An autonomous differential equation has the form dy/dt = f(y), where the rate of change depends only on the current value of y and not directly on time. This makes it possible to understand long-term behavior without finding an explicit formula for y(t). A phase line gives a compact visual summary of where solutions increase, decrease, and settle.
This is useful in physics, biology, economics, and any setting where a quantity changes according to its current state.
The key idea is to locate equilibrium points where f(y) = 0, then test the sign of f(y) on the intervals between them. If f(y) is positive, solutions move upward on the phase line, and if f(y) is negative, solutions move downward. Arrows pointing toward an equilibrium indicate stability, while arrows pointing away indicate instability.
This qualitative method predicts trends, limiting values, and threshold behavior without solving the differential equation explicitly.
Key Facts
- An autonomous differential equation has the form dy/dt = f(y).
- Equilibrium points occur where f(y) = 0.
- If f(y) > 0 on an interval, then y(t) is increasing there.
- If f(y) < 0 on an interval, then y(t) is decreasing there.
- An equilibrium is stable if nearby phase line arrows point toward it.
- An equilibrium is unstable if nearby phase line arrows point away from it.
Vocabulary
- Autonomous differential equation
- A differential equation in which the derivative depends on the dependent variable but not explicitly on the independent variable.
- Phase line
- A one-dimensional diagram that shows equilibrium points and the direction of solution movement along the dependent variable axis.
- Equilibrium point
- A value of the dependent variable where dy/dt = 0, so a constant solution can remain there.
- Stable equilibrium
- An equilibrium that nearby solutions approach as time increases.
- Unstable equilibrium
- An equilibrium that nearby solutions move away from as time increases.
Common Mistakes to Avoid
- Solving the equation before analyzing the phase line is unnecessary because autonomous equations often reveal long-term behavior from signs of f(y) alone.
- Putting arrows on the phase line based on the sign of y is wrong because arrows depend on the sign of dy/dt = f(y), not on whether y is positive or negative.
- Ignoring repeated equilibrium roots can give the wrong stability classification because the sign of f(y) may not change across a repeated root.
- Calling every equilibrium stable is wrong because stability depends on nearby arrow directions, not just on dy/dt being zero at that point.
Practice Questions
- 1 For dy/dt = y(4 - y), find the equilibrium points and determine whether y is increasing or decreasing on each interval.
- 2 For dy/dt = (y + 2)(y - 1)^2, find all equilibrium points and classify each as stable, unstable, or semistable using a phase line.
- 3 A population model has dy/dt = f(y), with f(y) < 0 for y < 3, f(y) > 0 for 3 < y < 8, and f(y) < 0 for y > 8. Describe the long-term behavior of solutions that start below 3, between 3 and 8, and above 8.