= = =

Numerous initial-value ordinary differential equations arise in engineering and science. Single ODEs governing a single dependent variable arise frequently, as do coupled systems of ODEs governing several dependent variables. Initial-value ODEs may be linear or nonlinear, first-or higher-order, and homogeneous or nonhomogeneous. In this chapter, the majority of attention is devoted to the general nonlinear first-order ordinary differential equation (ODE):

,dy )y =-=f(t,ydt (7.4)

where y' denotes the first derivative and f(t, y) is the nonlinear derivative function. The solution to Eq. (7.4) is the function y(t). This function must satisfy an initial condition at t = to, y(to) =Yo. In most physical problems, the initial time is t = 0.0 and y(to) = y(O.O). The solution domain is open, that is, the independent variable t has an unspecified (i.e., open) final value. Several finite difference methods for solving Eq. (7.4) are developed in this chapter. Procedures for solving higher-order ODEs and systems ofODEs, based on the methods for solving Eq. (7.4), are discussed.