Problem

Assume that the point (–1, .001) is on the graph of a function that satisfies the differential equation

Estimate the value of y when x = 1 using Euler's method. Use step sizes .1, .01, .001, and .0001. Then solve the differential equation to determine the exact value of y when x = 1.

Solution

Using the applet on the Web site, we get the follow results for the value of y when x = 1:

We would expect to see a limiting value, but as we do not yet. The limits of the applet do not accept smaller step sizes than .0001, so we must be content with the approximation we have.

We should be familiar with the differential equation: its solution is x² + y² = C, a circle centered at the origin. Since (–1, .001) is on the curve, use this information to solve for C.

Therefore, the exact solution when x = 1 is