|autonomous scalar ODE]]. > > A stationary point of the equation is a point $x^{*}\in \mathbb{R}$ such that $f(x^{*})=0.$ # Properties **Stability**: A stationary point $x^{*}$ is [[Stable Stationary Point of First Order Autonomous Scalar Ordinary Differential Equation|stable]] iff when the initial value of $x$ is close to $x^{*},$ then the solution to the initial value problem, $x(t),$ is close to $x^{*}$ for all $t.$ By [[Condition for Stability of Stationary Point of First Order Autonomous Scalar Ordinary Differential Equation]], $x^{*}$ is stable if $f'(x^{*})<0$ and unstable if $f'(x^{*})>0.$ # Applications **Examples**: See [[Stationary Points of Population Growth Model]].