Slope field

Suppose we have a first-order differential equation of the form \[ y'=f(x,y) \] where $f(x,y)$ is some expression in $x$ and $y$. The differential equation says that the slope of a solution curve at a point $(x,y)$ on the curve is $f(x,y)$. If we draw short line segments with slope $f(x,y)$ at several points $(x,y)$, the result is called a slope field (or direction field). These line segments indicate the direction in which a solution curve is heading, so the direction field helps us visualise the general shape of these curves.

Simulation

The following simulation shows the slope field for a given first-order differential equation $y'=f(x,y)$.

Things to try:

• Change the function $y'=f(x,y)$. Example: (x-1)(y^2-1)
• Activate the box Solution curve.
• Change the initial conditions $x_0,y_0$.

Worksheet exemplar

The following file contains activities and problems associated with the simulation.

• Slope field (pdf)
• Slope field (tex)