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.
Understanding Calculus: Autonomous Differential Equations
A useful feature of these equations is that the same value of y always has the same instantaneous rate. Imagine two experiments that reach the same population size at different times. In this model, both populations have the same tendency to change at that size.
Time shifts do not alter the shape of a solution curve. They only move the curve left or right on a graph of y against time.
This is why one solution can represent many starts that occur at different clock times. It also means that a graph should not cross itself in the y direction for a single solution, provided the rate function is well behaved.
Students often use separation of variables when an actual formula is needed. The idea is to gather all terms involving y on one side and the time term on the other. Integration then relates elapsed time to the change in y.
This process can be powerful, but the resulting formula may be difficult to solve for y. A phase line can give the most important answer much faster. It tells whether a starting value rises, falls, approaches a level, or moves away from a threshold.
It is important to check every interval separated by zeros of the rate function. A sign can change at a zero, though it does not always change. That detail determines the behavior.
Some equilibrium values have one-sided stability. Suppose arrows below a level point upward while arrows above it point upward too. A solution starting below approaches the level, but a solution starting above moves away.
This is called semistable behavior. It appears when the rate function touches zero without passing through it, as happens with a squared factor. Repeated zeros deserve special attention for this reason.
The derivative of the rate function gives another local test. If the derivative is negative at an equilibrium, nearby values tend to return.
If it is positive, nearby values tend to depart. This test is quick, but a sign chart remains more reliable when the derivative is zero or when the function has unusual features.
A common real example is a population with limited resources. At low population levels, births may exceed deaths, so the population grows. Near a carrying level, crowding reduces the net growth rate until it becomes zero.
This produces a stable long-term population in a simple model. Threshold models can be more serious. A population below a critical level may decline toward extinction, while one above that level may grow toward a healthier level.
Similar thresholds occur in chemical reactions, disease spread, temperature control, and financial feedback. When building or interpreting a model, pay attention to units and assumptions.
The rate must have units of y per unit time. The model assumes the rate is controlled only by the current state, so seasonal forcing, changing treatment, or outside shocks require a model that includes time or other variables.
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.