sign analysis of f(y)
Here is the small piece of arithmetic that makes the whole phase-line method run. For an autonomous equation dy/dt = f(y), the derivative dy/dt is literally just f(y). So the sign of f(y) — whether it is positive, negative, or zero — directly tells you whether the solution at that value of y is going up, going down, or standing still. No solving required; you just check signs.
The recipe is a sign chart, exactly like the ones from precalculus. First find where f(y) = 0; these are the equilibria, the boundaries. They divide the y-axis into intervals. In each interval, plug in any test value to find the sign of f there. Where f(y) > 0, the solution is increasing (dy/dt > 0), so any solution passing through that band rises. Where f(y) < 0, the solution is decreasing, so it falls. The sign cannot flip within an interval because f only changes sign at its zeros, so one test point settles the whole interval. Translate '+' into an up-arrow and '−' into a down-arrow and you have built the phase line.
This is why qualitative analysis feels almost free: the same sign-chart skill used to sketch a function's increase and decrease now reveals the long-term destiny of every solution of the equation. It also exposes how an equilibrium is approached — solutions slow down as they near a stable equilibrium because f, and therefore the speed dy/dt, shrinks to zero there. The whole story of rise, fall, and approach is encoded in the single sign pattern of f.
For dy/dt = (y − 1)(y − 3), zeros at y = 1 and y = 3. Test y = 0: f = (−1)(−3) = +3 > 0 (up). Test y = 2: f = (1)(−1) = −1 < 0 (down). Test y = 4: f = (3)(1) = +3 > 0 (up). So solutions rise below 1, fall between 1 and 3, rise above 3.
Sign of f(y) is the sign of dy/dt: + means rising, − means falling.
This shortcut works only because the equation is autonomous, so dy/dt equals f(y) with no explicit t. For a nonautonomous equation the sign of the right side can change with time, and a fixed sign chart no longer tells the whole story.