Construct and interpret slope (direction) fields of a first-order differential equation, sketch solution curves through given points, and relate the qualitative behaviour of solutions to the field

What this dot point is asking

QCAA wants you to understand a first-order differential equation $\frac{dy}{dx} = F(x, y)$ geometrically, before solving it. A slope field draws a short line segment at each grid point with gradient given by $F$, and the solution curves follow these segments. The assessable skill is reading and constructing the field and sketching solution curves that stay tangent to it.

The answer

What a slope field shows

A first-order differential equation $\frac{dy}{dx} = F(x, y)$ assigns a slope to every point $(x, y)$ in the plane. The slope field (direction field) is a grid of short segments, each drawn with the gradient $F(x, y)$ at its location. A solution curve is any curve that is tangent to these segments everywhere it passes.

Constructing the field

At each grid point, evaluate $F(x, y)$ to get the slope, then draw a short segment with that gradient. For example, if $\frac{dy}{dx} = x$, the slope at $(2, 1)$ is $2$, so the segment there rises steeply; along the $y$-axis ($x = 0$) all segments are horizontal.

Reading qualitative behaviour

The field reveals behaviour without an explicit formula. Where $F(x, y) = 0$, segments are horizontal, marking possible turning points or equilibria. Where $F$ is large, the field is steep. Regions where $F > 0$ have solutions increasing left to right; where $F < 0$, decreasing.

Equilibrium solutions

If $\frac{dy}{dx} = F(y)$ depends only on $y$, then any value $y = c$ with $F(c) = 0$ gives a constant (equilibrium) solution: a horizontal line that the field segments lie along. Nearby solution curves either approach (stable) or move away from (unstable) this equilibrium.

Sketching a solution through a point

Given an initial point, start there and draw a smooth curve that follows the direction of the nearby segments, always staying tangent to the field. The curve threads through the segments like a path following arrows. Different initial points give different members of the solution family.

Solutions do not cross

For a well-behaved equation, distinct solution curves never cross, because the slope at any point is uniquely determined by $F(x, y)$. Two curves crossing would require two different tangents at one point, which is impossible.