TABLE OF CONTENTS In 1 we introduce initial value problems through an example from mechanics; here we employ elements from the theory of ordinary differential equations (ODE) that will be worked out in more detail in later sections and chapters. Then, in 2, we introduce vector fields and systems of first order differential equations. Their classification and some quite general properties are treated in 3. In 4 we show how higher order differential equations can be reduced to systems of first order; this then justifies the exclusive treatment of the first order equations at a later stage. The discrete analogue of a differential equation, viz. the difference equation, is defined and described in 5. In 6 we indicate how the solution of an ODE can be approximated through discretisation, thus giving a difference equation. We also introduce the notion of consistency for measuring the discrepancy between the solution of an ODE and that of its discretised counterpart.
The study of ordinary differential equations (ODE) goes back to times when classical mechanics was being developed. This has permeated both notation and terminology. This appears most clearly in the independent variable t, which often corresponds to the physical concept of time, although also space or other notions play the r le of independent variable occasionally. Mechanical systems can often be quite helpful in interpreting results or even inspiring methods. Therefore we should like to start this chapter by introducing important ODE concepts with this interpretation in mind. In Chapter XI we...
