James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 9.2 Direction Fields and Euler’s Method 595
y
1
solution curve
y=L(x)
Euler’s Method
The basic idea behind direction fields can be used to find numerical approximations to
solutions of differential equations. We illustrate the method on the initial-value problem
that we used to introduce direction fields:
y9 − x 1 y ys0d − 1
0 1
FIGURE 11
First Euler approximation
y
1
1.5
0 0.5
1
x
x
The differential equation tells us that y9s0d − 0 1 1 − 1, so the solution curve has slope 1
at the point s0, 1d. As a first approximation to the solution we could use the linear approximation
Lsxd − x 1 1. In other words, we could use the tangent line at s0, 1d as a rough
approximation to the solution curve (see Figure 11).
Euler’s idea was to improve on this approximation by proceeding only a short distance
along this tangent line and then making a midcourse correction by changing direction as
indicated by the direction field. Figure 12 shows what happens if we start out along the
tangent line but stop when x − 0.5. (This horizontal distance traveled is called the step
size.) Since Ls0.5d − 1.5, we have ys0.5d < 1.5 and we take s0.5, 1.5d as the starting point
for a new line segment. The differential equation tells us that y9s0.5d − 0.5 1 1.5 − 2,
so we use the linear function
y − 1.5 1 2sx 2 0.5d − 2x 1 0.5
FIGURE 12
Euler approximation with step size 0.5
y
1
0 0.25
1
FIGURE 13
Euler approximation with step size 0.25
y
x
as an approximation to the solution for x . 0.5 (the green segment in Figure 12). If we
decrease the step size from 0.5 to 0.25, we get the better Euler approximation shown in
Figure 13.
In general, Euler’s method says to start at the point given by the initial value and proceed
in the direction indicated by the direction field. Stop after a short time, look at the
slope at the new location, and proceed in that direction. Keep stopping and changing
direction according to the direction field. Euler’s method does not produce the exact
solution to an initial-value problem—it gives approximations. But by decreasing the step
size (and therefore increasing the number of midcourse corrections), we obtain successively
better approximations to the exact solution. (Compare Figures 11, 12, and 13.)
For the general first-order initial-value problem y9 − Fsx, yd, ysx 0 d − y 0 , our aim is
to find approximate values for the solution at equally spaced numbers x 0 , x 1 − x 0 1 h,
x 2 − x 1 1 h, . . . , where h is the step size. The differential equation tells us that the slope
at sx 0 , y 0 d is y9 − Fsx 0 , y 0 d, so Figure 14 shows that the approximate value of the solution
when x − x 1 is
y 1 − y 0 1 hFsx 0 , y 0 d
slope=F(x¸, y¸)
(⁄, ›)
hF(x¸, y¸)
Similarly,
In general,
y 2 − y 1 1 hFsx 1 , y 1 d
y n − y n21 1 hFsx n21 , y n21 d
h
y¸
0 x¸
FIGURE 14
⁄
x
Euler’s Method Approximate values for the solution of the initial-value problem
y9 − Fsx, yd, ysx 0 d − y 0 , with step size h, at x n − x n21 1 h, are
y n − y n21 1 hFsx n21 , y n21 d n − 1, 2, 3, . . .
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