Problem

Let y be a function of x that satisfies the differential equation

and assume (0, 0) is a point on the graph of y. Use Euler's method with step sizes .01, .001, .0001 to estimate the value of y when x = .99. Do these estimates seem reliable?

Verify that

is a solution to the differential equation above, and examine the graph of this function near x = .99 to explain any discrepancies in the estimates given by Euler's method.

Solution

We'll start with the second part, verifying the answer, and graph it to see how it looks.

so this is a solution.

The graph looks quite crowded near the value x = .99. Let's see what we get for the approximations.

The exact value is

The estimated values seem to be varying quite a bit, considering the small step sizes. The graph exhibits the problem in using Euler's method to estimate the value at x = .99 for the function y. Near this point, the slopes become extremely large, and they change very quickly as the function values oscillate more and more quickly. We can creep up to x = 1 with extremely small steps, and still see wide swings in the function value, and in the slope: