ABSTRACT
Unfortunately, finding exact solution formulas for given initial-value problems involving either single differential equations or systems of equations is often not practical. In these cases, one may turn to the approximations yielded by numerical solutions. In earlier chapters, the Euler, Improved Euler and Runge-Kutta methods for finding numerical solutions were developed, but only for single first-order differential equations.
This chapter shows how to extend those numerical methods to handle first-order systems of differential equations and single higher-order differential equations. There are three parts to this chapter. The first is a brief review of the basic Euler method for single first-order equations. This followed by the extension to first-order systems. Finally, it is shown that the method for systems can be applied to the system corresponding to a higher-order equation to yield a method for numerically solving single higher-order differential equations. For clarity and brevity, discussion is limited to extending the basic Euler method. Extending the improved Euler and Runge-Kutta methods is similar (though more involved), and can be easily carried out by the reader on their own following the development given in this chapter.
