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. with initial condition y(0) = 0, use Euler’s method with
∆x = 0.1 to estimate y when x = 0.5.
SOLUTION:
Here are the relevant computations:
A Caution: Euler’s method approximates the solution by substituting short line
segments in place of the actual curve. It can be quite accurate when the step sizes
are small, but only if the curve does not have discontinuities, cusps, or
asymptotes.
For example, the reader may verify that the curve for the domain
solves the differential equation with initial condition y = −1 when x = 2.
The domain restriction is important. Recall that a particular solution must be
differentiable on an interval containing the initial point. If we attempt to
approximate this solution using Euler’s method with step size Δx = 1, the first
step carries us to point (3,−3), beyond the discontinuity at and thus outside