> [!NOTE] Definition (Stable 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]]. > > Let $x^{*}\in \mathbb{R}$ be a [[Stationary Point of First Order Autonomous Scalar Ordinary Differential Equation|stationary point]] of the equation. Then $x^{*}$ is stable iff for all $\varepsilon>0,$ there exists $\delta>0$ such that $|x_{0}-x^{*}|<\delta \implies |x(t)-x^{*}|<\varepsilon$for all $t\geq 0.$ # Properties **Analytic condition for stability**: 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 $f'(x^{*})>0.$