AHL 5.15—Slope fields

Slope Field: A slope field (also called a direction field) is a graphical representation of a first-order differential equation $\frac{dy}{dx}=f(x,y)$. At each point $(x,y)$ in the plane, a short line segment is drawn with slope equal to $f(x,y)$, giving a visual picture of how solution curves behave , without actually solving the equation.

Slope fields are one of the most powerful qualitative tools in the study of differential equations. Rather than finding an explicit formula for $y$, we can see how solutions behave across the entire plane at a glance.

Key features of a slope field:

• It consists of short line segments plotted at a grid of points $(x,y)$.
• Each segment's gradient (slope) equals the value of $f(x,y)$ at that point.
• Together, the segments reveal the flow of solution curves , where they rise, fall, level off, or converge.

To construct a slope field for $\frac{dy}{dx}=f(x,y)$:

1. Choose a grid of points $(x,y)$ across the region of interest.
2. For each point, evaluate $f(x,y)$ to find the slope at that point.
3. Draw a short line segment through that point with the calculated slope.
4. Repeat across the entire grid.