1550078481-Ordinary_Differential_Equations__Roberts_

36 Ordinary Differential Equations

Notice that a solution which passes through any point below the line y =

x-l is strictly increasing, since below y = x-l, y < x and y' = x-y > 0. We

also claim that as x increases all of the solutions approach the line y = x - l.

That is, all solutions approach the line y = x - 1 asymptotically.

--4 -2 0

x

2

Figure 2.1 Direction Field for y' = x - y

4

Verify that the general solution of the differential equation y' = x - y is

y = ce-x+x-l, where C is any arbitrary constant. Observe that y = x-l is

the solution corresponding to c = 0 and that for any c -=I-0, y = ce-x +x-l
approaches x - 1 as x approaches +oo. Figure 2.2 is a graph of the direction
field for y' = x - y and the solution curves y = ce-x + x - 1 for c = -2,

C = -1, C = 0, C = 1, and C = 2.

Perhaps the following analogy will help you better understand the relation-
ship of the direction field and a solution to an initial value problem. Think of

the rectangle Ras a football field and imagine that you are floating above the

football field in a hot air balloon and looking down upon the field. Suppose
flags have been placed on the field to form a grid work. And suppose a gentle
breeze starts to blow across the field. Each flag will point in the direction the
breeze is blowing at that point on the field. Since we are above the field, we
see only the top of the flag which will be a straight line. If a dandelion seed is
released at some point on the field (the initial point), then the seed will float
across the field. The path of the dandelion seed corresponds to the solution of
the initial value problem (3) consisting of the differential equation (1), which
corresponds to the direction field, and the initial condition (2), which corre-
sponds to the point at which the seed was released. At any point where the
path of the seed touches a flag, the path will be tangent to the flag.