Autonomous Equations

A phase-line approach to autonomous first-order equations, equilibria, stability, and exact solution curves.

A direction field supplies a local slope at every point; a solution curve must follow those slopes. A grid of short oriented arrows is crossed by a smooth solution curve that follows their local slopes. A dashed equilibrium line separates regions where the indicated slope directions differ.

Precise definition

An autonomous first-order differential equation has the form $y'=f(y)$: the rate depends on the current state $y$, not explicitly on the independent variable $t$. An equilibrium is a constant solution $y(t)=y_$ satisfying $f(y_)=0$. Non-equilibrium solutions can often be separated as $dy/f(y)=dt$ on intervals where $f(y)\ne0$.

Notation and mathematical language

Use $t$ for time, $y(t)$ for state, and $f(y)$ for the vector field. A phase line marks zeros of $f$ and arrows according to the sign of $f$: $f>0$ means $y$ increases and $f<0$ means $y$ decreases. Stability describes nearby solutions, not whether the equilibrium value itself changes.

Conceptual picture

Because every point at the same height has the same slope, horizontal translation of a solution curve produces another solution where defined. On the phase line, arrows pointing toward an equilibrium from both sides indicate asymptotic stability; arrows pointing away indicate instability; one-sided attraction gives semistability.

Fully worked example

Interpretation and application

Autonomous equations model populations, temperature feedback, chemical concentration, and one-dimensional control. The equilibrium classification predicts long-run behaviour without solving explicitly. That prediction remains conditional on the model: external time-dependent forcing would make the equation non-autonomous and can change the conclusion.