FIGURE N9–6b
We observe that, since y ′′ for 3 ln x equals the true curve is concave down and below the
Euler graph.
The last column in the table shows the true values (to three decimal places) of y. The Euler
approximation for 3 ln 3 is 3.85; the true value is 3.296. The Euler approximation with four steps
is not very good! However, see what happens as we increase the number n of steps:
n EULER APPROXIMATION ERROR
4 3.85 0.554
10 3.505 0.209
20 3.398 0.102
40 3.346 0.050
80 3.321 0.025Doubling the number of steps cuts the error approximately in half.
EXAMPLE 7
Given the d.e. = x + y with initial condition y(0) = 0, use Euler’s method with Δx = 0.1 to
estimate y when x = 0.5.
SOLUTION: Here are the relevant computations:
x y (SLOPE) · Δx = (x + y) · (0.1) = Δ y
P 0 0 0 0(0.1) = 0
P 1 0.1 0 (0.1)(0.1) = 0.01
P 2 0.2 0.01 (0.21)(0.1) = 0.021
P 3 0.3 0.031 (0.331)(0.1) = 0.033