Slope Fields

By the end of this lesson, you will be able to construct and interpret slope fields, match slope fields to differential equations, and sketch solution curves through specified points.

The Big Idea

A slope field (direction field) is a graphical representation of a differential equation $y' = f(x, y)$. At each point $(x, y)$, a short line segment is drawn with slope $f(x, y)$. Together, these segments reveal the "flow" of solutions — you can sketch solution curves by following the segments.

Slope fields are especially useful when an explicit solution is hard to find: the geometry of solutions is visible without solving the equation.

Constructing a Slope Field

For $y' = f(x, y)$:

1. Choose a grid of points $(x_i, y_j)$
2. At each point, compute $m = f(x_i, y_j)$
3. Draw a short line segment through $(x_i, y_j)$ with slope $m$

Reading a Slope Field

Solution curves follow the direction of the line segments — they are tangent to the slope field at every point.

Horizontal segments ($m = 0$): where $y' = 0$ — potential extrema of solutions.

Vertical segments ($m$ undefined): where $y' = \infty$ — vertical tangents.

Parallel segments along the same line/curve: indicates the equation has the form $y' = g(x)$ or $y' = h(y)$ (depends only on one variable).

Example 1 — Identifying Key Features

$y' = x$:

• Slope increases as $x$ increases
• Along $x = 0$: all segments horizontal
• Along $x = 1$: all segments have slope 1
• Doesn't depend on $y$ → same slope for all $y$ at a given $x$
• Solutions: $y = x^2/2 + C$ (parabolas)

Example 2 — Autonomous Equation

$y' = y(1-y)$:

• Depends only on $y$, not $x$ → horizontal translation doesn't change the slopes
• Horizontal segments at $y = 0$ and $y = 1$ (equilibria)
• Positive slopes for $0 < y < 1$; negative for $y > 1$ and $y < 0$
• Solutions: logistic curves approaching $y = 1$ or falling to $y = 0$

Example 3 — Matching Slope Fields to ODEs

Match each ODE to its slope field:

(a) $y' = y$: segments point up-right in QI, down-left in QIII; steeper for larger $|y|$.

(b) $y' = x - y$: segments horizontal along the line $y = x$; tilted otherwise.

(c) $y' = -x/y$: segments perpendicular to lines $y = cx$ — circular flow (circles are solutions).