Lectures on Ordinary Differential Equations

1

Differential Equations of the First Order in One Unknown Function

PART A.

THE CAUCHY-EULER APPROXIMATION METHOD

1.Definitions. Direction Fields

By a domain D in the plane we understand a connected open set of points; by a closed domain or region , such a set plus its boundary points. The most general differential equation of the first order in one unknown function is

where F is a single-valued function in some domain of its arguments. A differentiable function y(x) is a solution if for some interval of x, (x, y(x), y′(x)) is in the domain of definition of F and if further F(x, y(x), y′(x)) = 0. We shall in general assume that (1) may be written in the normal form:

where f(x, y) is a continuous function of both its arguments simultaneously in some domain D of the x-y plane. It is known that this reduction may be carried out under certain general conditions. However, if the reduction is impossible, e.g., if for some (x0, y0, y0′) for which F(x0, y0, y0′) = 0 it is also the case that [∂F(x0, y0, y0′)]/∂y′ = 0, we must for the present omit (1) from consideration.

A solution or integral of (2) over the interval x0 ≤ x ≤ x1 is a single-valued function y(x) with a continuous first derivative y′(x) defined on [x0, x1] such that for x0 ≤ x ≤ x1:

Geometrically, we may take (2) as defining a continuous direction field over D; i.e., at each point P: (x, y) of D there is defined a line whose slope is f(x, y); and an integral of (2) is a curve in D, one-valued in x and with a continuously turning tangent, whose tangent at P coincides with the direction at P. Equation (2) does not, however, define the most general direction field possible; for, if R is a bounded region in D, f(x, y) is continuous in R and hence bounded

where M is a positive constant. If α is the angle between the direction defined by (2) and the x-axis, (4) means that a is restricted to such values that

The direction may approach the vertical as P: (x, y) approaches the boundary C of D, but it cannot be vertical for any point P of