Exploring Direction Fields

This page allows you to explore direction fields and solutions for a few example first order differential equations. The equations are all of the form $y'(x) =f(x,y)$ . The direction field is a way of visualizing a differential equation. The direction field arrows show the direction of the "flow" of solutions $y(x)$ so that at any point in the x-y plane the direction of tangent curve will be along the direction field arrows. You can select one of the example differential equations from the drop down menu below the graph.

Numerical solutions (computer calculated approximations) can be explored by draging the marker on the graph, The "forward" solution (increasing x) is shown in blue and the reverse solution is shown in red.

All computer solution methods have limitations, and you will occasionally see what appears to be rather bizarre curves rather than nice smooth curves." See if you can identify what it is about the differential equation and the solution that might be a problem for the solution method. Hint: very large y or y' can be problem spots for the numerical methods applied here.