3: Direction fields

Section 1: Introduction and Fundamentals - 3: Direction Fields

Differential equations often lack explicit solution formulas. Direction fields (or slope fields) provide a powerful graphical method to visualize the behavior of solutions to first-order ordinary differential equations (ODEs) of the form $dy/dx = f(x, y)$, without actually solving the equation. They reveal the "flow" of solutions across the xy-plane.

Constructing a Direction Field:

1. Select Grid Points: Choose a grid of points $(x, y)$ within a region of interest in the plane.
2. Calculate Slopes: At each grid point $(x_0, y_0)$, compute the slope $m = f(x_0, y_0)$ given by the ODE.
3. Draw Line Segments: Draw a short line segment through $(x_0, y_0)$ with slope $m$. This segment represents the tangent to the solution curve passing through that point. Typically, only a small dash or arrowhead is drawn to indicate the direction.

Interpreting Direction Fields:

• Solution Curves: A solution curve to the ODE is a curve that passes through the direction field and is tangent to every line segment it touches. Imagine sketching a curve that flows with the indicated slopes.
• Visualizing Families: The entire field depicts the family of all possible solution curves. Different starting points (initial conditions) trace out different curves within this family.
• Qualitative Behavior: Direction fields reveal critical long-term trends:
  • Equilibrium Solutions: Points where $f(x, y) = 0$ consistently. Segments are horizontal here. Solution curves are horizontal lines (if stable or unstable). Look for rows of horizontal segments.
  • Convergence/Divergence: Observe if solution curves tend to approach (converge toward) or move away (diverge from) specific curves, lines, or points as $x$ increases or decreases. This indicates stability or instability of equilibria.
  • Asymptotes: Identify curves that solution curves approach but never cross.
  • Periodicity: Look for repeating patterns suggesting periodic solutions.
  • Sensitivity: Observe how solutions starting close together behave – do they stay close (stable) or diverge rapidly (sensitive dependence)?

Example Insight:

Consider $dy/dx = y - x$.

1. Equilibria: The line $y = x$ is where $dy/dx = 0$. Segments are horizontal along $y=x$.
2. Above $y=x$: For points where $y > x$, $f(x, y) > 0$. Segments slope upwards. Solution curves increase.
3. Below $y=x$: For points where $y < x$, $f(x, y) < 0$. Segments slope downwards. Solution curves decrease.
4. Behavior: Solution curves above $y=x$ rise but their slope decreases as they near the line. Curves below $y=x$ fall but their slope becomes less negative (shallower) as they near the line. Solution curves approach the line $y = x$ asymptotically as $x$ increases. This shows $y=x$ is a stable equilibrium solution. Curves diverge from $y=x$ as $x$ decreases.