Slope fields and qualitative behavior | Code Scholars Tutoring

Understand the idea

At each point $\displaystyle (x,y)$ a slope field draws a short segment with slope given by the differential equation. If the rate depends only on $\displaystyle y$ , slopes agree along horizontal lines. Zero slopes reveal equilibrium solution candidates, which must satisfy the equation. Trace solution curves tangent to the field; do not connect segment endpoints as if they were data values.

y'=F(x,y)\quad\Rightarrow\quad\text{slope at }(a,b)\text{ is }F(a,b)

Worked example

For $\displaystyle y'=x-y$ , find the slope at $\displaystyle (1,3)$ and the line of horizontal segments.

The slope is

\displaystyle 1-3=-2

. Horizontal segments occur where

\displaystyle x-y=0

, or

\displaystyle y=x

. This nullcline is not itself a solution because the line has slope

\displaystyle 1

, while the field has slope

\displaystyle 0

on it.

Try it yourself

For $\displaystyle y'=2-y$ , describe slopes above and below $\displaystyle y=2$ .

Show solution and reasoning

Above $\displaystyle 2$ slopes are negative; below $\displaystyle 2$ they are positive. The constant function $\displaystyle y=2$ is an equilibrium solution.