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A phase line is a one-dimensional diagram showing the equilibrium solutions of dy/dx = f(y) and arrows indicating whether y increases or decreases. This visual tool predicts the long-term behavior of all solutions without solving the equation explicitly.
Draw the y-axis and mark all equilibrium points (zeros of f). Use the sign of f(y) between equilibria to determine flow direction with arrows. Classify equilibria as stable (sink) or unstable (source) based on arrow behavior.
An autonomous equation is one where the right-hand side depends only on y, not on x: dy/dx = f(y). From your study of direction fields, you know that for autonomous equations the slope depends only on the height y — every horizontal strip in the direction field has the same slope. The phase line compresses this entire direction field into a single vertical axis, capturing everything you need to know about long-term behavior.
To draw a phase line: first find all equilibrium solutions, which are the values of y where f(y) = 0. Plot these points on the y-axis. Between them, determine the sign of f(y): if f(y) > 0, then dy/dx > 0 and y is increasing, so draw an upward arrow. If f(y) < 0, then dy/dx < 0 and y is decreasing, so draw a downward arrow. The result is a complete portrait of where every solution is headed without computing a single formula.
Stability of an equilibrium y* is read directly from the arrows. If arrows on both sides point *toward* y*, then nearby solutions converge to it — y* is a stable equilibrium (or "sink"). Perturbations decay and the system returns. If arrows on both sides point *away* from y*, then nearby solutions diverge — y* is an unstable equilibrium (or "source"). Small perturbations grow. If arrows point toward on one side and away on the other, y* is semi-stable — stable from one direction, unstable from the other.
Consider dy/dx = y(1 − y), the logistic equation. Setting f(y) = 0 gives equilibria at y = 0 and y = 1. For y slightly negative, f(y) < 0 so arrows point down. For 0 < y < 1, f(y) > 0 so arrows point up. For y > 1, f(y) < 0 so arrows point down. Reading the phase line: y = 0 has arrows pointing away on both sides — unstable. y = 1 has arrows pointing toward on both sides — stable. No matter what positive initial condition you start with, the solution eventually approaches y = 1. The phase line told you this without solving the equation at all. This qualitative power — predicting long-term behavior from the sign of f alone — is what makes phase line analysis essential before the richer phase portraits of two-dimensional systems.