In the mathematical description of a physical system the basic laws are usually expressed as equations connecting local quantities. The equations consist of products and sums of the quantities and their derivatives, the actual combination of forms depending upon which variables are chosen to describe the situation.

The effect of external factors on the system is represented by the tying of the values of some of the quantities to particular values of others. The latter quantities are usually called the independent variables, and the former the dependent ones. A complete description (solution) of the physical system is obtained when the values of the dependent variables are known for all values of (i.e. as a function of) the independent ones.

In cases where derivatives appear in the expression of the physical laws it is thus necessary to convert a differential equation into a standard, tabulated, or computable function. This may take the form of a series or an integral which can be evaluated numerically. Methods by which the conversion can be effected form the content of this and the next two chapters and also (for equations containing more than one independent variable) of chapters 9 and 10.

Equations of the types considered in the first three of these chapters – those containing only one independent variable – are called ordinary differential equations. The two other chapters deal with some methods of solving partial differential equations, i.e. ones containing derivatives of dependent variables with respect to more than one independent variable.