Section 18.2

Direction Fields

If gives the slope at every point, we can draw the shape of the solution before we find the formula.

Introduction

Many differential equations cannot be solved explicitly with a nice formula. However, the DE itself gives us local information: the slope.

By sketching small slope lines at many points on a grid, we create a Direction Field (or Slope Field) that reveals the "flow" of all possible solutions.

The Sketching Recipe

How to Sketch

Create a grid of points (x,y).
At each point, calculate the slope .
Draw a short line segment with that slope.
Repeat to see the pattern.

Tip: Look for Isoclines—curves where the slope is constant (e.g., if , slope is 0 wherever ).

Visual: Interactive Field

Interactive: Logistic Growth

Notice how solution curves (blue) follow the direction of the little slope lines.

Worked Examples

Example 1: Autonomous DE

Sketch the direction field for .

1. Analyze Slope:

The slope depends only on y (Autonomous).

If , slope is 0 (Horizontal).

If , slope is positive ($y=3 \to m=1$).

If , slope is negative ($y=1 \to m=-1$).

2. Visual:

All arrows on a horizontal line are parallel.

Solutions act like they are "repelled" from the equilibrium line .

Example 2: Depend on x

Analyze .

1. Isoclines:

Slope is constant when x is constant (Vertical lines).

At , slope is 0.

At , slope is 1 everywhere.

2. Solution Shape:

We know .

The field shows a family of Parabolas opening upward.

Example 3: Long Term Behavior

For , what happens as if ?

1. Equilibria:

and .

2. Sign Analysis:

If (like our initial condition 4): .

Actually wait, . So slope is positive.

So y starts at 4 and increases forever.

Limit is .

3. What if y(0)=1?

Here . Then is Negative.

Solution decreases towards 0.