ABSTRACT
Despite the broad range of powerful analytical tools presented throughout this book, many occasions cry out for the application of numerical methods for solving ordinary differential equations. For example, an exact solution may be unavailable, or may be of little practical value. 1 This situation occurs when power series solutions to linear second order equations are constructed. In general, the series are rather good approximations near the initial condition, but the Taylor expansions can soon require prohibitively many terms should the solution be required at some large distance from that point. For large systems of equations, an exact solution may exist (in vector form) but the subsequent algebraic manipulations may be overwhelming. Furthermore, numerical solutions should not be cast in a light of last resort, for they form the mathematician’s petri dish—a crucible in which he can conduct any number of experiments on his differential equation and, by proxy, the very thing he is trying to model. 2
