A direction field is a visual tool for understanding a differential equation without first finding an exact formula for its solutions. At many points in the plane, small line segments show the slope that a solution curve would have there. This makes it possible to see patterns such as growth, decay, leveling off, and divergence.
Direction fields matter because many real systems are easier to analyze graphically than to solve algebraically.
Equilibrium solutions occur when the rate of change is zero, so the solution stays constant over time. For an autonomous differential equation dy/dt = f(y), equilibria appear as horizontal lines where f(y) = 0. Nearby slope marks show whether solutions move toward an equilibrium, away from it, or toward it from one side and away from it from the other.
By reading the arrows and slopes around these lines, students can predict long-term behavior and classify equilibria as stable, unstable, or semistable.
Key Facts
- A direction field shows short slope marks for a differential equation dy/dx = f(x, y).
- For an autonomous equation dy/dt = f(y), slopes depend only on y, so each horizontal row has the same slope.
- An equilibrium solution satisfies dy/dt = 0, so f(y) = 0 and y(t) = constant.
- A stable equilibrium attracts nearby solutions as t increases.
- An unstable equilibrium repels nearby solutions as t increases.
- For dy/dt = ky, the equilibrium is y = 0 and the general solution is y = Ce^(kt).
Vocabulary
- Direction field
- A direction field is a graph of small slope segments that shows the local direction of solution curves for a differential equation.
- Solution curve
- A solution curve is a curve whose tangent slope at each point matches the differential equation.
- Equilibrium solution
- An equilibrium solution is a constant solution where the derivative is zero for all time.
- Stable equilibrium
- A stable equilibrium is an equilibrium that nearby solutions approach as time increases.
- Unstable equilibrium
- An unstable equilibrium is an equilibrium that nearby solutions move away from as time increases.
Common Mistakes to Avoid
- Treating every horizontal line as an equilibrium is wrong because only lines where dy/dt = 0 are equilibrium solutions.
- Drawing solution curves that cross each other is wrong for many standard differential equations because a single initial condition should determine one solution.
- Ignoring the sign of dy/dt is wrong because positive slopes mean y increases as time increases, while negative slopes mean y decreases.
- Classifying stability from only one side is incomplete because an equilibrium can attract from one side and repel from the other.
Practice Questions
- 1 For dy/dt = y(4 - y), find all equilibrium solutions and classify each as stable or unstable.
- 2 For dy/dt = (y + 2)(y - 3), test the intervals y < -2, -2 < y < 3, and y > 3 to determine the direction of motion and the stability of each equilibrium.
- 3 A direction field shows solution curves above y = 1 moving downward and solution curves below y = 1 moving upward. Explain what this tells you about y = 1 and the long-term behavior of nearby solutions.