Ordinary differential equations - Introduction
The differential equations arise in many areas of applied mathematics, physics, mechanics, technique, etc. Practically any problems of dynamics of machines and mechanisms (see, for example, on our site sections of the dynamic analysis of hydraulic systems, drives and transmissions, control systems) are described with their help.
There is a majority of methods of decision of differential equations through elementary or special functions. However, more often these methods either at all are not applicable, or lead to so complex solutions that it is easier and more expedient to use the approached numerical methods. The differential equations contain in huge quantity of problems various types essential nonlinearity, and functions entering into them and factors are given in the form of tables and-or experimental data, that actually completely excludes an opportunity of use of classical methods for their decision and analysis.
Now there is a majority of various numerical methods of solution of the ordinary differential equations (for example, Euler's, Runge-Kutta, Milne's, Adams's, Gear's metods, etc.) [1–6]. We‘ll be limited here to consideration of the most widely used in practice Euler's and Runge-Kutta methods. As to other mentioned methods they are in detail stated in the literature, see, for example: [1, 4] – Milne's method, [1, 3, 5] – Adams's method, [5, 6] – Gear's method.
We also do not stop here on questions of stability of computing processes; they are in detail given in the corresponding works [4, 5, 7].