On the Teaching of Differential Equations

Abstract

The simplest introduction to the theory of differential equations is offered by nature. Cover a thin bar magnet with a horizontal piece of cardboard, and sprinkle fine iron splinters over the latter. Each splinter after coming to rest will assume a definite direction, depending upon its location. Before us is displayed the direction field associated with a first order differential equation. The splinters fit together along lines, called the lines of force of the magnet in the plane of the cardboard. These lines represent the solutions of the differential equation. If the magnet covered by the horizontal cardboard is in a vertical position touching the cardboard at one pole, then each splinter will point toward the pole. If we choose this pole as the origin of a cartesian coordinate system, the splinter at the point (x, y) thus has the slope y/x. Hence the differential equation associated with the magnetic direction field reads y'(x)= y(x)/x for x$O. The splinters fit together along the straight lines through the origin. With the excep-