In general it is difficult, if not impossible, to find solutions to differential equations analytically. When this happens we rely on the numerical approximation of solutions, and in this chapter we discuss how numerical solutions to initial value problems of the form are found. This will provide insight into the numerical techniques used in the MATLAB programs dfield5, pline and pplane5.
The numerical schemes that we consider approximate solutions up to a specified accuracy, and in Section ?? we describe the basic ideas underlying the construction of such schemes. In this section we will see that in general there is an error between the numerical approximation and the analytic solution of the underlying initial value problem. One of the main tasks in the numerical analysis of ODEs is to derive bounds on this error, and in Section ?? we derive these bounds for the simplest numerical scheme, Euler’s method. Finally, in Section ?? we generalize this treatment and indicate how to derive error bounds for arbitrary numerical schemes. In particular it turns out that in general the fourth order Runge-Kutta method leads to much more accurate numerical approximations than does Euler’s method.