Numerical Methods

Abstract

In Chap. 2, we considered a few types of first-order differential equations for which we found exact solutions. Unfortunately, the great majority of DEs arising in mathematical models cannot be solved explicitly, so we have to apply other techniques in order to obtain details about the nature and properties of their solutions. A direction field sketch, for example (see Sect. 2.8), helps us understand the long term behavior of solution families. But if we require quantitative information about their evolution on finite intervals, we need to resort to numerical approximation techniques. In what follows, we introduce four such methods that appeal to the user because of their combination of simplicity and, in many cases, reasonable accuracy. All numerical results are computed with rounding at the fourth decimal place.