Summary

In these notes we have introduced some methods for solving ordinary differential equations. However, by far the most important lesson to be learned is that to be useful a method must be both accurate and stable. The latter condition is often the most difficult to fulfil and careful analysis of the equations to be solved may be necessary before choosing an appropriate method to use for finding a numerical solution. The stability conditions derived above tend to have the form $\delta t \le \omega^{-1}$ which may be interpreted as $\delta t$ should be less than the characteristic period of oscillation. This conforms with common sense. In fact we can write down a more general common sense condition: $\delta t$ should be small compared with the smallest time scale present in the problem. Finally, many realistic problems don't fall into the neat categories of (1.1). The simplest example is a damped harmonic oscillator. Often it is difficult to find an exact analytical solution for the stability condition. It pays in such cases to consider some extreme conditions, such as (e.g. ) very weak damping or very strong damping, work out the conditions for these cases and simply choose the most stringent condition. In non-linear problems the cases when the unknown, $y$, is very large or very small may provide tractable solutions.