Autonomous Differential Equations

Definition

An autonomous differential equation is a first-order ordinary differential equation in which the independent variable (typically time $t$) does not appear explicitly. It takes the general form: $\frac{dy}{dt} = g(y)$

• How to read: “The derivative dy over dt equals g of y.”
• Meaning: The rate of change depends only on the current state $y$, not on when it occurs—time-invariant dynamics.

Why It Matters

They allow us to model systems where the rules of the game don’t change over time, which is true for most physical processes. This time-invariance makes it possible to predict the ultimate stability of a system from its current state.

Core Concepts

• Equilibrium Solutions: Constant solutions $y = c$ where $g(c) = 0$. These represent stationary states of the system.

  • How to read: “The solution y equals c”; “g of c equals zero.”
  • Meaning: Equilibria are fixed points—system doesn’t move when $y$ hits a root of $g$.

• Phase Line Analysis: A one-dimensional geometric representation of the system. Equilibrium points are plotted, and arrows indicate the direction of flow (sign of $g(y)$) in the intervals between them.

  • How to read: “The function g of y.”
  • Meaning: Phase-line arrows show the sign of $g(y)$ between equilibria—quick visual stability analysis without solving the ODE explicitly.

• Stability Classification:

  • Stable (Attractor): Trajectories on both sides move toward the equilibrium point.
  • Unstable (Repeller): Trajectories on both sides move away from the point.
  • Semi-stable: Flow moves toward the point from one side and away from it on the other.

Connected Concepts

• Classification of Differential Equations
• Superposition Principle (Differential Equations)
• First-Order Linear Differential Equations