1.3 | Direction Fields


A direction field (or slope field) is a graphical representation of the solutions of a first-order differential equation. It is useful because it can be created without solving the differential equation analytically. The representation may be used to qualitatively visualize solutions, or to numerically approximate them. A direction field can be constructed as follows:

  • Choose a point in the \(xy\)-plane, say \((x_0, y_0)\) and plot this point
  • Draw a line segment going through \((x_0,y_0)\) with slope given by the value of the derivative at \((x_0,y_0)\).
  • Repeat the first two steps for other points in the \(xy\)-plane.
An autonomous differential equation is an ordinary differential equation of the form
\[
\frac{dy}{dt} = f(y).
\]
An isocline is a curve through points at which the parent function’s slope will always be the same, regardless of initial conditions.

E 1.3 Exercises