equilibrium solution

Among all the solutions of a differential equation, a few special ones never change at all — they sit perfectly still. A population exactly at the level its environment can sustain, a cup of coffee already at room temperature, a pendulum hanging straight down: each is in balance, with no tendency to move. The constant function describing such a frozen state is an equilibrium solution, and these few flat curves organize the behaviour of all the others.

Precisely, an equilibrium solution (or constant solution) is a solution that is a constant function, y(t) = c, so its derivative is zero everywhere. For an autonomous first-order equation y' = f(y), these are exactly the values c where f(c) = 0 — set the right-hand side to zero and solve. For the logistic equation y' = r y (1 - y/K), setting the right side to zero gives y = 0 and y = K, so those two constants are the equilibrium solutions: at y = K the growth rate is zero, the population neither rises nor falls. In a slope field they appear as horizontal lines that other integral curves approach or veer away from.

Equilibrium solutions are the skeleton of the long-term story. Other solutions are often drawn toward a stable equilibrium (a sink) or pushed away from an unstable one (a source), so knowing the equilibria and their stability tells you where everything ends up, frequently without solving the equation at all. One honest warning that recurs throughout the subject: when you solve a separable equation by dividing through, you must first set the divisor to zero, because that very act of dividing can silently delete exactly these constant solutions — the equilibria are the classic 'lost solutions.'