1550078481-Ordinary_Differential_Equations__Roberts_

34 Ordinary Differential Equations

In chapter 1 we gave examples of initial value problems with no solution,
with a unique solution, and with multiple solutions. Later, we will state a
theorem which will guarantee the existence of a solution to an initial value
problem of the form (3) and we will state a second theorem which will guar-
antee the existence and uniqueness of a solution.

2.1 Direction Fields

First, let us examine the geometric significance of the differential equation
(1) y' = f(x, y). At each point (x, y) in the xy-plane for which the function
f is defined, the differential equation defines a real value, f(x, y). This value
is the slope of the tangent line to every solution of the differential equation
which passes through the point (x, y). Thus, the differential equation specifies
the direction that a solution must have at every point (x, y) in the domain of

f. Imagine passing a short line segment of slope f ( x, y) through each point

( x, y) in the domain of f. The set of all such line segments is called the

direction field of the differential equation y' = f(x, y). Usually, the domain

of f contains an infinite number of points; and, therefore, we cannot possibly

draw the direction field. Instead, we choose some rectangle

R = {(x, y)I Xmin::; x::; Xmax and Ymin::; y::; Ymax}

which contains points of the domain of f; we select a set of points (xi, Yi)

contained in R; and for those points (xi, Yi) in the domain off, we construct

a short line segment at (xi, Yi) with slope f(xi, Yi)· We will call a graph

constructed in this manner the direction field of y' = f(x, y) in the rectangle R.

The direction field indicates subregions in R in which solutions are increasing
and decreasing, it often reveals maxima and minima of solutions in R , it
sometimes indicates the asymptotic behavior of solutions, and it illustrates
the dependence of solutions on the initial conditions.
Let (x, y) be a fixed point in the rectangle Rat which f(x, y) is defined.

• If y' = f(x, y) > 0, then the solution which passes through the point
  (x, y) is increasing.

• Similarly, if y' = f(x, y) < 0, then the solution which passes through the

point (x , y) is decreasing.

• When y' = f(x, y) = 0, we must consider the direction lines near (x, y),
  and there are five distinct cases to consider.
  ( 1) If the direction lines of points immediately to the left and right of ( x, y)

are also horizontal (y' = 0 to the right and left of (x, y)), then the solution

through ( x, y) is constant on some interval containing x.
(2) If the direction lines immediately below and to the left of (x, y) have

positive slope (y' > 0 below and to the left of (x , y)), then the solution through

the point (x, y) increases as it approaches (x, y) from the left.