## ABSTRACT

In the previous chapter, it was seen that a computer can easily generate a slope field for a given first-order differential equation, and that, using that slope field, one can sketch a fair approximation to the graph of the solution y=y(x) to a corresponding initial-value. From that graph, one can then find a approximation of y(x) for any desired x in the region of the sketched slope field. The obvious question now arises: Why not just let the computer do all the work and find that approximate value for y(x)?

In this chapter, the basic ideas developed in the previous chapter for slope fields are used to develope a method --- the “Euler method” --- for generating approximate “numerical solutions” to a first-order initial-value problem. An illustration of using the Euler method follows its development, and that is followed by a discussion of the accuracy of the method, which includes an example of how the method can fail “catastrophically” and a detailed analysis of how the accuracy is related to the step size.