On the Isoclinal Lines of a Differential Equation of the First Order

Extract

A differential equation may be regarded from two points of view, one purely analytical, the other geometrical. From the analytical point of view, a differential equation of the first order is merely a functional relation between x, y, and p (where p = dy/dx), and the problem of solving the equation is to find a function of x, say f(x), such that if f (x) and df(x)/dx are substituted for y and p in the equation, the result is an identity in x. In the geometrical method, on the other hand, x and y are treated as the co-ordinates of a point in a plane and p as a direction. The differential equation then attaches to every point in the plane a certain direction, which may be conveniently represented by an arrow drawn through the point. The problem of integration then resolves itself into finding a family of curves, such that, at every point (x′, y′), the direction of the curve at that point is the direction obtained by substituting x′ and y′ in the differential equation and solving for p. These curves are called the integral curves of the equation. This method owes its development chiefly to Lie.